Yves Achdou ˚and

Claude Le Bris

HOMOGENIZATION OF SOME PERIODIC HAMILTON-JACOBI EQUATIONS WITH DEFECTS

We study homogenization for a class of stationnary Hamilton-Jacobi equations in which the Hamiltonian is obtained by perturbing near the origin an otherwise periodic Hamiltonian. We prove that the limiting problem consists of a Hamilton-Jacobi equation outside the origin, with the same effective Hamiltonian as in periodic homogenization, supplemented at the origin with an effective Dirichlet condition that keeps track of the perturbation. Various comments and extensions are discussed.

1. Introduction. This work discusses the homogenization limit for a first order stationary Hamilton-Jacobi equation of the form

α u ε `H ´x ε , Du ε ¯" 0 in R d ,
and some related equations, where the Hamiltonian H, besides satisfying some classical assumptions, is obtained by a local, compactly supported, perturbation of a periodic Hamiltonian H per . We henceforth refer to such a perturbation as a local defect.

Homogenization theory in the presence of local defects within an otherwise periodic environment was first introduced in [START_REF] Blanc | A possible homogenization approach for the numerical simulation of periodic microstructures with defects[END_REF], in the first of a series of works by the second author, in collaboration with X. Blanc and P-L. Lions. It was further developed in [START_REF]Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures[END_REF][START_REF]On correctors for linear elliptic homogenization in the presence of local defects[END_REF] and other subsequent works by various authors, considering different classes of defects such as, in particular, interfaces between two different periodic media. In those works, the typical setting is that of a linear non-degenerate elliptic equation, first in divergence form and next in more general form. Only recently, some quasilinear equation was considered in [START_REF] Wolf | Homogenization of the p-laplace equation in a periodic setting with a local defect[END_REF].

On the other hand, extensions to the context of nonlinear Hamilton-Jacobi equations arising from optimal control theory were first addressed in a series of lectures [START_REF] Lions | Équations elliptiques ou paraboliques, et homogénéisation précisée [Elliptic or parabolic equations and precised homogenization[END_REF] at Collège de France by P-L. Lions, where some results obtained in collaboration with P. Souganidis, [START_REF] Lions | [END_REF], were described. Since the present contribution is very much related to [START_REF] Lions | Équations elliptiques ou paraboliques, et homogénéisation précisée [Elliptic or parabolic equations and precised homogenization[END_REF][START_REF] Lions | [END_REF], we will devote a special paragraph below to commenting our results in view of the latter references and to stressing the similarities and differences. In [START_REF] Cardaliaguet | Perturbation problems in homogenization of Hamilton-Jacobi equations[END_REF], P. Cardaliaguet, P. Souganidis and the second author considered the extension to a fully nonlinear equation of a specific probabilistic setting originally introduced, in the linear elliptic setting and from a computational perspective, in [START_REF] Anantharaman | A numerical approach related to defect-type theories for some weakly random problems in homogenization[END_REF]. In that setting, the defect is assumed to occur only with a small probability, and the homogenized limit is identified at the leading order in probability. More generally, an introductory account of the entire mathematical endeavor together with a large variety of possible applications and extensions, has appeared in [START_REF] Bris | Mathematics for the modeling of defects in materials[END_REF].

In a distinct line of works, the first author, in collaboration with N. Tchou in particular, studied the asymptotic behaviour of the solutions to some classes of Hamilton-Jacobi equations depending on a vanishing parameter, for which the limit is characterized by boundary value problems with transmission conditions involving effective ˚Université Paris Cité and Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions, (LJLL), F-75006 Paris, France, achdou@ljll-univ-paris-diderot.fr : École des Ponts and INRIA, 6-8 avenue Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne La Vallée, France, claude.le-bris@enpc.fr Hamiltonians. In particular, [START_REF] Achdou | Hamilton-Jacobi equations on networks as limits of singularly perturbed problems in optimal control: dimension reduction[END_REF] dealt with dimension reduction for Hamilton-Jacobi equations posed in thin domains converging to networks. When the thickness of the domain vanishes, one finds at the limit a Hamilton-Jacobi equation posed on the network with a special transmission condition at the vertices. The latter involves a socalled flux limiter which keeps track of the microscopic geometry of the thin domains. Subsequently, the works [START_REF]Effective transmission conditions for Hamilton-Jacobi equations defined on two domains separated by an oscillatory interface[END_REF] and [START_REF]Homogenization of a transmission problem with Hamilton-Jacobi equations and a twoscale interface. Effective transmission conditions[END_REF] were concerned with Hamilton-Jacobi equations in an environment consisting of two different homogeneous media separated by an oscillatory interface. The oscillations of the interface have small period and amplitude, and as both the latter parameters vanish, the interface tends to an hyperplane. At the limit, one finds on the flat interface an effective nonlinear transmission condition keeping memory of the previously mentioned microscopic oscillations. Similarly, G. Galise, C. Imbert and R. Monneau studied in [START_REF] Galise | A junction condition by specified homogenization and application to traffic lights[END_REF] a family of time dependent Hamilton-Jacobi equations on the simplest possible network composed of two halflines with a perturbation of the Hamiltonian localized in a small region close to the junction. Related homogenization problems with applications to traffic flows were discussed in [START_REF] Forcadel | Specified homogenization of a discrete traffic model leading to an effective junction condition[END_REF][START_REF] Khatib | Homogenization of a microscopic pedestrians model on a convergent junction[END_REF] by N. Forcadel and his coauthors. Key arguments in [START_REF] Achdou | Hamilton-Jacobi equations on networks as limits of singularly perturbed problems in optimal control: dimension reduction[END_REF][START_REF]Effective transmission conditions for Hamilton-Jacobi equations defined on two domains separated by an oscillatory interface[END_REF][START_REF]Homogenization of a transmission problem with Hamilton-Jacobi equations and a twoscale interface. Effective transmission conditions[END_REF][START_REF] Galise | A junction condition by specified homogenization and application to traffic lights[END_REF] are the construction of families of correctors that account for the localized pertubations of the environment and are defined in unbounded domains. We will see that the construction of such correctors also plays a key role in the present work. Note also that another common feature between the references [START_REF] Achdou | Hamilton-Jacobi equations on networks as limits of singularly perturbed problems in optimal control: dimension reduction[END_REF][START_REF]Effective transmission conditions for Hamilton-Jacobi equations defined on two domains separated by an oscillatory interface[END_REF][START_REF]Homogenization of a transmission problem with Hamilton-Jacobi equations and a twoscale interface. Effective transmission conditions[END_REF][START_REF] Galise | A junction condition by specified homogenization and application to traffic lights[END_REF] is that at the limit, the new effective transmission conditions are all posed on manifolds of codimension one. The study of such homogenization problems was indeed part of the significant research effort that took place in the last decade on the analysis of Hamilton-Jacobi equations posed on heterogeneous structures such as networks [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF][START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF][START_REF]Viscosity solutions for junctions: well posedness and stability[END_REF], bookletlike geometries or multidimensional junctions [START_REF] Barles | A Bellman approach for regional optimal control problems in R N[END_REF][START_REF]Quasi-convex Hamilton-Jacobi equations posed on junctions: the multi-dimensional case[END_REF][START_REF] Oudet | Équations de Hamilton-Jacobi sur des réseaux ou des structures hétérogènes[END_REF]. These problems all involve Hamilton-Jacobi equations on manifolds with nonlinear transmission conditions on submanifolds with codimension one. In sharp contrast, in the present work, the problem arising at the limit involves a boundary condition posed at the origin, which is generally not a manifold of codimension one. This causes technical difficulties and a priori prevents one from relying on the theories contained in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF][START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF][START_REF]Viscosity solutions for junctions: well posedness and stability[END_REF]. On the contrary, the problem arising at the limit falls into the class of stratified problems (and is a particularly simple example of such problems) introduced by A. Bressan and coauthors, see [START_REF] Bressan | Optimal control problems on stratified domains[END_REF] and later studied by G. Barles and E. Chasseigne, see [START_REF] Barles | An illustrated guide of the modern approches of hamilton-jacobi equations and control problems with discontinuities[END_REF]. By and large, these problems involve Hamilton-Jacobi equations with discontinuities in a more general geometric setting, namely stratifications of R d . The latter references contain in particular comparison principles for viscosity solutions of stratified problems, but, up to our knowledge, there is no literature yet on homogenization or singular perturbations leading to stratified problems, and in particular no proofs of convergence.

Finally, in connection with [START_REF]Effective transmission conditions for Hamilton-Jacobi equations defined on two domains separated by an oscillatory interface[END_REF][START_REF]Homogenization of a transmission problem with Hamilton-Jacobi equations and a twoscale interface. Effective transmission conditions[END_REF], the present contribution can be seen as a first step toward homogenization theory for Hamilton-Jacobi equations with two different possibly periodic media separated by an interface presenting localized defects of periodicity. We plan to address this aspect in a forecoming work.

As already mentioned, the present study (essentially) considers a single localized defect, supported say in a neighborhood of the origin, within a periodic medium. It identifies the homogenized limit in the case of a first order Hamilton-Jacobi equation arising from optimal control, thus with a convex Hamiltonian; we will assume that the Hamiltonian has agreeable mathematical properties, which will be made precise in the next section. It is natural to expect that at the limit, except possibly at the origin, the Hamilton-Jacobi equation involving the effective Hamiltonian obtained from periodic homogenization is satisfied. We will see that if the defect has the effect of attracting the optimal trajectories starting not too far from the origin, (which occurs when the running cost or potential of the optimal control problem displays a prominent downward bump within the periodic environment), then the homogeneity of the environment is broken in the homogenized limit. We will indeed prove that, in this case, the limit problem involves a Dirichlet boundary condition at the origin. Besides, it is well known that, in singular domains such as R d zt0u, uniqueness does not hold for the now classical formulation of Dirichlet problems due to H. Ishii, see e.g. [6, chapter V, section 4]. By contrast, uniqueness holds for the formulation of the Dirichlet problem that we find here from homogenization. We stress again that because of the effective Dirichlet condition, the limit is generally not an Hamilton-Jacobi equation on the whole ambient space.

We wish to emphasize that the results exposed here are to be considered in the light of those obtained earlier by P-L. Lions and P. Souganidis and summarized in [START_REF] Lions | Équations elliptiques ou paraboliques, et homogénéisation précisée [Elliptic or parabolic equations and precised homogenization[END_REF] (and, likewise, of [START_REF] Cardaliaguet | Perturbation problems in homogenization of Hamilton-Jacobi equations[END_REF] for randomized defects). In the control theoretic interpretation of that work, the typical local perturbation of the running cost is a bump oriented upwards so that the neighborhood of the origin becomes repulsive. In [START_REF] Lions | Équations elliptiques ou paraboliques, et homogénéisation précisée [Elliptic or parabolic equations and precised homogenization[END_REF], the presence of such a defect indeed does not affect the homogenized limit, but only possibly "the next order correction", that is the definition of the corrector function itself. In sharp contrast, if the defect makes the origin attractive, then it affects the homogenized limit itself. Note that it is not unexpected that signs, in the wide acceptation of that term, play a critical role in the context of nonlinear equations.

In the authors' view, the present work, together with the results of [START_REF] Lions | Équations elliptiques ou paraboliques, et homogénéisation précisée [Elliptic or parabolic equations and precised homogenization[END_REF], lay the groundwork for a complete theory for Hamilton-Jacobi equations in periodic environments with local defects. Many challenging questions, some of them presumably quite difficult, however remain unsolved.

Our article is more precisely organized as follows. In the rest of this introduction, we first make precise, in Section 1.1, the mathematical setting and assumptions of the problem we study. The Hamilton-Jacobi equation under consideration reads as (1.7) below. We next state and comment in Section 1.2 our main result, namely Theorem 1.1 which establishes for (1.7) the homogenized limit (1.11) through (1.14) made precise therein. Section 2 then contains the detailed proof of Theorem 1.1. The proof essentially falls in four steps, respectively contained in Sections 2.1 through 2.4. The final Section 3 contains several comments upon and extensions of the problem considered. First, in Section 3.1, we illustrate in the one-dimensional setting the general results of Section 1.2. We in particular consider, in Section 3.1.4, a probabilized variant of our problem inspired by some of the earlier works recalled above. Section 3.2 discusses the extension to higher dimensions of the considerations and results given in Section 3.1.

Acknowledgments. The first author is partially on academic leave at Inria for the years 2021-22 and 2022-23 and acknowledges the hospitality of this institution during this period. This research was partially supported by ANR (Agence Nationale de la Recherche) through project COSS ANR-22-CE40-0010-03. The research of the second author is partially supported by ONR under Grant N00014-20-1-2691 and by EOARD under Grant FA8655-20-1-7043. The final stage of the writing of this manuscript has been completed ultimately when the second author was visiting the Math+ Institute in Berlin. Partial financial support from the Deutsche Forschungsge-meinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy -The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689) -is thus gratefully acknowledged.

The first author would like to thank N. Tchou for helpful discussions.

1.1. Setting and assumptions. Let us define the problem and give the assumptions that will be used in the whole paper.

We consider Hamilton-Jacobi equations linked to infinite horizon optimal control problems in R d . The Hamiltonian H : R d ˆRd Ñ R is of the form

(1.1)
Hpx, pq " max aPA ´´p ¨f px, aq ´ px, aq ¯.

Here, A is a compact metric space, f : R d ˆA Ñ R d is a bounded and continuous function, Lipschitz continuous with respect to its first variable uniformly with respect to its second variable, i.e. for any x, y P R d and a P A, |f px, aq ´f py, aq| ď L f |x ´y|, for some positive constant L f . We also suppose that there exists some radius r f ą 0 such that for any x P R d , tf px, aq, a P Au contains the ball B r f p0q, which implies that the trajectories are locally strongly controllable.

We suppose that the function : R d ˆA Ñ R is bounded and continuous and that here is a modulus of continuity ω such that for any x, y P R d and a P A, | px, aq ´ py, aq| ď ω p|x ´y|q. We next assume that, except in a neighborhood of the origin, the dynamics f and cost coincide with periodic functions. Let T d " R d {Z d denote the torus of R d . We assume that there exist R 0 ą 0, f per : T d ˆA Ñ R d and per : T d ˆA Ñ R such that, if |x| ą R 0 , then f px, aq " f per px, aq and px, aq " per px, aq. Hence, the suprema M f and M introduced above are indeed maxima. Let H per stand for the related periodic Hamiltonian:

Define M f " sup xPR d ,
(1.5) H per px, pq " max aPA ´´p ¨fper px, aq ´ per px, aq ¯.

