Francois Hamel

François Hamel

Andrej Zlatoš

Speed-up of combustion fronts in shear flows

Introduction and main results

In this paper we analyze the asymptotic limit of traveling fronts and of their speeds for reaction-diffusion equations in the presence of strong shear flows. More precisely, we consider the model

v t + A α(y) v x = ∆v + f (v), t ∈ R, (x, y) ∈ R × R N -1 (1.1)
when A ≫ 1. The physical dimension N ≥ 1 is arbitrary. The reaction function f : [0, 1] → R is assumed to be of class C 1,δ ([0, 1]) for some δ > 0 and to satisfy ∃ θ ∈ [0, 1) : f = 0 on [0, θ] ∪ {1}, f > 0 on (θ, 1), f is non-increasing near 1. (1.

2)

The function α : R N -1 → R is assumed to be of class C 1,δ (R N -1 ) and (1, . . . , 1)-periodic (for the sake of simplicity; other periods are handled identically) so that α ∈ C 1,δ (T N -1 ). The coefficient A α(y) is then the x-component of an incompressible shear flow A q(x, y) with amplitude A and flow profile q(x, y) = (α(y), 0, . . . , 0).

We are interested in the asymptotic strong-flow regime A → +∞. Equation (1.1) arises in models of flame propagation, especially when 0 < θ < 1, and the quantity 0 ≤ v ≤ 1 then stands for normalized temperature (see [START_REF] Ya | Stabilization of solution of the Cauchy problem for equations encountred in combustion theory[END_REF]). The real number θ is then the ignition temperature, below which the reaction is suppressed. When θ = 0, this equation also arises in chemical, biological, and ecological models, and v typically stands for the density of a substance or a species (see [START_REF] Fife | Mathematical aspects of reacting and diffusing systems[END_REF][START_REF] Murray | Mathematical Biology[END_REF][START_REF] Shigesada | Biological Invasions: Theory and Practice, Oxford Series in Ecology and Evolution[END_REF]).

We will be interested in traveling fronts for (1.1), that is, solutions of the form v(t, x, y) = u(x -ct, y).

Here the front profile u is (1, . . . , 1)-periodic in y ∈ R N -1 and c ∈ R is the front speed. These solutions, which are invariant in the frame moving with speed c along the x-axis, play an essential role in the study of large-time behavior of the processes modeled by (1.1). If 0 < θ < 1, then it is known [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF][START_REF] Xin | Existence and nonexistence of travelling waves and reaction-diffusion front propagation in periodic media[END_REF] that for each amplitude A ∈ R there exists a unique speed c = c * (Aq, f ) and a unique (up to shifts in x) profile u : R × T N -1 → (0, 1), which connects 0 to 1. That is u is a classical solution of the elliptic equation

∆u + (c -A α(y)) u x + f (u) = 0 in R × T N -1 , 0 ≡ u(+∞, •) < u < u(-∞, •) ≡ 1 in R × T N -1 , (1.3)
where the above limits 0 and 1 are uniform in T N -1 . Moreover, u is decreasing in x.

On the other hand, if θ = 0, then there is a minimal speed c * (Aq, f ) such that problem (1.3) has a solution u with speed c if and only if c ≥ c * (Aq, f ). Moreover, for each such c there is a decreasing-in-x solution, and all solutions are such if f ′ (0) > 0 [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF].