Let α be a positive scalar and ε be a small positive parameter that will eventually vanish. It is well known, see e.g. [START_REF] Bardi | Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Systems & Control: Foundations & Applications[END_REF], that the value function u ε of the following optimal control problem: (1.6)

u ε pxq " inf ż 8 0 e ´αt ˆyptq ε , aptq ˙dt subject to $ ' & ' % yptq " x `şt 0 f ´ypτq ε , apτ q ¯dτ, a P L 8 pR `q aptq P A, a.e.
is the unique viscosity solution in BUCpR d q of (1.7)

α u ε `H ´x ε , Du ε ¯" 0 in R d .
Our goal is to study the asymptotic behaviour of u ε as ε Ñ 0. Homogenization of the periodic Hamilton-Jacobi equation involving H per instead of H is well understood since the pioneering work [START_REF] Lions | Homogenization of Hamilton-Jacobi equations[END_REF]. In the periodic case, the homogenized equation is

(1.8) α u `HpDuq " 0 in R d ,
where the effective Hamiltonian is characterized as follows: for any p P R d , Hppq is the unique real number such that there exists a periodic corrector χ per,p , i.e. a viscosity solution χ per,p P CpT d q of (1.9) H per py, p `Dχ per,p q " Hppq in R d .

In general, the latter periodic corrector is not unique even up to the addition of a scalar constant. It is well known that, under the assumptions made above, H is convex on R d , Lipschitz continuous and coercive. Therefore, there exists a vector p 0 P R d , possibly non unique, such that (1.10) Hpp 0 q " min qPR d

Hpqq.

We choose such a vector p 0 and fix it for all what follows.

1.2. The main result. Our main result is the following:

Theorem 1.1. We consider the solution u ε of (1.7). As ε Ñ 0, the family u ε converges locally uniformly to the unique function u P BUCpR d q defined by the following properties:

1. u is a viscosity solution of

(1.11) α u `HpDuq " 0 in R d zt0u,
with the effective Hamiltonian H defined above by periodic homogenization, see (1.9) 2. The condition (1.12) α up0q `E ď 0 holds, where E is the effective Dirichlet datum defined in paragraph 2.1 below. In addition, if φ P C 1 pR d q is such that u´φ has a local maximum at the origin, then

(1.13) α up0q `HpDφp0qq ď 0.

3. If φ P C 1 pR d q is such that u ´φ has a local minimum at the origin, then ´E{α. Note that this formulation is stronger than that due to H. Ishii, which expresses that at the origin, u is both a viscosity subsolution of α up0q `minpHpDup0qq, Eq ď 0 and a viscosity supersolution of α up0q `maxpHpDup0qq, Eq ě 0, see for example [6, Chapter V, Section 4], [START_REF] Barles | An introduction to the theory of viscosity solutions for first-order Hamilton-Jacobi equations and applications[END_REF] and [START_REF] Ishii | A short introduction to viscosity solutions and the large time behavior of solutions of Hamilton-Jacobi equations[END_REF]. Indeed, while the supersolution condition (1.14) coincides with that of Ishii, the subsolution condition (1.12)-(1.13) is stronger. A consequence of the difference between the two formulations regards uniqueness. Reference [6, Chapter V, Section 4] contains a counterexample for the uniqueness of viscosity solutions in the sense of Ishii of Dirichlet problems posed in R d zt0u. Note also that an interior cone condition is sufficient for uniqueness, see (4.30) in the latter reference, but this condition is not satisfied by R d zt0u. By contrast, and as we will see in the proof of Theorem 1.1 in Section 2.4 below, uniqueness holds for (1.11), (1.12), (1.13), (1.14).

Remark 1.2. The conditions (1.11) through (1.14) fall into the general notion of stratified solutions of Hamilton-Jacobi equations, see [START_REF] Bressan | Optimal control problems on stratified domains[END_REF][START_REF] Barles | An illustrated guide of the modern approches of hamilton-jacobi equations and control problems with discontinuities[END_REF], and correspond to the partition of R d into the sets R d zt0u and t0u. The final step in the proof of Theorem 1.1 needs a comparison result which can be found in [START_REF] Barles | An illustrated guide of the modern approches of hamilton-jacobi equations and control problems with discontinuities[END_REF] in a much more general setting. In order to keep the paper self-contained, we will give a short proof of the comparison principle, see Section 2.4, because it is quite simple in the particular case under consideration.

Remark 1.3. With the same assumptions on H, Theorem 1.1 may be easily generalized to the homogenization of Hamilton-Jacobi equations of the form

α u ε `H ´x ε , Du ε ¯" bpxq in R d ,
where b P BUCpR d q. The effective Dirichlet datum E would then depend on b only through its value at the origin. Similarly, the result can be easily extended to problems of the form

α u ε `Hε px, Du ε q " 0 in R d ,
where H ε px, pq " max aPA p´p ¨fε px, aq ´ ε px, aqq and

ε px, aq " 1 px, x ε , aq ` 0 p x ε , aq, f ε px, aq " f 1 px, x ε , aq `f0 p x ε , aq,
the functions 0 , f 0 are as above (they model the perturbation located near the origin) and 1 and f 1 are smooth with respect to their first argument and periodic with respect to the second one.

Other generalizations to different settings may be considered. While those to evolutive problems seem fairly easy, extensions to viscous Hamilton-Jacobi equations seem more challenging.

Remark 1.4. As mentioned in the introduction, the homogenization of periodic Hamilton-Jacobi equations in the presence of a defect has been first addressed by P-L. Lions and P. Souganidis, see the results announced in [START_REF] Lions | Équations elliptiques ou paraboliques, et homogénéisation précisée [Elliptic or parabolic equations and precised homogenization[END_REF][START_REF] Lions | [END_REF]. The assumptions made therein imply that E ď min pPR d Hppq and that the defect does not show up in the homogenized limit, i.e. u is a viscosity solution of α u `HpDuq " 0 in R d . We therefore recover as a particular case of our setting the homogenized limit obtained in [START_REF] Lions | Équations elliptiques ou paraboliques, et homogénéisation précisée [Elliptic or parabolic equations and precised homogenization[END_REF][START_REF] Lions | [END_REF], under the particular assumptions made there.

For example, in the case when p 0 " 0 in (1.10), we find that u is then identically equal to ´1 α Hp0q " ´1 α min p Hppq.

On the contrary, if

(1.15) E ą Hp0q " min pPR d Hppq,
then the defect is visible in the homogenized limit. Indeed, it can first be proven (see § 3.2.1 below) that

(1.16) lim |x|Ñ8 upxq " ´1 α Hp0q.
Combining (1.12), (1.15) and (1.16), we then see that u is not a constant function, so it differs from the homogenization limit in the absence of defect. We also claim that in this particular case, the global infimum of u is reached at the origin and only at the origin. Indeed, (1.12), (1.15) and (1.16) altogether imply that there exists a minimizer x 0 of u. If x 0 " 0, then (1.11) yields that α upx 0 q ě ´Hp0q, thus α upx 0 q ą ´E ě α up0q, and we reach a contradiction. Note also that, in [START_REF] Lions | Équations elliptiques ou paraboliques, et homogénéisation précisée [Elliptic or parabolic equations and precised homogenization[END_REF], thanks to the assumption made, the authors are able to construct correctors χ p in the whole space R d for all vectors p P R d (p stands for the gradient of a smooth test-function at the origin). Their construction uses arguments from the theory of optimal control. These correctors, which differ from those used in periodic homogenization, can be used for proving the convergence to the homogenized problem by means of the now classical method of perturbed test-functions, see [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF].

In contrast, when E ą min pPR d Hppq, we will have to adjust the strategy proposed in [START_REF] Lions | Équations elliptiques ou paraboliques, et homogénéisation précisée [Elliptic or parabolic equations and precised homogenization[END_REF] in order to keep relying on the theory of optimal control : using arguments somewhat reminiscent of those proposed in [START_REF] Achdou | Hamilton-Jacobi equations on networks as limits of singularly perturbed problems in optimal control: dimension reduction[END_REF], we will construct correctors associated to suitable piecewise affine functions rather than smooth or linear functions.

2. Homogenization of (1.7): the proof of Theorem 1.1. Recall that our goal is to understand the asymptotic behavior of u ε as ε tends to 0. First, using either comparison principles, see for example [6, Chapter II, Theorem 3.5] or arguments from the theory of optimal control, we see that From this estimate and (1.7), we infer from the coercivity of the Hamiltonian that u ε is Lipschitz continuous in R d with a Lipschitz constant independent of ε.

In order to study the asymptotic behaviour of u ε , we consider

upxq " lim sup εÑ0 u ε pxq, (2.1) upxq " lim inf εÑ0 u ε pxq. (2.2)
Note that, from the observation above on the regularity of u ε , u and u coincide respectively with the half-relaxed semi-limits lim sup x 1 Ñx,εÑ0 u ε px 1 q and lim inf x 1 Ñx,εÑ0 u ε px 1 q, that are classically used in the homogenization of Hamilton-Jacobi equations. It is clear that the functions u and u are bounded and Lipschitz continuous.

We now observe that, in R d zt0u, the homogenized equation does not keep track of the defect of periodicity and is identical to (1.8). This is not surprising, since the support of the defect shrinks to t0u ˆA as ε Ñ 0, while the related Hamiltonian does not become singular.

Proposition 2.1. The functions u and u are respectively a bounded subsolution and a bounded supersolution of (1.11)

in R d zt0u.
Proof. The proof is classical and relies on perturbed test-functions techniques, see [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF].

The strategy of proof for Theorem 1.1. Below, the remainder of the proof of Theorem 1.1 is done in four different steps. Accordingly, Section 2 is cut into four parts. The four steps, which will be rapidly summarized below, mostly rely on the theory of viscosity solutions to first order Hamilton-Jacobi equations, except the third one, which also contains arguments from the theory of optimal control.

1. The first step consists of constructing the ergodic constant E associated to the defect and a related corrector w. It will be proved that E ě Hpp 0 q. An important difficulty is that the corrector w must be a function defined in the whole space R d , which makes it necessary to impose some condition at infinity. We will see that the latter amounts to the fact that w is the locally uniform limit as R Ñ `8 of a family pw R q Rą0 of solutions of problems with state constraints posed in the balls B R p0q. From the optimal control theory viewpoint, these problems, refered to as truncated cell problems, account for trajectories that remain close to the defect at the microscopic scale. Proposition 2.2 below contains information on the growth of w at infinity. 2. In the second step, we prove that the upper-limit u satisfies conditions (1.12) and (1.13). While (1.13) is a rather easy consequence of Proposition 2.1, the proof of (1.12) relies on Evans' method of perturbed test-functions, see [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF].

The construction of the perturbed test-function involves the above mentioned solution w R to the truncated cell problem in the ball B R p0q. 3. The third step, the most involved one, consists of proving that the lower-limit u satisfies condition (1.14). Most of the work concerns the situation in which E ą Hpp 0 q. Here again, one needs to construct a suitable perturbation of a test-function by a corrector. The crucial theoretical novelty is the choice of the test-function. Instead of using smooth or affine test-functions as it is classically done, we consider suitable piecewise affine test-functions, in the spirit of [START_REF] Achdou | Hamilton-Jacobi equations on networks as limits of singularly perturbed problems in optimal control: dimension reduction[END_REF]. Their behaviour at infinity makes it possible to construct associated correctors in the whole space R d . The key intermediate result is Proposition 2.6 which states the existence of such correctors with a strictly sublinear growth at infinity. Its proof uses as an important ingredient the function w introduced in the first step and Proposition 2.2. Note also that the proof of Proposition 2.6 relies on control theoretical arguments in the same spirit as in [START_REF] Lions | Équations elliptiques ou paraboliques, et homogénéisation précisée [Elliptic or parabolic equations and precised homogenization[END_REF]. The choice of the above mentioned piecewise affine test-functions is precisely made so that the latter arguments can be applied when E ą Hpp 0 q. 4. The fourth step of the proof mostly consists of deducing from the previously obtained results that u " u, by means of a comparison principle. Once this is done, the results announced in Theorem 1.1 follow.

2.1. The ergodic constant associated to the defect. We next introduce ingredients which play a key role in the effective boundary condition at the origin.

2.1.1. Ergodic constants for state-constrained problems in truncated domains. In order to understand the effect of the defect on the asymptotics of u ε , we start by solving truncated cell problems in balls centered at the origin; these are associated to state constrained boundary conditions. From the optimal control theory viewpoint, these problems account for trajectories that remain close to the defect at the microscopic scale. For λ ą 0, R ą 0, we know from e.g. [START_REF] Soner | Optimal control with state-space constraint. I[END_REF][START_REF] Capuzzo-Dolcetta | Hamilton-Jacobi equations with state constraints[END_REF] that there exists a unique function w λ,R P CpB R p0qq such that λw λ,R `Hpy, Dw λ,R q ď 0 in B R p0q, (2.3) λw λ,R `Hpy, Dw λ,R q ě 0 in B R p0q, (2.4) the above inequalities being understood in the sense of viscosity. An equivalent way to write (2.3)-(2.4) is the following: λw λ,R `Hpy, Dw λ,R q " 0 in B R p0q, (2.5) λw λ,R `HÒ py, Dw λ,R q " 0 on BB R p0q, (2.6) in the sense of viscosity, where, for y P BB R p0q,

(2.7)
H Ò py, pq " max aPA, f py,aq¨yď0

´´p ¨f py, aq ´ py, aq ¯.

is the Hamiltonian associated to the admissible trajectories that do not exit the ball Bp0, Rq through y. The function w λ,R is the value function of the following infinite horizon state constrained optimal control problem in B R p0q,

w λ,R pzq " inf ż 8 0 e ´λt ˆyptq ε , aptq ˙dt subj. to $ ' ' ' & ' ' ' % (2.8)
yptq " z `şt 0 f ´ypτq ε , apτ q ¯dτ yptq P B R p0q, a P L 8 pR `q aptq P A, a.e. which the reader may compare to (1.6). Since is bounded on R d ˆA (because, in particular, A is compact and coincides with per out of B R0 p0q), λ}w λ,R } L 8 pB R p0qq is bounded uniformly in λ and R. More precisely, min py,aqPR d ˆA py, aq ď λw λ,R ď max py,aqPR d ˆA py, aq. This and the uniform coercivity of H imply with (2.5) that }Dw λ,R } L 8 pB R p0qq is bounded uniformly in λ and R.

Using Ascoli-Arzelà theorem, we may suppose that up to the extraction of a sequence, as λ Ñ 0, λw λ,R tends uniformly on B R p0q to some ergodic constant ´ER which is bounded from above and below uniformly in R, and that w λ,R ´wλ,R p0q tends uniformly on B R p0q to some function w R such that w R p0q " 0 and which is Lipschitz continuous with a Lipschitz constant independent of R. By classical results on the stability of viscosity solutions of state constrained problems, w R is a viscosity solution of

Hpy, Dw R q ď E R in B R p0q, (2.9) Hpy, Dw R q ě E R in B R p0q. (2.10)
The comparison principle for state constrained problems, see [START_REF] Soner | Optimal control with state-space constraint. I[END_REF][START_REF] Capuzzo-Dolcetta | Hamilton-Jacobi equations with state constraints[END_REF], yields the uniqueness of E R such that (2.9)-(2.10) has a solution. Thus, lim λÑ0 }λw λ,R ÈR } CpB R p0qq " 0 (uniform convergence and not only for a subsequence).

2.1.2. The ergodic constant and the cell problem. We deduce for example from (2.8) that R 1 ě R 2 ñ λw λ,R1 ď λw λ,R2 , and passing to the limit as λ Ñ 0, we obtain the monotonicity property of the ergodic constants E R :

(2.11)

R 1 ě R 2 ñ E R1 ě E R2 . Since E R is bounded from above independently of R (2.11) implies that (2.12) E " lim RÑ8 E R exists in R.
Similarly, since w R p0q " 0, w R is Lipschitz continuous on B R p0q with a Lipschitz constant independent of R, we may construct by Ascoli-Arzelà theorem and a diagonal extraction argument a sequence pR n q nPN , R n Ñ `8 as n Ñ 8, such that w Rn tends to some function w locally uniformly in R d ; we then see that wp0q " 0 and w is a Lipschitz continuous viscosity solution of (2.13)

Hpy, Dwq " E in R d .
Let us now zoom out and pass to the macroscopic scale by considering the function

w ε : x Þ Ñ εwp x ε q;
it is clearly a viscosity solution of Hp x ε , D x w ε q " E, and it is Lipschitz continuous with the same constant as w. Hence, after the extraction of a sequence, we may assume that w ε converges locally uniformly to some Lipschitz function W on R d . As for Proposition 2.1, a standard argument yields that W is a viscosity solution of HpDW q " E. This implies the important inequality

(2.14) E ě min pPR d
Hppq " Hpp 0 q, the right hand side equality holding because of (1.10). Proposition (2.2) below gives some information on the behaviour of w at infinity. It will be useful for proving that u satisfies (1.14).

Proposition 2.2. If E ą Hpp 0 q " min pPR d Hppq, then

(2.15) lim |y|Ñ`8 wpyq ´p0 ¨y " `8,
where w is the corrector constructed in (2.13).

If E " Hpp 0 q " min pPR d Hppq, then wpyq ´p0 ¨y is bounded from below.

Proof.

Step 1. Let us start by proving the desired results in the case when p 0 " 0. With R 0 defined in Section 1.1, we already know that, for all R ą R 0 , the corrector w R solution to (2.9)-(2.10), is Lipschitz continuous with a Lipschitz constant C ą 0 independent of R.

For R 1 ą R 0 , let Q R1 be the cube

Q R1 " ty P R d , }y} 8 ď R 1 u. It contains B R0 p0q. It is clear that, for all R ą R 0 and y P BQ R1 , (2.16) |w R pyq| ď C ? dR 1 ,
using the facts that w R p0q " 0 and that C is a Lipschitz constant for w R . We distinguish two cases:

1. E ą Hp0q. Let pe i q i"1,...,d be the canonical basis of R d associated with the system of coordinates py i q i"1,...,d . Since E ą Hp0q, we may choose δ, 0 ă δ ă E ´Hp0q. Now, because of the convergence of E R to E, we have for R large enough (that we can always suppose larger than ? dR 1 , so that the ball

B R p0q contains Q R1 ), (2.17) E ě E R ą E ´δ.
Since Hp0q " min p Hppq, the continuity and the coercivity of H together with E ´δ ą Hp0q imply that for all i P t1, . . . , du, there exists pp i,δ , p i,δ q P R 2 , p i,δ ă 0 ă p i,δ such that (2.18) Hpp i,δ e i q " Hpp i,δ e i q " E ´δ.

Consider now the functions

w i,δ pyq " c i,δ `pi,δ y i `χper,p i,δ ei pyq, (2.19) w i,δ pyq " c i,δ `pi,δ y i `χper,p i,δ ei pyq. (2.20)
where c i,δ and c i,δ are scalars chosen such that, for any y P BQ R1 , (2.21) w i,δ pyq ă ´C? dR 1 ď w R pyq and w i,δ pyq ă ´C? dR 1 ď w R pyq,

recalling that Q R1 Ă B R p0q.
Moreover, since H coincides with H per in pR d zQ R1 q ˆRd , we see from (1.9) that Hpy, Dw i,δ pyqq " E ´δ, (

Hpy, Dw i,δ pyqq " E ´δ, (2.23) in the sense of viscosity in R d zQ R1 . Since w R is a viscosity supersolution of (2.10), we may use a comparison principle in B R p0qzQ R1 and deduce from (2.17) and (2.21)-(2.22)-(2.23) that for R large enough and for all y P B R p0qzQ R1 , w i,δ pyq ď w R pyq, and w i,δ pyq ď w R pyq.

To summarize, we have proved that, for all y P B R p0qzQ R1 :

(2.24) max w i,δ pyq, w i,δ pyq, i " 1, . . . , d ( ď w R pyq.

By passing to the limit in R (possibly after the extraction of a sequence), we deduce that for any

y P R d zQ R1 , (2.25) max w i,δ pyq, w i,δ pyq, i " 1, . . . , d ( ď wpyq.
Since the correctors χ per,p i,δ ei are bounded functions, it is clear from (2.19)-(2.20) that there exists a constant c such that for all y P R d zQ R1 , max w i,δ pyq, w i,δ pyq, i " 1, . . . , d (

ě min ! |p i,δ |, |p i,δ |, i " 1, . . . , d ) }y} 8 ´c.
This yields the desired result, namely (2.15) in the case when p 0 " 0 and E ą Hp0q.

2. E " Hp0q. For any c, the function wpyq " χ per,0 pyq `c thus satisfies

Hpy, Dwq " E, in B R p0qzQ R1 ,
in the sense of viscosity. Moreover, it is possible to choose c such that wpyq " χ per,0 pyq `c ă ´C? dR 1 for all y P BQ R1 . By a comparison principle, it follows that w ď w R in B R p0qzQ R1 . By passing to the limit in R (possibly after the extraction of a sequence), we deduce that w is bounded from below, which concludes the proof of Proposition 2.2 when p 0 " 0.

Step 2. We need to prove the result for a general p 0 . The idea consists of suitably shifting the Hamiltonians. More explicitly, we consider the new running costs and Hamiltonians: r px, aq " px, aq `p0 ¨f px, aq, (2.26) Ą per px, aq " per px, aq `p0 ¨fper px, aq, (2.27) r Hpx, pq " Hpx, p `p0 q " sup aPA ´´p ¨f px, aq ´r px, aq ¯, (2.28) Ć H per px, pq " H per px, p `p0 q " sup aPA ´´p ¨fper px, aq ´Ą per px, aq ¯, (2.29) which satisfy all the stuctural assumptions made on , per , H and H per in Section 1.1.

It is clear that the new effective Hamiltonian r

Hppq " Hpp `p0 q is obtained from Ć H per by solving the shifted cell problem:

(2.30)

H per py, p 0 `p `DČ χ per,p q " r Hppq in R d , the solutions of which are of the form Č χ per,p " χ per,p`p0 , where χ per,p`p0 is a solution of the original periodic cell problem (1.9) associated to p`p 0 . Note that 0 P arg min r H. It is straightforward to realize that r w R pyq " w R pyq ´p0 ¨y is a viscosity solution of the state constrained problem:

r Hpy, D r w R q ď E R in B R p0q, (2.31) r Hpy, D r w R q ě E R in B R p0q. (2.32)
and that r w R tends to r wpyq " wpyq ´p0 ¨y locally uniformly. We can apply the results proven above to r H and r w because r H reaches its minimal value at 0. This yields the desired result in the general case.

2.2.

The function u is a subsolution of (1.11) and satisfies (1.12)-(1.13).

Proposition 2.3. The upper limit u satisfies (1.12). Proof. Let us proceed by contradiction and assume that (2.33) αup0q `E " θ ą 0.

Using w R defined in paragraph 2.1.2 (recall w R p0q " 0), we define

φ ε,R " up0q `εw R p x ε q.
We deduce from (2.10) that φ ε,R is a viscosity supersolution of

αφ ε,R pxq `H ´x ε , Dφ ε,R ¯ě αεw R ´x ε ¯`E R ´E `θ in B εR p0q.
There exists r ą 0 such that E R ´E ě ´θ 4 for any R ě r. Let us fix such a value of R.

Having fixed R, we see that for ε 0 sufficiently small and any ε such that 0 ă ε ă ε 0 , αεw R pyq ě ´θ 4 for any y P B R p0q.

We deduce that, for any ε ă ε 0 ,

αφ ε,R `H ´x ε , Dφ ε,R ¯ě θ 2 in B εR p0q.
Next, using (2.1), consider a vanishing sequence 0 ă ε n ă ε 0 such that u εn p0q tends to up0q. We know that u εn satisfies in the sense of viscosity

α u εn `H ˆx ε n , Du εn ˙ď 0 in B εnR p0q.
The comparison principle for Hamilton-Jacobi equations with state constraints, see [START_REF] Soner | Optimal control with state-space constraint. I[END_REF][START_REF] Capuzzo-Dolcetta | Hamilton-Jacobi equations with state constraints[END_REF] and [6, Th. 5.8, Chapter IV, page 278], then implies that

φ εn,R ´θ 2α ě u εn in B εnR p0q,
or, put differently, up0q `εn w R p x ε n q ´θ 2α ě u εn pxq for any x P B εnR p0q.

Taking x " 0 and letting n tend to `8 yields up0q ´θ 2α ě up0q, the desired contradiction.

The next proposition states that u is a viscosity subsolution of (1.11) (which is already known from Proposition 2.1) and satisfies condition (1.13).

Proposition 2.4. The function u is a viscosity subsolution of αv `HpDvq ď 0 in the whole space R d .

Proof. We know from Proposition 2.1 that u is Lipschitz continous in R d . Hence, since u is a viscosity subsolution of αv `HpDvq ď 0 in R d zt0u, it satisfies αupxq HpDupxqq ď 0 at almost every x P R d , see [6, Prop. 1.9, Chapter I, page 31] and its proof. But H is convex. Therefore, from [6, Prop. 5.1, Chapter II, page 77], u is a viscosity subsolution of αv `HpDvq ď 0 in the whole space R d .

2.3.

The function u is a supersolution of (1.11) and satisfies (1.14). We already know from Proposition 2.1 that u is a bounded supersolution of (1.11).

Let φ P C 1 pR d q be such that u ´φ has a local minimum at the origin. We wish to prove that αup0q `max `E, HpDφp0qq ˘ě 0. The proof differs depending whether HpDφp0qq ą E or HpDφp0qq ď E. It is based on control theoretic arguments and partly inspired from the ideas proposed by P-L Lions and P. Souganidis when they dealt with the case E " Hpp 0 q, see [START_REF] Lions | Équations elliptiques ou paraboliques, et homogénéisation précisée [Elliptic or parabolic equations and precised homogenization[END_REF]. However, various new arguments, reminiscent of those used in [START_REF] Achdou | Hamilton-Jacobi equations on networks as limits of singularly perturbed problems in optimal control: dimension reduction[END_REF], will be needed to address the more difficult case E ą Hpp 0 q.

Let us start by considering p P R d such that Hppq ą E. It is obvious that p " p 0 because E ě Hpp 0 q " min q Hpqq. From the convexity and the coercivity of H, we see that there exists a unique unit vector e colinear to p ´p0 such that R Q t Þ Ñ Hpp `teq is strictly decreasing in a neighborhood of t " 0, and a unique vector p such that 1. p ´p0 is colinear to p ´p0 2. Hppq " Hppq 3. t Þ Ñ Hpp `teq is strictly increasing in a neighborhood of t " 0. We easily check that pp ´p0 q ¨e ă 0 and that pp ´p0 q ¨e ą 0.

The following observation will be useful for proving that u satisfies (1.14).

Remark 2.1. Note that if the function x Þ Ñ upxq´up0q´p¨x has a local minimum at x " 0, then the function x Þ Ñ upxq ´up0q ´minpp ¨x, p ¨xq also has a local minimum at x " 0.

We now assume that, for any λ ą 0 and y P R d , there exists an optimal trajectory pz λ , a λ q of the optimal control problem χ λ per,p pyq " inf This assumption on the existence of such an optimal trajectory is made for simplicity. In the full generality, we may find trajectories as close to optimal as needed and what follows remains true with such trajectories.

The value function χ λ per,p is a continuous function defined on R d {Z d ; let xχ λ per,p y stand for its mean value. It is well known, see [START_REF] Lions | Homogenization of Hamilton-Jacobi equations[END_REF], that ´λxχ λ per,p y tends to Hppq as λ Ñ 0 and that after the extraction of a sequence, y Þ Ñ χ λ per,p pyq ´xχ λ per,p y tends to χ per,p uniformly, where χ per,p is a corrector associated with p for the periodic homogenization problem. We recall that in periodic homogenization, the corrector may not be unique, even up to the addition of a constant.

Similarly, let us assume that there exists pz λ , ãλ q, an optimal trajectory for the optimal control problem χ λ per, ppyq " inf The next lemma gives a uniform upper bound on the time spent by the optimal trajectories z λn in the half-space tx P R d : e ¨x ď ru for r ą 0, as λ n converges to 0, and a symmetric estimate concerning zλn . This lemma will be used in the proof of Proposition 2.6 below.

Lemma 2.5. If Hppq ą E, let pλ n q nPN and p λn q nPN be sequences converging to 0 such that y Þ Ñ χ λn per,p pyq ´xχ λn per,p y converges to χ per,p uniformly and y Þ Ñ χ λn per, ppyq xχ λn per, py converges to χ per, p uniformly. There exists a positive constant c that only depends on Hppq, such that for any T ą 0, there exist subsequences still denoted by pλ n q nPN and p λn q nPN satisfying the following properties: for the unit vector e introduced in the beginning of Section 2.3, e ¨pz λn ptq ´yq ě cpt ´1q and e ¨pz λn ptq ´yq ď ´cpt ´1q, for all n and t P r0, T s.