The question of existence of traveling fronts for shear flows was first considered in [START_REF] Berestycki | Multidimensional traveling-wave solutions of a flame propagation model[END_REF][START_REF] Berestycki | Traveling fronts in cylinders[END_REF] in infinite cylinders with bounded cross sections and Neumann boundary conditions, and the case with periodic boundary conditions in y can be treated similarly. Since then, a considerable amount of research has been directed at the problem, in particular, first results on stability of fronts and long-time convergence of large classes of solutions to them appeared in [START_REF] Aronson | Multidimensional nonlinear diffusions arising in population genetics[END_REF][START_REF] Berestycki | Stability of traveling fronts in a curved flame model, Part I: Linear analysis[END_REF][START_REF] Ya | Stabilization of solution of the Cauchy problem for equations encountred in combustion theory[END_REF][START_REF] Roquejoffre | Stability of traveling fronts in a curved flame model, Part II: Non-linear orbital stability[END_REF][START_REF] Roquejoffre | Eventual monotonicity and convergence to traveling fronts for the solutions of parabolic equations in cylinders[END_REF]. Existence of fronts in the case of general, not necessarily shear, flows which are periodic in all spatial variables (on periodic domains in R N ) has also been studied. These pulsating fronts are of the type v(t, z) = U(z • e -ct, z), with z = (x, y), e ∈ R N a unit vector, and U periodic in the last N arguments z = (z 1 , . . . , z N ) ∈ R N , and their existence was first proved in [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF][START_REF] Xin | Existence of planar flame fronts in convective-diffusive periodic media[END_REF]. We also refer to [START_REF] Smaily | Min-max formulae for the speeds of pulsating traveling fronts in periodic excitable media[END_REF][START_REF] Hamel | Formules min-max pour les vitesses d'ondes progressives multidimensionnelles[END_REF][START_REF] Heinze | Variational principles for propagation speeds in inhomogeneous media[END_REF] for variational min-max type formulas for the unique and minimal propagation speeds of traveling and pulsating fronts, and to [START_REF] Berestycki | Reaction-Diffusion Equations and Propagation Phenomena[END_REF][START_REF] Xin | Analysis and modeling of front propagation in heterogeneous media[END_REF] for many further references on propagation phenomena for reaction-diffusion equations.

The main question addressed in this paper is related to the behavior of the traveling fronts u of (1.3) and the propagation speeds c * (Aq, f ) in strong shear flows, that is as the amplitude A goes to +∞. We especially aim at quantifying the speed-up induced by the underlying strong advection. Roughly speaking, in combustion models, a stronger advection is going to increase the width of the reaction zone, and hence enhance the propagation speed. It is clear that if α ≤ β, then, for each A ≥ 0, the corresponding unique or minimal speeds associated with α and β are ordered, that is c * (Aq α , f ) ≤ c * (Aq β , f ), where q α (x, y) = (α(y), 0, . . . , 0) and q β (x, y) = (β(y), 0, . . . , 0), see [START_REF] Berestycki | Reaction-Diffusion Equations and Propagation Phenomena[END_REF][START_REF] Berestycki | Traveling fronts in cylinders[END_REF]. However, in general the speeds for flows A α(y) and A ′ α(y) cannot be compared if A = A ′ . Therefore, the average effect of the amplitude A and the limiting behavior as A → +∞ are a priori not clear. Nevertheless, the comparison principle immediately shows lim sup

A→+∞ c * (Aq, f ) A ≤ max T N-1 α (1.4)
(see [START_REF] Berestycki | Reaction-Diffusion Equations and Propagation Phenomena[END_REF][START_REF] Berestycki | Traveling fronts in cylinders[END_REF]). Furthermore, it also follows from [START_REF] Kiselev | Enhancement of the traveling front speeds in reaction-diffusion equations with advection[END_REF] that

lim inf A→+∞ c * (Aq, f ) A ≥ T N-1 α(y)dy, (1.5)
with equality only if α is constant. (In [START_REF] Kiselev | Enhancement of the traveling front speeds in reaction-diffusion equations with advection[END_REF] only mean-zero α were considered, with the above bound being 0, but the general case follows from c * (Aq, f

) = c * (A(q -p), f ) + Aβ if p(x, y) = (β, 0, • • • , 0).
) That is, the front speed in the presence of a strong shear flow is between two linear functions of the amplitude of the flow. The main result of this paper shows that this speed-up is truly asymptotically linear as A → +∞, that is, lim A→+∞ c * (Aq, f )/A exists. Furthermore, we characterize this limit, in both cases 0 < θ < 1 and θ = 0, in terms of solutions to a second-order degenerate equation.

We start with the positive ignition temperature case 0 < θ < 1. The existence of the limit lim A→+∞ c * (Aq, f )/A in this case was stated as an open problem in [START_REF] Berestycki | The influence of advection on the propagation of fronts in reaction-diffusion equations[END_REF], where it was proved for KPP reactions (see below).