Proof. We will omit the index n, except at the end of the proof. We focus on the first assertion (concerning z λ ) since the proof of the second assertion (on zλ ) is similar. Since Hpp 0 q ď E ă Hppq, there exists q " p 0 `π0 pp ´p0 q, with 0 ă π 0 ă 1 such that E ă Hpqq ă Hppq. The trajectory pz λ , a λ q is then strictly suboptimal for the optimal control problem :

χ λ per,q pyq " inf z,a "ż 8 0 e ´λt
´ per pzptq, aptqq `q ¨fper pzptq, aptqq ¯dt We deduce from the latter two inequalities that

´"χ λ per,q pz λ q ´xχ λ per,q y ´χλ per,p pz λ q `xχ λ per,p y ı t 0 ďpq ´pq ¨pz λ ptq ´yq ´λtpχ λ per,q pz λ ptqq ´χλ per,p pz λ ptqqq `2Cλt 2 .

(2.38)

We know that for two correctors χ per,p and χ per,q of the periodic problems (1.9) respectively associated with p and q, it is possible to find some subsequence λ such that ' χ λ per,q ´xχ λ per,q y converges uniformly to χ per,q ' χ λ per,p ´xχ λ per,p y converges uniformly to χ per,p ' λpχ λ per,q ´χλ per,p q converges uniformly to ´Hpqq `Hppq. Note that 0 ă Hppq ´Hpqq ă Hppq and that χ per,q and χ per,p are uniformly bounded by a constant which depends on Hppq. Hence the left hand side of (2.38) is bounded from below by ´c0 , for a suitable constant c 0 depending on Hppq only.

From (2.38) and the observations above, we deduce that there exists a constant c 1 depending on Hppq only, such that for all T ą 0, we may select a subsequence pλ n q such that pp ´qq ¨pz λn ptq ´yq ď pHpqq ´Hppqqt `c1 for all t P r0, T s and all n. The desired result follows.

The next proposition deals with the existence of a corrector associated to the piecewise linear function y Þ Ñ minpp ¨y, p ¨yq, in the whole space R d . In view of Remark 2.1, this corrector will be useful for proving that u satisfies (1.14). As already mentioned, this strategy differs from that used in [START_REF] Lions | Équations elliptiques ou paraboliques, et homogénéisation précisée [Elliptic or parabolic equations and precised homogenization[END_REF] in which the corrector is associated to the linear function y Þ Ñ p ¨y, under stronger assumptions.

Proposition 2.6. For any p P R d such that Hppq ą E, let e and p be defined as in the beginning of Section 2. For this purpose, we distinguish two cases: Case 1: E ą Hpp 0 q. We first observe that y Þ Ñ min ´pp´p 0 q¨y, pp´p 0 q¨y ¯is bounded from above. Therefore, y Þ Ñ min ´pp´p 0 q¨y `χper,p pyq, pp´p 0 q¨y `χper,p pyq īs also bounded from above. Since Hppq ą E ą Hpp 0 q, we deduce from this and (2.15) in Proposition 2.2 that for |y| large enough, wpyq ´p0 ¨y ´c is larger that min ´pp ´p0 q ¨y `χper,p pyq, pp ´p0 q ¨y `χper,p pyq ¯, hence σpyq " min ´p ¨y `χper,p pyq, p ¨y `χper,p pyq ¯.

This yields (2.47). Case 2: E " Hpp 0 q. From the second conclusion of Proposition 2.2, we know that there exists a a constant C ą 0 independent of R such that, for all y P R d , (2.48) wpyq ´p0 ¨y ą ´C.

Hence, (2.44), which also reads (2.49) wpyq ´p0 ¨y ´c ă min ´pp ´p0 q ¨y `χper,p pyq, pp ´p0 q ¨y `χper,p pyq then yields that |e ¨y| is bounded by a constant uniform in R. In turn, this implies that min ´pp ´p0 q ¨y `χper,p pyq, pp ´p0 q ¨y `χper,p pyq īs bounded above uniformly in R. Combining this with (2.48), we deduce that for a possibly different value of the constant C, wpyq ´c ´min ´p ¨y `χper,p pyq, p ¨y `χper,p pyq ¯ě ´C.

Inserting this minoration into (2.43) yields (2.47).

Our third step consists of finding a supersolution to (2.41)-(2.42). This is easy, because for q P R d , |q| large enough and y 0 P R d such that ´q ¨y0 is large enough, y Þ Ñ q ¨py ´y0 q is indeed a supersolution of (2.41)-(2.42).

The existence of a solution χ R p, p to (2.41)-(2.42) is then obtained by Perron's method. It can be proved in a classical way that this solution is unique. Besides, we have the following representation formula:

(2.50) χ R p, ppyq " inf z,a,τ $ ' & ' % ż τ 0
´ pzptq, aptqq `Hppq ¯dt `min ´p ¨zpτ q `χper,p pzpτ qq, p ¨zpτ q `χper,p pzpτ qq ¯, / . / -, subject to z 1 ptq " f pzptq, aptqq, aptq P A for almost t ą 0, zp0q " y, τ " inftt ą 0 : |zptq| " Ru.

Now, our goal is to obtain χ p, p as a locally uniform limit of χ Rn p, p for some sequence R n tending to `8. For that purpose, we will use (2.47) which readily gives a bound from below on χ R p, p, but we also need an accurate bound from above. Our fourth step therefore consists of obtaining this bound by using the representation formula (2.50) and Lemma 2.5.

With c given in Lemma 2.5, which depends only on Hppq, let T ą 0 be such that cpT ´1q ą 2R. We denote by λ any term in the sequence appearing in Lemma 2.5. With pz λ , a λ q introduced above, let τ λ be the first time at which z λ hits BB R p0q, which is smaller than T thanks to Lemma 2.5. From (2.50), we deduce that χ R p, ppyq ď ¨ż τ λ 0 ` pz λ ptq, a λ ptqq `Hppq ˘dt `min ´p ¨zλ pτ λ q `χper,p pz λ pτ λ qq, p ¨zλ pτ λ q `χper,p pz λ pτ λ qq

¯‹ ' ď ż τ λ 0 `
pz λ ptq, a λ ptqq `Hppq ˘dt `p ¨zλ pτ λ q `χper,p pz λ pτ λ qq "A `p ¨y where, for brevity, we have set

A " ż τ λ 0
´ pz λ ptq, a λ ptqq `p ¨fper pz λ ptq, a λ ptqq `Hppq ¯dt `χper,p pz λ pτ λ qq.

Observing that, from Lemma 2.5, the measure of tt : z λ ptq P B R0 p0qu is bounded, and that coincides with per outside B R0 p0q, we first obtain that, for a positive constant c 1 depending only on Hppq, A ď

ż τ λ 0
´ per pz λ ptq, a λ ptqq `p ¨fper pz λ ptq, a λ ptqq `Hppq ¯dt `χper,p pz λ pτ λ qq `c1 "

ż τ λ 0 e ´λt
´ per pz λ ptq, a λ ptqq `p ¨fper pz λ ptq, a λ ptqq ¯dt `e´λτ λ χ per,p pz λ pτ λ qq `τλ Hppq `c1 `B, where B " p1 ´e´λτ λ qχ per,p pz λ pτ λ qq `ż τ λ 0 p1 ´e´λt q

´ per pz λ ptq, a λ ptqq `p ¨fper pz λ ptq, a λ ptqq ¯dt.

But τ λ is bounded uniformly in λ (it does depend on R), so lim λÑ0 p1 ´e´λτ λ q " 0, and we see that for λ smaller than a constant depending on R, |B| ď c 1 , hence

A ď ż τ λ 0 e ´λt
´ per pz λ ptq, a λ ptqq `p ¨fper pz λ ptq, a λ ptqq ¯dt `e´λτ λ χ per,p pz λ pτ λ qq `τλ Hppq `2c 1

Using (2.36), the right hand side in the latter inequality can be written χ λ per,p pyq ´e´λτ λ χ λ per,p pz λ pτ λ qq `e´λτ λ χ per,p pz λ pτ λ qq `τλ Hppq `2c 1 .

Next, since lim λÑ0 λxχ λ per,p y " ´Hppq, we get that for λ small enough,

A ď χ λ per,p pyq ´e´λτ λ χ λ per,p pz λ pτ λ qq `e´λτ λ χ per,p pz λ pτ λ qq ´λτ λ xχ λ per,p y `3c 1 " χ λ per,p pyq ´xχ λ per,p y ``´e ´λτ λ `1 ´λτ λ ˘xχ λ per,p y ´e´λτ λ `χλ per,p pz λ pτ λ qq ´xχ λ per,p y ´χper,p pz λ pτ λ qq ˘`3c 1

The uniform convergence of χ λ per,p ´xχ λ per,p y to χ per,p then implies that for λ small enough A ď χ per,p pyq `4c 1 .

We have proven that for all R,

(2.51) χ R p, ppyq ď χ per,p pyq `p ¨y `4c 1 .

Similarly, using Lemma 2.5, we see that there exists a positive constant c1 such that (2.52) χ R p, ppyq ď χ per, ppyq `p ¨y `4c 1 .

This concludes our fourth step. We deduce from (2.51) and (2.52) that there exists a constant c 2 independent of R, (which depends only on Hppq) such that for all y P B R p0q,

(2.53) σpyq ď χ R p, ppyq ď min pχ per,p pyq `p ¨y, χ per, ppyq `p ¨yq `c2 .

From (2.53) and the fact that χ R p, p is locally Lipschitz with a Lipschitz constant independent of R, we can find a sequence pR n q n which tends to `8 such that χ Rn p, p tends to some χ p, p locally uniformly, and by passing to the limit in (2.41), (2.53) (using (2.47)), we obtain that χ p, p satisfies (2.39)-(2.40).

Proposition 2.7. If there exists φ P C 1 pR d q such that 0 is a local minimizer of u ´φ and HpDφp0qq ą E, then (2.54) αup0q `HpDφp0qq ě 0.

Proof. We can always assume that φp0q " up0q and that u ´φ has a strict local minimum at the origin. For brevity, let us set p " Dφp0q. Because Hppq ą E, we know that p " p 0 and we can apply Proposition 2.6, with e and p defined as above. Suppose by contradiction that (2.55) αup0q `Hppq " ´θ ă 0, and consider the perturbed test-function

φ ε pxq " φp0q `εχ p, pp x ε q,
where χ p, p is the function appearing in Proposition 2.6. The definition of φ ε , (2.40) and (2.55) imply that αφ ε `Hp x ε , Dφ ε q ď αεχ p, pp x ε q´θ in the sense of viscosity in R d .

From this and (2.39), we deduce that there exists r 0 ą 0 and ε 0 ą 0 such that for all 0 ă ε ă ε 0 and 0 ă r ă r 0 , φ ε is a viscosity subsolution of (2.56) αφ ε `Hp x ε , Dφ ε q ď ´θ 2 in B r p0q.

On the other hand, since 0 is a strict local minimizer of u ´φ, there exists r 1 ą 0 and a function k : p0, r 1 s Ñ p0, 1s, such that lim rÑ0 kprq " 0 and for any r P p0, r 1 s, φpxq ď upxq ´kprq on BB r p0q.

From (2.39), we know that for x " 0, εχ p, pp x ε q " minpp ¨x, p ¨xq `|x|o εÑ0 p1q. Using (2.2), this implies that first fixing r ą 0 small enough, we have for ε small enough, (2.57) φ ε pxq ă u ε pxq ´kprq 2 on BB r p0q.

From (2.56) and (2.57) and since u ε is a viscosity solution of (1.7), the comparison principle yields that it is possible to choose r ą 0 such that, for ε small enough, φ ε pxq ď u ε pxq ´kprq 2 in B r p0q. By choosing a sequence ε n such that u εn p0q tends to up0q, we deduce that φp0q `kprq 2 ď up0q, the desired contradiction. Proposition 2.8. If there exists φ P C 1 pR d q such that 0 is a local minimizer of u ´φ and HpDφp0qq ď E, then

(2.58)
αup0q `E ě 0.

Proof. As above we may assume without loss of generality that φp0q " up0q and that 0 is a strict local minimizer of u ´φ. Let us again set p " Dφp0q.

If Hppq ą Hpp 0 q " min q Hpqq, we set e and p as above. For η, η ą 0, the function x Þ Ñ upxq ´up0q ´minppp 0 `p1 `ηqpp ´p0 qq ¨x, pp 0 `p1 `ηqpp ´p0 qq ¨xq has a local minimum at at x " 0. Given ε ą 0, we may choose η and η such that Hpp 0 `p1`ηqpp´p 0 qq " Hpp 0 `p1`ηqpp´p 0 qq " E `ε. The same argument as in the proof of Proposition 2.7 with χ p0`p1`ηqpp´p0q,p0`p1`ηqp p´p0q , the corrected version of the piecewise affine function y Þ Ñ minppp 0 `p1`ηqpp´p 0 qq¨y, pp 0 `p1`ηqpp´p 0 qq¨yq, yields αup0q `Hpp 0 `p1 `ηqpp ´p0 qq ě 0, that is αup0q `E `ε ě 0.

Letting ε tend to 0, we obtain (2.58).

On the other hand, if Hppq " min q Hpqq " Hpp 0 q, we then consider two cases: 1. If p " p 0 , then from the convexity of " H, the set tt P R, Hpp 0 `tpp ´p0 qq " Hpp 0 qu is an interval rβ, γs, with β ď 0 and γ ě 1. Let e be the unique unit vector aligned with p ´p0 such that t Þ Ñ Hpp 0 `γpp ´p0 q `teq is non increasing near t " 0. For η, η ą 0, the function x Þ Ñ up0q `minppp 0 `pγ ὴqpp ´p0 qq ¨x, pp 0 `pβ ´ηqqpp ´p0 q ¨xq touches u from below at x " 0. Given ε ą 0, we may choose η and η such that Hpp 0 `pγ `ηqpp ´p0 qq " Hpp 0 `pβ ´ηqpp ´p0 qq " E `ε. The same argument as in the proof of Proposition 2.7 with χ p0`pγ`ηqpp´p0q,p0`pβ´ηqpp´p0q , the corrected version of the piecewise affine function y Þ Ñ minppp 0 `pγ `ηqpp ´p0 qq ¨y, pp 0 `pβ ´ηqqpp ´p0 q ¨yq, yields that αup0q `E `ε ě 0. Then (2.58) is obtained by letting ε tend to 0.

2. If p " p 0 , we may choose any non zero vector q ε such that Hpq ε q " E `ε: let e be the unique unit vector aligned with q ε ´p0 such that t Þ Ñ Hpq ε `teq is non increasing near t " 0 and let qε be the unique vector different from q ε such that qε ´p0 aligned with q ε ´p0 , and Hpq ε q " E `ε. The function x Þ Ñ up0q `minpq ε ¨x, qε ¨xq touches u from below at x " 0, which yields αup0q `E `ε ě 0 as in the proof of Proposition 2.7, using the corrector χ qε,qε . Letting ε tend to 0, we obtain (2.58).

We have proved that u satisfies (1.14). In order to prove Theorem 1.1, there only remains to establish that u " u.

2.4. End of the proof of Theorem 1.1. We consider two cases: Case 1. Suppose first that there does not exist any function φ P C 1 pR d q such that u ´φ has a local minimum at the origin and that HpDφp0qq ď E. Hence, from Proposition 2.7, for any function φ P C 1 pR d q such that u ´φ has a local minimum at the origin, αup0q `HpDφp0qq ě 0.

We deduce from this and Lemma 2.1 that u is a bounded supersolution of (1.8) in R d . On the other hand, from Proposition 2.4, u is a bounded subsolution of (1.8) in R d . Hence, from the comparison principle, [6, Th. 2.12, Chapter III, page 107], we deduce that for any x P R d , upxq ď upxq. From the definitions (2.1)-(2.2) of u and u, we deduce that u " u. Case 2. If, on the contrary, there exists φ P C 1 pR d q such that u ´φ has a local minimum at the origin and that HpDφp0qq ď E, Proposition 2.8 yields that αup0q`E ě 0. On the other hand, from Proposition 2.3, we know that αup0q`E ď 0. From the definitions (2.1)-(2.2) of u and u, we deduce that up0q " up0q. This allows us to apply the comparison principle in R d zt0u, [6, Remark 2.14, Chapter III, page 109], because u and u are respectively a bounded supersolution and a bounded subsolution of (1.11) in R d zt0u. Therefore, for any x P R d , upxq ď upxq, and finally, u " u.

In both cases, we deduce that the whole family u ε converges locally uniformly to a solution of (1.11) through (1.14), which concludes the proof of Theorem 1.1.

Miscellaneous comments and extensions.

This section contains various comments about, and illustrations of the general result proven above.

3.1.

A simple and explicit one-dimensional example. We begin by providing in this Section 3.1 some illustrations of the results in a simple one-dimensional setting.

Upon considering this simple situation, we expect to illustrate as sharply as possible, the homogenization process described by Theorem 1.1. We also intend to (a) make explicit in those simple settings the key quantities involved in Theorem 1.1, in particular the constant E and the Dirichlet condition (1.12)-(1.13)-(1.14) and (b) provide some additional light on some qualitative aspects of the problem that are best exposed using simple examples and that will be generalized in higher dimensions later on.

We will exploit the peculiarity of the considered one-dimensional setting. In the course of our arguments, we will make several simplifying assumptions about the type of Hamiltonian H and the shape of the defect(s) that we consider: see for instance (3.2), (3.7), (3.31) below, etc. Even if we expect that several of our arguments may carry over when some of these simplifying assumptions are relaxed, it might be the case that not everything can be generalized. We do not claim that the techniques we are employing below, which are specific to the particular setting, carry over to higher dimensions and other Hamiltonians. We only intend to illustrate, on some simple enough cases, some phenomena that we find interesting, with no sake of generality whatsoever.

We consider in this section what is, to some extent, the simplest possible setting where the homogenization process we have established in the general setting occurs. We pose the Hamilton-Jacobi equation on the real line R. We pick a separate Hamiltonian H (by this term, we mean, throughout this section, an Hamiltonian that depends separately of x and p), with a potential part that reads as the sum (3.1)

" per ` 0 , of a periodic function per and a local defect 0 . Both per and 0 are assumed smooth. We also assume that 0 is compactly supported, with, say,

(3.2) Supp 0 Ă r´1 2 , 1 2 s.
In order to observe an actual perturbation by 0 of the homogenized equation, we further assume

(3.3) inf R p per ` 0 q ă inf R per .
The role of this condition will become clear below but it is already intuitive that its spirit is to make sure that the defect indeed locally lower the periodic environment (or put differently in the optimal control interpretation, diminishes the periodic cost) and therefore indeed shows up in the homogenized limit. Condition (3.3) is our only actual assumption, all the other assumptions above and below (smoothness of the data, compact support of 0 , and others to come) being only used to simplify the algebraic expressions manipulated and to spare the reader unnecessary technicalities. As for the kinetic part, we choose |p|, that is a simple, commonly used convex nonlinearity.

In our notation of Section 1.1, our choices of course correspond to A " r´1, 1s and f px, aq " f per px, aq " a. Put differently, there is neither oscillation nor defect in this kinetic part. We thus manipulate (3.4) Hpx, pq " |p| ´ per pxq ´ 0 pxq throughout this Section 3.1.

Because this positive constant is irrelevant in our arguments and can be easily reinstated in our results, we choose α " 1 in (1.7) and keep this value throughout the section.

3.1.1.

A downward defect inserted in a flat environment. To start with, let us temporarily further simplify the above model. We take per " 0 and recall the trivial example already introduced in [11, Section 6]. The Hamilton-Jacobi equation considered then reads as

(3.5) u ε pxq `ˇp u ε q 1 pxq ˇˇ" 0 ´x ε ¯,
and should be thought as the perturbation, by the local potential 0 , of the periodic equation (3.6) upxq `ˇu 1 pxq ˇˇ" 0.

As is well known, the only bounded C 1 solution of (3.6) is u " 0. We may likewise make explicit the bounded solution u ε to (3.5). For this purpose, and just to keep algebraic manipulations and expressions simple, we assume, in addition to (3.2)-(3.3), that 0 satisfies (3.7) 0 p0q " inf xPR 0 pxq and $ & % p 0 q 1 pxq ă 0, @x Ps ´1 2 , 0r X Supp 0 , p 0 q 1 pxq ą 0, @x Ps0, 1 2 r X Supp 0 .

Precisely under such conditions, it is easy to verify that the unique C 1 solution to (3.5) reads as

(3.8) u ε pxq " $ ' ' ' ' & ' ' ' ' % e x ˆ 0 p0q `ż 0 x e ´t 0 ˆt ε ˙dt ˙for x ă 0, e ´x ˆ 0 p0q `ż x 0 e t 0 ˆt ε ˙dt ˙for x ą 0,
Indeed, a simple computation allows to check that, for instance if x ą 0, the function gpxq " e x bpxq ´bp0q ´ż x 0 e t bptq dt is such that gp0q " 0 and g 1 pxq " e x b 1 pxq. Hence, if b reaches its global minimum at x " 0, is non-increasing for x ă 0 and non-decreasing for x ą 0, then the function g satisfies gp0q " 0, is non-increasing for x ă 0 and non-decreasing for x ą 0. This clearly implies that g ě 0 on R. Applying this to b " 0

´. ε ¯we deduce that

e x 0 ´x ε ¯´ 0 p0q ´ż x 0 e t 0 ˆt ε ˙dt ě 0.
This means that, u ε being defined by (3.8), pu ε q 1 pxq is non-negative for x ą 0, hence pu ε q 1 pxq " |pu ε q 1 pxq|. In addition, since u ε pxq `pu ε q 1 pxq " 0 `x ε ˘, we obtain equation (3.5) for x ą 0. A similar argument holds for x ă 0. Using the explicit expression (3.8) of the solution to (3.5), we now easily identify its limit u as ε Ñ 0. Since the function 0 has compact support, the sequence of functions 0

´. ε ¯converges strongly to zero in L p for any p ă 8. Hence, in any such space, the locally uniform limit u of u ε , is (3.9) upxq " 0 p0q e ´|x| .

Put differently, the homogenized limit u is a bounded (in fact, vanishing at infinity) solution to (3.10)

$ & % upxq `ˇu 1 pxq ˇˇ" 0 for x " 0, up0q " 0 p0q.

It is enlightening to reconcile this particular result with the general result stated in Theorem 1.1. Since the periodic Hamiltonian identically vanishes here, its homogenized limit is evidently H " 0. The first line of (3.10) thus agrees with (1.11) (for the choice of constant α " 1 of course). Far more interesting is the Dirichlet condition up0q " 0 p0q posed at the origin. In the particular setting of the present section, (1.12) and (1.14) respectively read as up0q ď ´E and up0q `max pE, 0q ě 0. The effective Dirichlet datum E is itself obtained, in whole generality, as the limit, first as λ Ñ 0 and next as R Ñ `8, of ´λw λ,R , where w λ,R is the solution to (2.3)-(2.4). In our setting again, w λ,R is actually indeed independent of R and is easily obtained from the explicit expression (3.8). Using a rescaling argument (setting w λ,R pxq " u ε"1 pλxq for the rescaled defect λ ´1 0 pλ ´1xq), we realize that λ w λ,R `ˇp w λ,R q 1 ˇˇ" 0 in R.

(3.11) w λ,R pxq " $ ' ' ' ' ' & ' ' ' ' ' % λ ´1 e λx
Taking the limit λ Ñ 0 of ´λ w λ,R readily yields the effective Dirichlet condition E " ´ 0 p0q. But, since we have assumed that 0 p0q ă 0, E is positive. Thus up0q ď ´E and up0q `max pE, 0q ě 0 combine with one another into (3.13) up0q " ´E " 0 p0q .

We have thus recovered in (3.10) the general limit (1.11)-(1.12)-(1.13)-(1.14) stated in Theorem 1.1. In addition, taking the limit, as λ Ñ 0 and say for x ą 0, of w λ,R pxq ´wλ,R p0q " λ ´1 ˜pe ´λx ´1q 0 p0q `e´λx

ż λx 0 e t 0 `λ´1 t ˘dt ¸, yields w λ,R pxq ´wλ,R p0q Ñ ´ 0 p0q x `ż x 0 0 ptq dt,
and we thus find that, as predicted by (2.15) in Proposition 2.2, lim xÑ`8 wpxq " `8.

A similar argument using the first line of (3.11) confirms the same limit in ´8.

Remark 3.1. We note in passing that, had we assumed, say, that the local defect 0 is everywhere nonnegative, satisfies % p 0 q 1 pxq ą 0, @x Ps ´1, 0r X Supp 0 , p 0 q 1 pxq ă 0, @x Ps0, 1r X Supp 0 , instead of (3.7) (and assumed also, again a technicality, that 0 is even), then an easy adaptation of the above algebraic expressions shows that, mutatis mutandis, This explicit calculation thus shows that, instead of solving (3.10), the homogenized limit u vanishes over the real line, just like it is the case for the periodic, null, Hamiltonian. As briefly mentioned in the introduction, the presence of this "upward" defect 0 ě 0 only affects the next order term. This relates to the "other regime" considered in the works [START_REF] Lions | [END_REF][START_REF] Cardaliaguet | Perturbation problems in homogenization of Hamilton-Jacobi equations[END_REF].

(3.15) u ε pxq " $ ' ' ' ' & ' ' ' ' %
Remark 3.2. We also store for future use (see Remark 3.3 and Section 3.2.1) the observation that the homogenized limit u given by (3.9) does not have compact support, and therefore only agrees at infinity x " ˘8 with the (here, trivial) homogenized limit of the periodic case, solution to (3.6).

3.1.2. Periodic unperturbed environment: the homogenized limit. We now reinstate a non trivial (that is, non constant) periodic potential per in (3.1), which, without loss of generality, we assume of period 1. We also assume that there exists only one point per unit interval where per reaches its minimum : per px 0 q " inf R per . This is again just for simplicity. We recall that we additionally assume (3.2)-(3.3) and the separated form (3.4) of the Hamiltonian H. Before we insert any defect 0 in the periodic environment described by per , we need to lay some groundwork and make explicit the homogenized limit in the absence of defect.

Homogenized equation. For convenience, we now recall some basic facts. We consider the one-dimensional equation (3.16) u per,ε pxq `ˇp u per,ε q 1 pxq ˇˇ" per ´x ε ¯, (obviously a particular case of (1.8)) where the subscript per in u per,ε refers to the fact that the right-hand side is only the periodic cost per . Actually, using elementary facts from the theory of viscosity solutions, u per,ε is indeed periodic (of period ε); we will use this property below, but at this time this is irrelevant.

The homogenized Hamiltonian H arising from (3.16) is characterized, for any p P R, as the unique real number Hppq such that there exists a periodic corrector χ per,p , viscosity solution to (3.17) ˇˇp `pχ per,p q 1 pyq ˇˇ" Hppq ` per pyq in R , which is a particular case of (1.9). Mimicking the explicit calculations performed in [START_REF] Lions | Homogenization of Hamilton-Jacobi equations[END_REF] that address the case of the quadratic Hamiltonian |p| 2 ´ per pyq instead of |p| ´ per pyq, it is easy to identify the periodic solution χ per,p and the homogenized Hamiltonian It follows that the (periodic) homogenized equation, defined by (1.8) in whole generality, here reads as u `Hpu 1 q " 0 , for the particular Hamiltonian (3.18). We immediately notice that its unique BUC viscosity solution is the constant function

u " inf R per ,
which we henceforth denote by u per to distinguish it from the homogenized limit u in the presence of defect, which we are going to manipulate shortly.

An alternative "direct" proof. Our next step is to remark that in the present setting, besides being a consequence of the general periodic homogenization theory, the limit u per,ε Ñ inf R per may alternatively and independently be obtained from the original equation (3.16), as soon as we have some elementary information on the solution u per,ε . The interest of following this alternative route is to better understand the phenomena at play and prepare the ground for the homogenization limit in the presence of a defect that will be addressed in Section 3.1.3 below.

It is immediate to deduce from (3.16) that its solution u per,ε consists of the combination of two different branches, respectively solving (3.19) u per,ε pxq `pu per,ε q 1 pxq " per ´x ε ¯when pu per,ε q 1 pxq ě 0 and (3.20) u per,ε pxq ´pu per,ε q 1 pxq " per ´x ε ¯when pu per,ε q 1 pxq ď 0 .