Theorem 1.1 Assume (1.2) with 0 < θ < 1. Then there exists γ * (q, f ) ≥ T N-1 α(y)dy (with equality only if α is constant) such that the unique speeds c * (Aq, f ) of the problem (1.3) satisfy lim A→+∞ c * (Aq, f ) A = γ * (q, f ). (1.6) Furthermore, there exists U ∈ L ∞ (R×T N -1 ) such that ∇ y U ∈ L 2 (R×T N -1 )∩L ∞ (R×T N -1 ) and U is a (distributional) solution of          ∆ y U + (γ -α(y)) U x + f (U) = 0 in D ′ (R × T N -1 ), 0 ≤ U ≤ 1 a.e. in R × T N -1 , lim x→+∞ U(x, y) ≡ 0 uniformly in T N -1 , lim x→-∞ U(x, y) ≡ 1 uniformly in T N -1 , (1.7)
with γ = γ * (q, f ), and each sequence A n → ∞ has a subsequence along which the functions U An (x, y) = u An (A n x, y) (with u An the unique solution of (1.3) for A = A n , translated in x so that max y∈T N-1 u An (0, y) = θ) converge a.e. to a solution of (1.7) with γ = γ * (q, f ) and U non-increasing in x. Finally, if there is r > 0 such that

α ∈ C ∞ (T N -1 ) and 1≤|ζ|≤r |D ζ α(y)| > 0 for all y ∈ T N -1 , 1 (1.8
)

then the pair (γ, U) solving (1.7) is unique up to shifts of U in x, with U ∈ C 1,δ (R × T N -1 ) satisfying 0 < U < 1 and U x < 0 on R × T N -1 .
Remark 1.2 This result continues to hold when the ignition reaction f is only Lipschitz and α ∈ C δ (T N -1 ) for some δ > 0. Indeed, our proof does not use the extra smoothness, and [START_REF] Zlatoš | Generalized traveling waves in disordered media: Existence, uniqueness, and stability[END_REF]Theorem 1.6] shows that there still exists a unique solution u to (1.3), which is decreasing in x due to being the limit of solutions with the same property and due to the strong maximum principle applied to u x .

1 For ζ = (ζ 1 , . . . , ζ N -1 ) ∈ N N -1 , we let |ζ| = ζ 1 + • • • + ζ N -1 and D ζ α(y) = ∂ |ζ| α ∂y ζ 1 1 •••∂y ζ N -1 N -1 (y).
Remark 1.3 The condition (1.8) ensures that the differential operator L = ∆ y +(γ-α(y))∂ x from (1.7) satisfies Hörmander's hypoellipticity condition for any γ ∈ R, namely that the smooth vector fields ∂ y 1 , . . . , ∂ y N-1 , (γ -α(y))∂ x and their commutators up to order r + 1 span all of R N at any point (x, y). Because of this, we will be able to apply the results from [START_REF] Rothschild | Hypoelliptic differential operators and nilpotent groups[END_REF] to questions of regularity of solutions of (1.7).

Remark 1.4

It remains an open question whether uniqueness of the pair (γ, U) solving (1.7) as well as 0 < U < 1 and U x < 0 hold for all non-constant α. Note that any y-independent not necessarily continuous U, taking values in [0, θ] ∪ {1} and having the prescribed limits, along with γ = α, is a solution for constant α.

On the other hand, for zero ignition temperature θ = 0 the following conclusion holds. We define for θ ′ ∈ (0, 1/4] the "cut-off" ignition reaction f θ ′ (u) := f (u)χ(u/θ ′ ), where χ : R → [0, 1] is a smooth non-decreasing function with

χ(v) = 0 if v ≤ 1 and χ(v) = 1 if v ≥ 2.
Theorem 1.5 Assume (1.2) with θ = 0 and also f ∈ C 1,1 ([0, 1]). Then the limit

γ * (q, f ) := lim θ ′ →0 γ * (q, f θ ′ )
exists (with γ * (q, f θ ′ ) from Theorem 1.1) and (1.6) holds for the minimal speeds c * (Aq, f ) of problem (1.3). Furthermore, for any γ ≥ γ * (q, f ), there exists a solution U of (1.7) such that

∇ y U ∈ L 2 (R × T N -1 ) ∩ L ∞ (R × T N -1
) and U is non-increasing in x. Lastly, if (1.8) holds, then all solutions U of (1.7) are classical and satisfy γ ≥ γ * (q, f ).