We now introduce

(3.21) g per,ε pxq " e ´x ż x ´8 e t per ˆt ε ˙dt,
which is not an unexpected function given the calculations in Section 3.1.1 above. It is evidently a periodic function, which satisfies g per,ε `pg per,ε q 1 " per ´. ε ¯and thus xg per,ε y " x per y. We deduce that in the branch described by (3.19), u per,ε pxq " g per,ε pxq `α e ´x for some constant α (depending on ε but this is irrelevant). Should the branch (3.19) extend to `8, we would therefore obtain, using periodicity and shifting to `8, that u per,ε pxq " g per,ε pxq for all x in that branch.

Since xg per,ε y " x per y while, in view of (3.16), u per,ε ď per

´. ε ¯everywhere, this implies g per,ε " per ´. ε ¯. Given the differential equation satisfied by g per,ε , this may only hold when per is constant. We reach a contradiction with the assumption made on per .

Similarly to (3.21), we introduce the function h per,ε pxq " e x A symmetric argument allows to exclude that the branches extend to ´8. We conclude that the solution u per,ε consists of an effective, infinite alternation of the two branches (3.19) and (3.20). In our particular periodic setting, we could have obtained this fact simply using the uniqueness, thus the periodicity, of the viscosity solution to (3.16). But the specific argument we have developed will be useful later, in the presence of a defect.

We next remark that, because of the generic properties of viscosity solutions, a transition, at some x, from the branch (3.20) at the left of x to the branch (3.19) at its right, may only occur when pu per,ε q 1 is continuous at x, nonpositive before and nonnegative after x. Thus pu per,ε q 1 pxq " 0 and x is a local minimizer of u per,ε . Using the equation, u per,ε pxq " per pxq. Since we know that u per,ε ď per everywhere, the three facts altogether imply that per reaches its minimum at x. For simplicity, we have assumed that such a minimizer is unique per period. So we exactly know that, per period, there is one transition (3.20)-(3.19) (at the minimizer x " arg min per ) and, thus also, one transition (3.19)-(3.20) (thus necessarily above the level min R per ).

An interesting corollary of the previous observation is the following. Since u per,ε ě inf R per everywhere, |pu per,ε q 1 | " per ´uper,ε is thus bounded independently of ε. It follows from these two facts together with the periodicity of u per,ε (with period ε) that, necessarily, u per,ε Ñ inf R per as ε vanishes. In dimension 1, this provides us with an independent and alternative proof of the homogenization limit, as announced above.

3.1.3. A downward defect in this periodic environment. We now insert the defect 0 . The equation under consideration is

(3.22) u ε pxq `ˇp u ε q 1 pxq ˇˇ" per ´x ε ¯` 0 ´x ε ¯.
Using the exact same argument as above in (3.19)-(3.20), the two auxiliary functions g per,ε and h per,ε , along with the functions g 0,ε pxq " e ´x ż x ´8 e t 0 ˆt ε ˙dt,

h 0,ε pxq " e x ż 8
x e ´t 0 ˆt ε ˙dt, we may similarly conclude that no branch can extend to ˘8. Recall indeed that 0 vanishes at infinity so does not modify the argument performed in the periodic case above. We can thus claim that u ε is an infinite alternation of the two branches. Again similarly as above, we know that the junction (3.20)- (3.19) (in this order from left to right) may only occur at local minimizers of p per ` 0 q

´. ε ¯, and that at such points, u ε " p per ` 0 q

´. ε ¯. Outside Supp 0

´. ε ¯, this thus occurs at some minimizers of per

´. ε ¯, at which u ε coincides with per `. ε ˘.

We next recall that, precisely at all minimizers of per ´. ε ¯, we also have the equality u per,ε " per ´. ε ¯, where u per,ε denotes the solution in the absence of defects. Thus, at any minimizer of per ´. ε ¯outside Supp 0

´. ε ¯where u ε coincides with per `. ε ˘, we also have u ε " u per,ε . Between two such minimizers, using a comparison principle for the equation, we deduce that u ε " u per,ε . So u ε " u per,ε at least everywhere outside a bounded interval containing the origin and presumably slightly exceeding Supp 0 `. ε ˘. Our next task is thus to fully identify the homogenized limit of u ε . Since we do not have as much information on u ε as we used to have on u per,ε in the previous section, we need to make a small detour (in fact by defining the homogenized problem itself and next backpedalling) .

We introduce the function (3.23) upxq "

$ ' ' & ' ' %
x per y ´e´|x|

´x per y ´inf R p per ` 0 q ¯for |x| ď µ, inf R per for |x| ě µ ,
for µ defined by (3.24) e ´µ " x per y ´inf R per x per y ´inf R p per ` 0 q .

This function u is indeed the homogenized limit of u ε , as we will see below. For the time being, we just remark that (i) when 0 " 0 and more generally exactly when condition (3.3) is not satisfied, then µ " 0 and u " inf R per is indeed the homogenized limit we found for u per,ε . (ii) when per " 0, then µ " `8, upxq " pinf R 0 q e ´|x| and we recover the homogenized limit (3.9) if we additionally assume (3.7) as we did in Section 3.1.1.

We also remark that u solves

$ ' & ' % upxq `H `u1 pxq ˘" 0 for x " 0, up0q " inf R p per ` 0 q,
where H is defined in (3.18). In this specific setting, we again recover, as in Section 3.1.1, the general limit (1.11)-(1.12)-(1.13)-(1.14) stated in Theorem 1.1, this time with the specific value

(3.25) E " ´inf R p per ` 0 q
of the effective Dirichlet condition. Expression (3.25) is obviously a generalization of (3.13). And (1.12) and (1.14) again combine with one another to yield the above Dirichlet condition, precisely because of the condition (3.3). We omit here the detailed verification that the explicit value of E given in (3.25) may indeed be independently obtained using the sequence of approximate correctors, as in the general theory using (2.3)-(2.4), similarly to what we did above in Section 3.1.1 in the particular setting where per identically vanishes.

In order to prove that u defined in (3.23) is indeed the limit of u ε , let us consider

v ε pxq " e ´x ż x 0 e t ˆt ε ˙dt `e´x inf R .
As ε vanishes, this function pointwise (and in fact locally uniformly) converges to

x per y p1 ´e´x q `e´x inf R " upxq, as defined by (3.23), for all 0 ă x ă µ. Since evidently per ´x ε

¯ě inf R per , per ´x ε ¯´v ε pxq ě inf R per ´e´x ż x 0 e t ˆt ε ˙dt ´e´x inf R ,
where, as ε vanishes, the right-hand side likewise converges to inf

R per ´x per y p1 ´e´x q ´e´x inf R " pinf R per ´inf R q ´px per y ´inf R q p1 ´e´x q.
Note that this quantity is nonnegative when

x ď µ, since pinf R per ´inf R q ´px per y ´inf R q p1 ´e´x q ě pinf R per ´inf R
q ´px per y ´inf R q p1 ´e´µ q " 0, and only vanishes at x " µ.

If we now take 0 ă x ă µ, outside Supp 0

´. ε ¯, and slightly bounded away from µ (by some irrelevant constant, say of order ε), we thus have, for ε sufficiently small, ´x ε ¯´v ε pxq ě pactually " q per ´x ε ¯´v ε pxq ą 0.

If follows that, in that region, pv ε q 1 pxq " ´x ε ¯´v ε ą 0 and thus v ε is solution to (3.22). We may proceed similarly for ´µ ă x ă 0 and the other branch, that of type (3.20). The function v ε that we have constructed may then be continued close to and beyond ˘µ. Eventually, it yields the global solution u ε .

Remark 3.3. In echo to Remark 3.2, we notice that, here, the two homogenized limits obtained respectively for per and per ` 0 exactly agree outside a bounded interval surrounding the origin, unless per is constant. The latter case is indeed the only case where the homogenized Hamiltonian H is not flat around p " 0 (it is |p|). The agreement at infinity of the two homogenized solutions, respectively in the absence and in the presence of defects, reflects this property. We will further investigate this question in higher dimensions in Section 3.2.1 below.

A randomized variant.

As briefly mentioned in Section 1, a randomized variant of the theory of local defects has been introduced by A. Anantharaman and the second author in [START_REF] Anantharaman | A numerical approach related to defect-type theories for some weakly random problems in homogenization[END_REF]. In that work, the equation under consideration was a linear elliptic equation in conservation form. The adaptation of the setting considered therein to our equation (3.22) is as follows.

We consider as a perturbation of the underlying periodic environment encoded in per , the random running cost

S px, ωq " ÿ kPZ X η k pωq 0 px ´kq,
where tX η k pωqu kPZ is a sequence of i.i.d. random variables that all follow a Bernoulli law of parameter η, that is, X η k pωq " 0 or " 1 with probability 1 ´η and η respectively. The parameter η ą 0 is to be thought of as a small parameter, which will eventually be sent to zero. This models that the periodic environment encoded in per is only slightly perturbed.

After rescaling in ε of the potential S , the equation for which we study homogenization thus reads as

u ε px, ωq`ˇˇpu ε q 1 px, ωq ˇˇ" per ´x ε ¯` S ´x ε , ω ¯" per ´x ε ¯`ÿ kPZ X η k pωq 0 ˆx ´kε ε ˙.
We intend to identify the homogenized limit ε Ñ 0 for this equation in the regime when the parameter η vanishes. Here, to keep things simple and allow for analytic calculations, we only consider the case per " 0 of a flat unperturbed environment, that is

(3.26) u ε px, ωq `ˇp u ε q 1 px, ωq ˇˇ" ÿ kPZ X η k pωq 0 ˆx ´kε ε ˙,
using a "shape function" 0 for the defects that satisfies (3.7).

Remark 3.4. Let us emphasize that, even though the specific algebraic manipulations below depend on the technical assumption that per " 0, the general result that we are going to obtain carries over to other cases.

Extensions to other ambient dimensions will be addressed in Section 3.2.

In order to understand the homogenized limit, we need a preliminary step, namely the case of two localized (downward) defects, instead of a single one at the origin as in Section 3.1.3. Intuitively, S px, ωq is the random superposition of an infinite number of such defects. Since "He who can do more can do less", we need to first understand the case of two defects, that is:

(3.27) z ε pxq `ˇp z ε q 1 pxq ˇˇ" 0 ´x ε ¯` 0 ˆx ´ε ε ˙.
It is then easy to realize that the solution reads as

(3.28) z ε pxq " inf pu ε pxq , u ε px ´εqq ,
where u ε is the solution for one single defect at the origin, which we made explicit in (3.8). In order to realize that (3.28) holds, we just need to establish that, in Supp 0 ´ε ¯, we have u ε pxq ď u ε px ´εq. A similar argument will prove a symmetric result on Supp 0 ˆx ´ε ε ˙, while, outside the two supports, both u ε p¨q and u ε p¨´εq solve (3.27). Let us recall that, because of (3.2), the two supports Supp 0 and Supp 0 p¨´1q are disjoint. Given the explicit expression (3.8) of u ε and its monotonicity on R `, we then have that

(3.29) u ε pxq ď e ´ε a ˆ 0 p0q `ż ε a 0 e t 0 ˆt ε ˙dt ˙,
for a ď 1 2 and all 0 ă x ă ε a. On the other hand, again for all 0 ă x ă ε a, we have, by the same argument but this time looking at the explicit expression of u ε px ´εq that , (3.30) u ε px ´εq ě e ε a´ε

ˆ 0 p0q `ż ε ε´ε a e ´pt´εq 0 ˆt ´ε ε ˙dt ˙.
Since a ď 1 2 , we have e ε a´ε ď e ´ε a . On the other hand, for simplicity let us further assume that (3.31) 0 is an even function , so that the two integrals appearing in the right-hand sides of (3.29) and (3.30) are identical. We deduce that u ε pxq ď u ε px ´εq for all 0 ă x ă ε a, that region being the part of Supp 0 ´ε ¯at the right of the origin. A similar argument allows to conclude for all the other regions. Note that a simple estimation of the terms in (3.29) and (3.30) shows that, had we not assumed (3.31) for simplicity, the argument above would also hold true provided the two defects are separated from a distance q ε for an integer q chosen sufficiently large. We will see a similar argument in Section 3.2.2 below.

Equation (3.28) readily implies that, returning to the random setting (3.26), (3.32) u ε px, ωq " inf kPZ tX η k pωq u ε px ´kεqu , since any k P Z for which X η k pωq " 1 contributes to lowering the solution, while others k do not. The "min-formula" (3.32) is the main ingredient in the rest of our argument and will be generalized in Section 3.2.

Remark 3.5. Note that we work here with the solution u ε of the original equation (3.26), prove it explicitly reads as (3.32) and proceed from there to find its homogenized limit. In fact, we could, using a similar string of arguments, identify the solution to the discounted problem (3.33) δ w δ py, ωq `ˇp w δ q 1 py, ωq ˇˇ" ÿ kPZ X η k pωq 0 py ´kq , for δ ą 0, that replaces the corrector problem in the context of stochastic homogenization. The similarity between (3.26) and (3.33) is evident. Using a simple rescaling (exactly as we did in the specific deterministic case (3.11)-(3.12) above), any explicit expression of the solution u ε of the former equation yields an expression of the solution w δ to the latter equation. Taking the limit lim δÑ0 δ w δ , we may then identify the value H η p0q of the homogenized Hamiltonian in this stochastic setting. Note that the same remark applies to the multidimensional context we will address in Section 3.2.2 below, but we will not repeat it there.

Since we already know that, in the homogenized limit, u ε given by (3.8) converges to upxq " 0 p0q e ´|x| in (3.9) (later generalized in (3.23)), we temporarily replace (3.32) by (3.34)

u ε pxq " 0 p0q sup kPZ ! X η k pωq e ´|x´kε|
) ,

where we have used that 0 p0q ă 0. Take now x " 0. The key quantity appearing on the right-hand side of (3.34) is in fact the random variable

(3.35) sup kPZ ! X η k pωq e ´|k|ε ) ,
and to get a grasp on this quantity, let us only consider therein the positive indices, so that we focus our attention on

Z η ε pωq " sup kPZ,kě0
X η k pωq e ´kε ( .

Since the function k Ñ e ´kε is decreasing with respect to k ą 0, it is evident that Z η ε " e ´kε for the first index k such that X η k " 1. But now, since the X η k are i.i.d. and all follow a Bernoulli law, it is well known that the law of this first index is geometric. More precisely, we have (3.36) Z η ε pωq " e ´kε with probability p1 ´ηq k η.