When θ = 0 in (1.2) and the nonlinearity f is of the Kolmogorov-Petrovsky-Piskunov (KPP) [START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF] type, that is f (u) ≤ f ′ (0)u for all u ∈ [0, 1], then the limit γ * (q, f ) given in Theorem 1.5 can be expressed as

γ * (q, f ) = max w∈H 1 (T N-1 )\{0} ∇w 2 L 2 (T N-1 ) ≤f ′ (0) w 2 L 2 (T N-1 ) T N-1 α(y) w(y) 2 dy T N-1
w(y) 2 dy (1.9) (see [START_REF] Berestycki | The influence of advection on the propagation of fronts in reaction-diffusion equations[END_REF][START_REF] Heinze | Large convection limits for KPP fronts[END_REF][START_REF] Zlatoš | Sharp asymptotics for KPP pulsating front speed-up and diffusion enhancement by flows[END_REF]). Using this, one can conclude [START_REF] Heinze | Large convection limits for KPP fronts[END_REF][START_REF] Zlatoš | Sharp asymptotics for KPP pulsating front speed-up and diffusion enhancement by flows[END_REF] lim

M →+∞ lim A→+∞ c * (Aq, Mf ) A = max y∈T N-1 α(y)
and lim

M →0 + 1 √ M lim A→+∞ c * (Aq, Mf ) A - T N-1 α(y)dy = 2 f ′ (0) sup w∈H 1 (T N-1 )\{0} T N-1 w(y)dy=0 T N-1 α(y)w(y)dy ∇w L 2 (T N-1
) .

The first of these limits does not depend on f and so it holds for any f with f ′ (0) > 0. Indeed, then we can find two KPP reactions g, h such that g ≤ f ≤ h, and the claim follows from the fact that the limits for g, h are equal and from the well known relation c * (Aq, g) ≤ c * (Aq, f ) ≤ c * (Aq, h) [START_REF] Berestycki | Traveling fronts in cylinders[END_REF]. Similar results hold for more general heterogeneous equations with periodic coefficients and KPP reactions. We refer to [START_REF] Bages | How travelling waves attract the solutions of KPP equations[END_REF][START_REF] Berestycki | The speed of propagation for KPP type problems. I -Periodic framework[END_REF][START_REF] Berestycki | Analysis of the periodically fragmented environment model: II -Biological invasions and pulsating traveling fronts[END_REF][START_REF] Hamel | Qualitative properties of KPP and monostable fronts: monotonicity and exponential decay[END_REF][START_REF] Hamel | Uniqueness and stability properties of monostable pulsating fronts[END_REF][START_REF] Mallordy | A parabolic equation of the KPP type in higher dimensions[END_REF][START_REF] Weinberger | On spreading speeds and traveling waves for growth and migration in periodic habitat[END_REF] for further results on existence and qualitative properties of KPP traveling and pulsating fronts and to [START_REF] Audoly | Réaction-diffusion en écoulement stationnaire rapide[END_REF][START_REF] Berestycki | The influence of advection on the propagation of fronts in reaction-diffusion equations[END_REF][START_REF] Berestycki | Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena[END_REF][START_REF] Smaily | Pulsating traveling fronts: Asymptotics and homogenization regimes[END_REF][START_REF] Smaily | The speed of propagation for KPP reaction-diffusion equations within large drift[END_REF][START_REF] Ryzhik | KPP pulsating front speed-up by flows[END_REF][START_REF] Zlatoš | Sharp asymptotics for KPP pulsating front speed-up and diffusion enhancement by flows[END_REF] for results on the asymptotics of the minimal speeds in strong shear and periodic flows in the KPP case.

We also note that for cellular flows in two spatial dimensions, the speed-up is known to be O(A 1/4 ) rather than linear (see [START_REF] Audoly | Réaction-diffusion en écoulement stationnaire rapide[END_REF][START_REF] Constantin | Bulk burning rate in passive-reactive diffusion[END_REF][START_REF] Kiselev | Enhancement of the traveling front speeds in reaction-diffusion equations with advection[END_REF][START_REF] Novikov | Boundary layers and KPP fronts in a cellular flow[END_REF][START_REF] Ryzhik | KPP pulsating front speed-up by flows[END_REF][START_REF] Zlatoš | Reaction-diffusion front speed enhancement by flows[END_REF]), and characterization of those two-dimensional periodic q for which c * (Aq, f ) is unbounded in A (i.e., speed-up occurs) was obtained in [START_REF] Ryzhik | KPP pulsating front speed-up by flows[END_REF][START_REF] Zlatoš | Reaction-diffusion front speed enhancement by flows[END_REF]. In fact, these two-dimensional results hold for general reactions [START_REF] Zlatoš | Reaction-diffusion front speed enhancement by flows[END_REF].

Remark 1.6 In the case of KPP reactions and shear flows with T N-1 α(y)dy ≥ 0, it is also known that c * (Aq, f ) is non-decreasing with respect to A ≥ 0 and c * (Aq, f )/A is nonincreasing with respect to A ≥ 0, see [START_REF] Berestycki | The influence of advection on the propagation of fronts in reaction-diffusion equations[END_REF][START_REF] Berestycki | Reaction-Diffusion Equations and Propagation Phenomena[END_REF]. However, the monotonicity of the quantities c * (Aq, f ) and c * (Aq, f )/A for general non-KPP reactions f when either θ = 0 or 0 < θ < 1, is still not known. These questions are much more intricate than in the KPP case, due to the lack of explicit formulas for the propagation speeds. However, Theorems 1.1 and 1.5 show that the limit γ * (q, f ) of the normalized speeds c * (Aq, f )/A can still be identified as the unique or minimal speed of fronts for a degenerate elliptic problem.

We thank Tom Kurtz and Daniel Stroock for useful discussions and pointers to references. FH is indebted to the Alexander von Humboldt Foundation for its support. His work was also supported by the French Agence Nationale de la Recherche through the project PREFERED. AZ was supported in part by NSF grants DMS-1113017 and DMS-1056327, and by an Alfred P. Sloan Research Fellowship. Part of this work was carried out during visits by FH to the Departments of Mathematics of the Universities of Chicago and Wisconsin and by AZ to the Faculté des Sciences et Techniques, Aix-Marseille Université, the hospitality of which is gratefuly acknowledged.

2 Existence of solutions (γ, U ) of (1.7) for θ ∈ [0, [START_REF] Aronson | Multidimensional nonlinear diffusions arising in population genetics[END_REF] In this section, we prove the existence of solutions (γ, U) of (1.7) as claimed in Theorems 1.1 and 1.5. Such solutions are obtained as the A → +∞ limit of the solutions (c A , u A ) of (1.3), after a suitable normalization and a scaling in x. The scaling, given in (2.12) below, makes the first-order coefficient bounded but it makes the equation degenerate as A → +∞. The degeneracy of the limiting equation makes the analysis of the properties of the limiting solutions more complicated than in the usual regular case.

We will assume in this section that

T N-1 α(y)dy = 0, (2.10)
which we can do without loss of generality because if p(x, y) = (β, 0, • • • , 0), then c * (Aq, f ) = c * (A(q -p), f ) + Aβ and the two u A are identical. Moreover, we will also assume that α is not identically 0. Otherwise, c * (Aq, f ) = c * (0, f ) does not depend on A and the existence results in both theorems are immediate with γ * (q, f ) = 0 and U(x, y) = χ (-∞,0) (x) (for γ = 0) or U(x, y) = V (x) such that V x = -f (V )/γ, V (+∞) = 0, and V (-∞) = 1 (for γ > 0 in Theorem 1.5), which obviously exists. Set

γ * A = c * (Aq, f ) A > 0
for each A ≥ 1, where c * (Aq, f ) is the unique (resp. minimal) speed of traveling fronts for (1.3) when 0 < θ < 1 (resp. θ = 0). Notice first that, because of (1.4) and (1.5), and since the speeds c * (Aq, f ) are positive and continuous in A (see [START_REF] Berestycki | Reaction-Diffusion Equations and Propagation Phenomena[END_REF][START_REF] Berestycki | Traveling fronts in cylinders[END_REF]), it follows that