If we now let ε Ñ 0, we obtain Z η ε pωq Ñ 1 almost surely. It is easy to see that a similar argument applies to (3.35), since only |k| matters there and maxpX η ´k, X η k q are i.i.d. Bernoulli variables of parameter 1 ´p1 ´ηq 2 , which scales as η, when η is small. Thus, eventually we may guess that u ε p0, ωq Ñ 0 p0q almost surely in the limit ε Ñ 0. The actual proof of this fact necessitates to realize that we have replaced u ε given by (3.8) by its homogenized limit upxq " 0 p0q e ´|x| . The error committed in this approximation may easily be controlled (at least in this case where everything is explicitly known by the formulae (3.8) and (3.9)) and proven to be irrelevant in the limit process above. In addition, the asymptotics that we have found for x " 0 in (3.34) readily carry over to any x P R, since all what matters is the decay of the function k Ñ e ´|x´kε| away from x. We have thus obtained (3.37) a.s. lim εÑ0 u ε px, ωq " 0 p0q.

The convergence (3.37) is both disappointing and intuitive. It is disappointing because, the homogenized limit being flat, there is not much interesting mathematical phenomenon to discuss. The convergence (3.37) is, on the other hand, intuitive, because as soon as a defect occurs at a location kε for some index k such that kε Ñ 0 in the homogenized limit, this defect is brought to the origin by the rescaling and everything happens as if we had a deterministic localized defect there. For instance, even the probability of having at least one defect in the 1 ? ε first indices (say at the right of the origin) is

(3.38) ÿ 0ďkď 1 ? ε p1 ´ηq k η « 1 ´p1 ´ηq 1{ ? ε εÑ0 ÝÑ 1,
for η fixed, so all this happens with probability one. A similar argument again applies if we consider both sides of the origin, and likewise when we consider any point x since all points play identical roles. In the end, the limit (3.37) is flat. It is also independent of η. This comes in sharp contrast to the case of a similar random coefficient inserted in an elliptic equation, which is the setting studied in the work [START_REF] Anantharaman | A numerical approach related to defect-type theories for some weakly random problems in homogenization[END_REF]. This striking difference may be intuitively explained by the fact that, at the macroscopic scale, an elliptic equation only sees suitable averages of the oscillatory coefficient, while an Hamilton-Jacobi type equation sensitively sees extremal values of that coefficient. Put in an even more simplified language, what we observe is nothing but the difference between lim

N Ñ8 1 N N ÿ k"1 X η k pωq and lim N Ñ8 sup 1ďkďN tX η k pωqu ,
for the i.i.d. Bernoulli random variables X η k of parameter η. The leftmost limit scales as η while the rightmost limit is independent of η .

The net conclusion of the above discussion is that, in the model we have considered so far, the probability of having defects is so large that, in the limit, there are almost surely defects everywhere. In order to get a much more interesting regime, we need to relate the small parameter ε, which measures the distances at the microscopic scale, with the parameter η, which measures the amount of random perturbation present in the environment (recall that, for η " 0, we recover the unperturbed environment). A possible way to obtain this property is to ensure that, for instance, there is, asymptotically, a fixed, small, proportion of defects per unit macroscopic length (or, in higher dimensional settings, volume). More precisely, we now take (3.39) η " η ε, for η fixed.

The scaling law (3.39) is precisely adjusted so that, in contrast to what we observed above in (3.38) for η fixed, the probability of seeing at least one defect over a unit macroscopic length around the origin scales as (3.40)

ÿ 0ďkď 1 ε p1 ´ηεq k ηε « 1 ´p1 ´ηεq 1{ε εÑ0 ÝÑ 1 ´e´η ηÑ0 « η ! 1.
If we revisit our calculations above with the particular value (3.39), we may mimic the argument step by step. We realize that the key quantity from (3.36) is now (3.41) Z η ε pωq " e ´kε with probability p1 ´ηεq k η ε.

A simple calculation shows that, for all µ ě 0,

P `Zη ε ď e ´µ˘"
p1 ´ηεq µ{ε « e ´η µ , as ε Ñ 0, that is, for all 0 ď t ď 1, (3.42) P pZ η ε ď tq « t η , as ε Ñ 0.

Upon differentiating with respect to t, we obtain that, in the limit ε Ñ 0, the law is η t η´1 . This suffices to establish that the limit is not deterministic. Put differently, (i) when η ą 0 is fixed (or η " ε) the limit u ε is deterministic and flat at the level 0 p0q, (ii) when η " 0 (or η ! ε) the limit u ε is deterministic and flat at the level 0 (that is the unperturbed solution), and (iii) when exactly η scales as ηε in (3.39), then the limit is flat, and its value is a random variable in the interval r 0 p0q, 0s.

3.2. Generalization to higher dimensions. Motivated by the phenomena we have just observed in the one-dimensional setting of Section 3.1, we consider two specific questions in a higher dimensional situation: (i) the behavior when |x| Ñ `8 of the homogenized solution in presence of a defect at the origin, as compared to that of the homogenized solution in the unperturbed periodic setting (ii) the random superposition of point defects. The two issues are respectively studied in Sections 3.2.1 and 3.2.2 below.

3.2.1. Influence of the defect at infinity. We return to the general setting of Section 1.1 with, in ambient dimension d ě 1, a general Hamiltonian H (defined in (1.1)) satisfying the classical properties made precise in that section, in particular (1.2) through (1.4) and that is the perturbation of a periodic Hamiltonian H per (defined in (1.5)) in the following sense: f " f per and " per outside a neighborhood of the origin.

We denote by u the homogenized solution provided by Theorem 1.1. In order to avoid any confusion, we denote by u per the homogenized solution associated with the periodic Hamiltonian H per in equation (1.8). We now intend to prove that (3.43) |upxq ´uper pxq| Ñ 0 as |x| Ñ `8.

This property of course mathematically encodes that the local defect (that is, the difference between pf, q and pf per , per q assumed compactly supported near the origin) only affects the homogenized solution locally. In Section 3.1, we have seen that it is true in the one-dimensional setting and for the separate Hamiltonian considered, and even that, if per is not constant, then upxq " u per pxq outside a neighborhood of the origin.

The proof of (3.43) turns out to be rather simple. Let us consider, say, the first canonical vector e 1 in R d , and the function u n pxq " upx ´n e 1 q obviously solution to the same problem as (1.11) through (1.14), but with an effective Dirichlet condition set at the point x n " n e 1 instead of the origin. The sequence of functions u n inherits, uniformly in n P N, of the properties of u itself. We therefore have uniform almost everywhere pointwise bounds on u n and Du n . It follows that, up to an extraction (and a diagonal argument) which we henceforth omit to explicitly denote, u n locally uniformly converges to some function v. Fix now a unit ball B 1 py 0 q around an arbitrary point y 0 P R d . For any sufficiently large n P N, the point x n lies outside that ball, and the function u n thus solves (in the viscosity sense) the periodic homogenized equation (1.11), that is αu n `HpDu n q " 0 on that ball. Given the above convergence and using the result of stability of viscosity solutions for that equation, so does the limit v. By uniqueness of the bounded uniformly continuous viscosity solution to (1.11) posed on the entire space R d , we thus know that v " u per . Since the limit is unique, we conclude that the sequence u n itself converges to u per uniformly locally. The convergence (3.43) follows.

Two remarks are in order.

First, we note that we have only used the homogenized form of the equation provided by Theorem (1.1) and not the assumption itself that f " f per and " per outside a neighborhood of the origin. So, if Theorem 1.1 turns out to hold true for some defect 0 that does not necessarily have compact support (or alternatively and more generally, in some specific settings, for more general pf, q that converge to pf per , per q in a milder sense), the convergence (3.43) still holds true. On the other hand and as briefly mentioned above, unless peculiar conditions are met, there is no reason for u and u per to exactly agree outside a bounded domain.

Second, both above facts may be intuitively understood when resorting to the optimal control interpretation of the equations and solutions at play (in the spirit of our arguments of Section 2.3). When starting farther and farther away from the defect located at the origin, the optimal trajectory has lesser reason to visit the neighborhood of the origin. As |x| Ñ `8, the value function therefore becomes decreasingly affected by the presence of the defect. Thus the convergence (3.43). Additionally and depending upon the specific landscape, such an excursion toward the origin may come, or not, at a price. If for points x located far enough from the origin, this price becomes too high, then the optimal paths leaving x do not visit the support of the defect. In this case, the exact equality u " u per may hold outside a bounded domain. This is what happens for a nontrivial one-dimensional setting as that of Section 3.1.2, but not for that of Section 3.1.1 or for a higher dimensional setting where some specific paths escaping from the wells of per can be cost-advantageous. In the latter situations, only (3.43) holds.

3.2.2. Randomized variant. We devote this section to randomized lattices of points defects in R d , d ą 1. We focus our attention to the case when the background environment is constant. The extension to random perturbations of periodic Hamiltonians is left for future work, because it is not obvious that a counterpart of formula (3.32) along with the observation that the supports of the two functions 0 ˆx ´ε q k ε and 0 ˆx ´ε q k 1 ε ˙for different indices k " k 1 do not overlap and the latter two functions are nonpositive, so, for all x P R d and ε ą 0, hence

0 ˆx ´ε q k ε ˙` 0 ˆx ´ε q k 1 ε ˙" min " 0 ˆx ´ε q k ε ˙, 0 ˆx ´ε q k 1 ε ˙ .
We now consider, for almost all ω, the equation

(3.49) α U ε px, ωq `Hε S px , DU ε , ωq " 0.
We intend to first prove that this equation has a unique, bounded uniformly continuous solution U ε px, ωq in the viscosity sense, and next study the limit of this solution as ε vanishes. For this purpose, under a suitable assumption on q, we will in fact explicitly characterize the solution U ε as

(3.50) U ε px, ωq " inf kPZ d tX η k pωq u ε px ´ε q kqu ,
where u ε pxq is the unique viscosity solution in BUCpR d q to (1.7), that is

αu ε `H ´x ε , Du ε ¯" 0 in R d .
The expression (3.50) evidently generalizes (3.32) to the present multi-dimensional context.

In order to establish (3.50), it is obviously sufficient to understand the setting where only two defects are present and are respectively localized, say, at the origin x 0 " 0 and at x k " ε q k, for some k P Z d zt0u. We thus need to manipulate the two Hamiltonians H ε k and H ε 0 (the latter is obtained by setting k " 0 in (3.45)). In this simplified setting, we claim that the solution to (3.51) α V ε px, ωq `max tH ε 0 , H ε k u px, DV ε , ωq " 0 actually reads as (3.52)

V ε px, ωq " min " X η 0 pωq u ε pxq `p1 ´Xη 0 pωqq u , X η k pωq u ε px ´ε q kq `p1 ´Xη k pωqq u * ,
where u ε solves (1.7) in the sense of viscosity, and the constant function u " ´Hp0q{α corresponds to the situation when there is no defect.

If we temporarily admit that (3.52) holds true, then the generalization to infinitely many randomized defects is immediate and yields (3.50), where we note that all the terms in p1 ´Xη k pωqqu originally present in (3.52) may be discarded because, with full probability, one at least of the X η k pωq, k P Z d , has value one and u ε p¨´ε q kq ď u by the comparison principle (using the non-positiveness of 0 ).

In order to establish (3.52), we proceed as follows. First, when X η 0 pωq " X η k pωq " 0, H ε 0 p¨, p, ωq " H ε k p¨, p, ωq " H ppq, the constant function u is the solution to (3.51) and (3.52) holds. Second, if X η 0 pωq " 1 while X η k pωq " 0, then H ε 0 p¨, p, ωq " H ´ε , p ¯and H ε k p¨, p, ωq " H ppq, thus, given the non-positiveness of 0 , max tH ε 0 p¨, p, ωq , H ε k p¨, p, ωqu " H ´ε , p and the solution is indeed u ε . We conclude "symmetrically" when X η 0 pωq " 0 and X η k pωq " 1. The only interesting case is the third one, when X η 0 pωq " X η k pωq " 1, and we wish to prove that (3.53) V ε px, ωq " min " u ε pxq , u ε px ´ε q kq * .

To address the latter case, we partition R d into three non-overlapping regions, namely

A ε 0 " Supp 0 ´ε ¯, A ε k " Supp 0 ´ε ´q k ¯" ε q k `Supp 0 ´ε ¯, and the remaining part pA ε 0 Y A ε k q c of R d . From the non-positiveness of 0 , we notice that (3.54) max tH ε 0 , H ε k u " $ & % H ε 0 in A ε 0 , H ε k in A ε k , H ε 0 " H ε k " H in pA ε 0 Y A ε k q c .
We are going to prove that for ε sufficiently small, (3.55)

" u ε pxq ď u ε px ´ε q kq in A ε 0 , u ε px ´ε q kq ď u ε pxq in A ε k .
Assume temporarily that (3.55) is true. Since u ε and u ε p¨´ε q kq are respectively viscosity solutions to (1.7) and that same equation translated from ´ε q k, and since they are both locally Lipschitz continuous, we know from [6, Prop. 1.9, Chapter I] that they are solutions almost everywhere of those equations, respectively. It follows from (3.54) and (3.55) together, that min u ε pxq , u ε px ´ε q kq ( is solution almost everywhere to (3.51). We also know that this minimum is a viscosity supersolution of the equation. Finally, since the minimum of two Lipschitz continuous functions is also Lipschitz continuous and since the Hamiltonian is convex, we know from [6, Prop. 5.1, Chapter II] that this almost everywhere, Lipschitz continuous, solution is also a viscosity subsolution. We thus conclude that it is a viscosity solution and that (3.53). Consequently, if (3.55) holds for all k P Z d zt0u, then (3.50) holds.

The inequations in (3.55) in fact only rely upon an interplay between the properties of the solution u ε for the problem with one defect (essentially that u ε pxq grows as x departs from the defect) and the distance between the two contiguous defects considered. At this point, we may assume that α " 1, just for alleviating notation.

To start with, we observe that the solution u to the Dirichlet problem (1.11)-(1.14) in the sense of Theorem 1.1, (a) is radially symmetric and (b) is a strictly increasing function of the radius |x|. The radial symmetry comes from the uniqueness of that solution stated in Theorem 1.1. The increasing character comes from our assumption (3.46). Indeed, because of this assumption and (1.12), we have up0q ď ´E ă ´Hp0q. By continuity, upxq ă ´Hp0q on a neighborhood B R p0q for some radius R ą 0. The continuity of H then implies that |Dupxq| ě δ ą 0 for some δ ą 0 and almost all x P B R p0q. By radial symmetry, this amounts to ˇˇˇB u Br ˇˇˇě δ ą 0 for almost all r ď R. But the notion of viscosity solution then implies that, on B R p0q, Bu Br , which is strictly positive around the origin (given that we have already established that u reaches its global infimum there) can change sign and become negative at most once. If this is the case, then u decreases on the region where Bu Br ď ´δ, so the inequality upxq ă ´Hp0q is all the more true. We may thus enlarge the ball B R p0q and since |Du| cannot be arbitrarily small on this region, the viscosity solution u cannot become increasing again. Therefore, u remains decreasing as |x| grows to infinity. On the other hand, we also already know from (3.43) that |upxq ´u| Ñ 0 at infinity and from the non-positiveness of 0 that u ď u, so we reach a contradiction.

In conclusion, Bu Br ě δ ą 0 everywhere outside the origin and u is indeed a strictly increasing, radially symmetric function.

We now return to u ε , for which we will apply a similar argument, using the uniformity of properties with respect to ε. To start with, we notice that, because 0 is assumed radially symmetric and H is isotropic, u ε is also radially symmetric. Next, since 0 is nonpositive, comparison yields as above that u ε ď u " ´Hp0q in R d . On the other hand, by Theorem 1.1, u ε p0q Ñ up0q ď ´E ă ´Hp0q.

Let us fix an arbitrary positive constant β such that 0 ă 2β ă E ´Hp0q. There exists a positive radius R 0 such that upxq ă ´Hp0q´2β on B R0 p0q. From the uniform convergence of u ε to u on B R0 p0q, we know that there exists ε 0 ą 0 such that for any ε such that 0 ă ε ď ε 0 , (3.56) u ε pxq ă ´Hp0q ´β, on B R0 p0q.

Step 1. Let us start by studying u in B R0 p0qzB ε p0q. Given that H ε coincides with H outside Supp 0 ´ε ¯, (3.56) implies that there exists some δ ą 0, independent of ε ď ε 0 , such that Since we know, from the notion of viscosity solution and from (3.57), that Du ε cannot be arbitrarily small on B R0 p0qzB ε p0q, Bu ε Br may change sign at most once in this ring, and if it is the case, then the jump is from positive to negative as |x| grows.

Hence, if lim rÑε´B u ε Br prq ă 0, then Bu ε Br ď ´δ in B R0 p0qzB ε p0q. This implies that we may find a larger R 0 still satisfying u ε pxq ď ´Hp0q ´β for x P B R0 p0qzB ε p0q.

Repeating the argument, we obtain that we may indeed choose R 0 " `8 and that lim |x|Ñ8 u ε pxq " ´8, which contradicts the convergence of u ε to ´Hp0q at infinity.

Therefore, lim rÑε´B u ε Br prq ą 0 and Bu ε Br may change sign at most once on B R0 p0qzB ε p0q to become negative, say at some radius R 1 ă R 0 , where (this is therefore the only possible case) it jumps: We claim that this situation cannot occur. Indeed, if it was the case, then u ε pxq ă ´Hp0q ´β for |x| " R 0 and we would be able to enlarge R 0 while keeping the inequality true. Then u ε would stay decreasing in the whole region |x| ě R 1 , which again would contradict the convergence of u ε to ´Hp0q at infinity.

To summarize, we have proven that (3.60) Bu ε Br ě δ, in the ring tx : ε ď |x| ď R 0 u.

Step 2. Let us now study u ε in B ε p0q.

Assume first that u ε has a local maximum at 0. Then, u ε p0q ď ´Hp0q ` 0 p0q, and the function u ε cannot have a minimum at some x, 0 ă |x| ă ε. Indeed, if it was the case, u ε pxq ě ´Hp0q ` 0 pxq ą ´Hp0q ` 0 p0q ě u ε p0q, which is contradictory. Therefore, the minimum of u ε in B ε p0q is achieved at |x| " ε, which is impossible because lim rÑε´B u ε Br prq ą 0. We have proved by contradiction that u ε does not have a local maximum at 0.

On the other hand, the semi-concavity and the radial symmetry of u ε imply that Buε Br has a limit at 0, which is nonpositive.

Combining the latter two points yields that u ε has a derivative at 0 and that Buε Br p0q " 0. This implies that (3.61) u ε p0q " ´Hp0q ` 0 p0q, and from the convergence of u ε p0q to up0q, that (3.62) E " Hp0q ´ 0 p0q.

We have also proved that ´E is the minimal value of u ε on B ε p0q. Note that (3.61) and (3.62) may be obtained in an easier way by using arguments from the theory of optimal control.

Step 3. Our next step consists of proving that it possible to choose a positive integer q such that We deduce from (3.64) and (3.65) that if q ą 1 `M {δ, then min qεď|x|ďR0 u ε pxq ě max |x|ďε u ε pxq. Moreover, u ε cannot reach a value smaller than max |x|ďε u ε pxq at some y such that |y| ą R 0 , because, if it was the case, then the infimum of u ε | t|x|ěR0u would be achieved by some x, R 0 ă |x|, and max |x|ďε u ε pxq ě u pxq ě ´Hp0q, which is a contradiction for ε ď ε 0 . Hence (3.63) holds for q ą 1 `M {δ and ε ď ε 0 .

It is then easy to obtain (3.55) for any ε ď ε 0 , with (3.66) q " q `2, q ą 1 `M {δ.

Finally, the explicit expression (3.50) of the solution to (3.49) is established, it is a straightforward adaptation of our results of Section 3.1.4 to indeed identify the limit as ε vanishes. All our arguments and conclusions hold mutatis mutandis with minor modifications.

Generalization. The results obtained above can be generalized to a class of nonradially symmetric situations under the following assumptions:

' The Hamiltonian H defined by Hppq " max aPA ´f paq ¨p ´ paq is a C 2 , convex, globally Lipschitz function which reaches its minimum at p " 0. This implies that Hp0q " ´min aPA paq " ´min aPA:f paq"0 paq ' The function 0 : R d ˆA Ñ R ´is smooth, and for all a P A, 0 p¨, aq is supported in the unit ball B 1 p0q Ă R d . The perturbed Hamiltonian is Hpy, pq " max aPA f paq ¨p ´ paq ´ 0 py, aq ' The ergodic constant E defined in Section 2.1 satisfies (3.46) ' There exists two Hamiltonians H 1 and H 2 which are radially symmetric, C 2 , convex, globally Lipschitz fonctions defined on R d and two smooth, radially symmetric functions 0,1 and 0,2 , defined on R d with values in R ´, supported in B 1 p0q, such that the following conditions are satisfied:

the pairs pH 1 , 0,1 q and pH 2 , 0,2 q satisfy all the assumptions made in the radially symmetric setting. In particular, as we have seen above, the related ergodic constants E 1 and E 2 are given by E 1 " H 1 p0q ´ 0,1 p0q and E 2 " H 2 p0q ´ 0,2 p0q there holds H 1 ppq ď Hppq ď H 2 ppq for all p P R d , 0,1 pxq ě 0 px, aq ě 0,2 pxq for all x P R d , a P A. Moreover, H 1 p0q " Hp0q " H 2 p0q and 0,1 p0q " 0,2 p0q. The corresponding perturbed Hamiltonians are H 1 py, pq " H 1 ppq ´ 0,1 pyq and H 2 py, pq " H 2 ppq ´ 0,2 pyq. Obviously, (3.67) H 1 py, pq ď Hpy, pq ď H 2 py, pq, and it can be checked that the assumptions imply that E 1 " E " E 2 . Using (3.67) and comparison principles, it is possible to prove that (3.52) still holds in this situation, provided that q is chosen large enough, and all the results proved in the radially symmetric setting remain valid. Note that the assumptions made are rather strong: in particular they require that a Þ Ñ 0 p0, aq be constant.

(1. 14 )

 14 α up0q `max `E, HpDφp0qq ˘ě 0. Remark 1.1. The conditions (1.11) through (1.14) may be seen as a weak formulation of a Dirichlet boundary value problem comprising the Hamilton-Jacobi equation (1.11) in the singular open set R d zt0u and the Dirichlet boundary condition up0q "

  ´max

  yPR d Hpy, 0q ď α u ε pxq ď ´min yPR d Hpy, 0q.

`

  λ´1 t ˘dt ¸for x ą 0, (actually independent of R) is the bounded C 1 solution to the equation(3.12)

e ´x ż x ´8 e t 0 ˆt ε ˙dt for x ă 0, e x ż ` 8 x e ´t 0

 80 ˆt ε ˙dt for x ą 0.

  x per y ´inf R per , |p| ´x per y for |p| ě x per y ´inf R per .

  in the branch(3.20), u per,ε pxq " h per,ε pxq `β e x . It immediately follows that (3.20) cannot extend to `8 either, since the asymptotic blow-up implied by the term e x would contradict the boundedness of the solution.

( 3 .

 3 57) |Du ε pxq| ě δ ą 0, for almost all ε ď |x| ď R 0 . Therefore, for almost all ε ď |x| ď R 0 , (3.58) ˇˇˇB u ε Br ˇˇˇě δ.

for ε ď |x| ă R 1 ,

 1 Bu ε Br ď ´δ for R 1 ă |x| ď R 0 .

( 3 .

 3 63) min|x|ěqε u ε pxq ě max |x|ďε u ε pxq. First, we know that there exists M ą 0 independent of ε such that }Du ε } 8 ď M . This and (3.61)-(3.62) imply that (3.64) max |x|ďε u ε pxq ď ´E `M ε. On the other hand, (3.60) implies (3.65) min qεď|x|ďR0 u ε pxq ě u ε p|y| " εq `δpq ´1q ě ´E `δpq ´1q .

  aPA |f px, aq| and M " sup xPR d ,aPA | px, aq|. It is easy to check that the Hamiltonian H defined in (1.1) has the following properties: H is convex with respect to its second argument, and for any x, y, p, q P R d , ) implies the coercivity of H w.r.t. its second variable uniformly in its first variable, i.e. lim |p|Ñ8 inf xPR d Hpx, pq " `8.

	(1.2)	Hpx, pq ě r f |p| ´M ,
	(1.3)	|Hpx, pq ´Hpy, pq| ď L f |p||x ´y| `ω p|x ´y|q,
	(1.4)	|Hpx, pq ´Hpx, qq| ď M f |p ´q|.
	Property (1.2

  [START_REF] Achdou | Hamilton-Jacobi equations on networks as limits of singularly perturbed problems in optimal control: dimension reduction[END_REF]. Then there exists χ p, p P CpR d q such that Dχ p, pq " Hppq in R d ,(2.40) where (2.40) is understood in the sense of viscosity.Proof. Fix χ per,p and χ per, p as in Lemma 2.5. For a radius R ą R 0 `1 that will eventually converge to `8, consider the Dirichlet problem pyq are three Lipschitz continuous viscosity subsolutions of H per py, Dvq ď Hppq in B R p0qzB R0`1 p0q. We deduce from [6, Prop. 1.9, Chapter I, page 31] that (2.45) H py, Dσpyqq ď Hppq for almost all y P B R p0qzB R0`1 p0q. Dσq ď Hppq in B R p0q.Hence, σ is a subsolution of the Dirichlet problem (2.41)-(2.42). Our second step is to establish the existence of a constant C ą 0 independent of R ą R 0 `1 such that, for all y P B R p0q,(2.47) σpyq ´min ´p ¨y `χper,p pyq, p ¨y `χper,p pyq ¯ě ´C.

	(2.39)	lim |y|Ñ8	|χ p, ppyq ´minpp ¨y, p ¨yq| |y|	" 0,
	(2.41)		H py, H `y, Dχ R p, p˘" Hppq in B R p0q,
	Combining this with (2.44), σ actually satisfies (2.45) for almost all y P B R p0q. Then,
	since H is convex with respect to its second argument, [6, Prop. 5.1, Chapter II, page
	77] can be applied and yields that σ is a viscosity subsolution of
	(2.46)	H py,

χ R p, ppyq " min ´p ¨y `χper,p pyq, p ¨y `χper,p pyq ¯on BB R p0q. (2.42) As a first step, let us construct a subsolution to (2.41)-(2.42). For a constant c ą 0 that will be chosen below, we set (2.43) σpyq " min ´wpyq ´c, p ¨y `χper,p pyq, p ¨y `χper,p pyq ¯, where w is the viscosity solution of (2.13) constructed in Section 2.1.2, which is Lipschitz continuous in R d and thus satisfies (2.13) almost everywhere in R d from [6, Prop. 1.9, Chapter I, page 31]. We next choose c such that, for any y P B R0`1 p0q, wpyq ´c ă min ´p ¨y `χper,p pyq, p ¨y `χper,p pyq ¯, thus (2.44) σpyq " wpyq ´c, for any y P B R0`1 p0q. On the other hand, since E ă Hppq, w, y Þ Ñ p ¨y `χper,p pyq and y Þ Ñ p ¨y χper,p On the other hand, it is clear that σpyq ď min ´p ¨y `χper,p pyq, p ¨y `χper,p pyq ¯for any y P BB R p0q.

  holds in this situation.where tX η k pωqu kPZ d is a set of i.i.d. real valued, Bernoulli random variables of parameter η, that is, a multi-dimensional analogue of the sequence introduced in Section 3.1.4. We consider the case when ' H is a C 2 , convex, globally Lipschitz and radially symmetric function of p (it therefore reaches its minimum at p " 0) ' 0 is smooth, non positive and radially symmetric function, supported in the unit ball centered at the origin, and such that arg min 0 " t0u and 0 p0q ă 0 ' the ergodic constant E defined in Section 2.1 satisfies (originally set at the origin) is shifted at each of the positions ε q k, and only occurs there with the probability η, encoded in the random variable X k pωq. The parameter q is a positive integer that will be adjusted in the course of the proof, see (3.66) below. We then form the random Hamiltonian

	Radially symmetric situations. We consider the Hamiltonians
	(3.44)	H py, pq " Hppq ´ 0 pyq ,
	and for each k P Z d ,
	(3.45)	H ε k px , p , ωq " Hppq ´ 0	ˆx ´ε q k ε	˙Xη k pωq,
	(3.46)		E ą Hp0q.
	We will see later that (3.46) is in fact a consequence of the assumptions made
	on 0 . The Hamiltonian H ε k corresponds to a rescaled Hamiltonian similar to H the local defect (3.47) H ε S px , p , ωq " sup kPZ d H ε k px , p , ωq,	´x ε	, p ¯, when
	which also reads as
	(3.48)	H ε S px , p , ωq " Hppq	´ÿ kPZ d	X η k pωq 0	ε ˆx ´ε q k	˙.
	This is the d-dimensional counterpart of the one-dimensional Hamiltonian considered
	in Section 3.1.4. The expression (3.48) proceeds from the definition (3.45)-(3.47)