0 < γ := inf A≥1 γ * A ≤ sup A≥1 γ * A =: γ < +∞. (2.11)
Now, let (c A , u A ) A≥1 be a family of solutions of (1.3), and define

γ A = c A A .
We will only consider solutions with u A,x < 0 (which is always possible from [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF]) and γ A uniformly bounded in A ≥ 1. Notice that by (2.11), the family (γ A ) A≥1 is automatically bounded from below by a positive constant, and it is also bounded from above when 0 < θ < 1. For each A ≥ 1, we set

U A (x, y) = u A (Ax, y) for all (x, y) ∈ R × T N -1 . (2.12)
The functions

U A satisfy          ∆ y U A + A -2 U A,xx + (γ A -α(y)) U A,x + f (U A ) = 0 in R × T N -1 , 0 < U A < 1 in R × T N -1 , U A,x < 0 in R × T N -1 , U A (+∞, •) ≡ 0, U A (-∞, •) ≡ 1 uniformly in T N -1 . (2.13) By standard elliptic estimates, each function U A is of class C 3,δ (R × T N -1 ).
We first derive some uniform integral and pointwise gradient estimates for the functions U A . [START_REF] Berestycki | On the method of moving planes and the sliding method[END_REF]) over (-M, M) × T N -1 and pass to the limit as M → +∞. Since U A,x (±∞, •) = 0 from standard elliptic estimates, since α has zero average and since f ≥ 0 in [0, 1], we get that f (U A ) ∈ L 1 (R × T N -1 ) and

Lemma 2.1 For any family

(c A , u A ) A≥1 of solutions of (1.3) such that u A,x < 0 and (c A /A) A≥1 is bounded, the functions U A defined by (2.12) satisfy sup A≥1 U A,x L 1 (R×T N-1 ) + ∇ y U A L 2 (R×T N-1 ) + ∇ y U A L ∞ (R×T N-1 ) < +∞. Proof. First, since U A,x < 0 in R × T N -1 and U A (+∞, •) = 0, U A (-∞, •) = 1, it follows immediately that U A,x L 1 (R×T N-1 ) = 1. Next integrate (2.
R×T N-1 f (U A (x, y)) dx dy = γ A
for each A ≥ 1. Now multiply (2.13) by U A , integrate over (-M, M) × T N -1 and pass to the limit as M → +∞. It follows that

R×T N-1 |∇ y U A | 2 + A -2 U 2 A,x dx dy = R×T N-1 f (U A (x, y)) U A (x, y) dx dy - γ A 2 ≤ γ A 2 .
Therefore, the assumption of boundedness of the family (γ A ) A≥1 yields the boundedness of the family (

∇ y U A L 2 (R×T N-1 ) ) A≥1 . Lastly, since U A ∈ C 3 (R × T N -1
) and the family (γ A ) A≥1 is assumed to be bounded, it then follows from [START_REF] Berestycki | Gradient estimates for elliptic regularizations of semilinear parabolic and degenerate elliptic equations[END_REF] that sup

A≥1 ∇ y U A L ∞ (R×T N-1 ) < +∞,
which completes the proof of Lemma 2.1.

We next prove the claims in the second sentence of Theorem 1.1 when 0 < θ < 1 with some γ > 0 which is the limit of a subsequence of c * (A n q, f )/A n . The claim as stated will be established when later we prove (1.6). Proposition 2.2 Assume 0 < θ < 1. Then any A n → +∞ has a subsequence along which the (unique) solution U An of (2.13), normalized as in Theorem 1.1, converges a.e. to U and γ An converges to γ, with (γ, U) a solution of (1.7) such that 0 < γ ≤ max T N-1 α and

∇ y U ∈ L 2 (R × T N -1 ) ∩ L ∞ (R × T N -1 ). Proof.
Step 1: general properties of the limit function U. The assumptions of Lemma 2.1 are satisfied since 0 < θ < 1. Because of (1.4) and (2.11), there is a subsequence of A n (which we again call A n ) and a real number γ such that 0 < γ ≤ max

T N-1
α and γ An → γ as n → +∞.

(2.14)

On the other hand, 0

< U An < 1 in R × T N -1 and the sequence (U An ) n∈N is bounded in W 1,1 loc (R × T N -1
) by Lemma 2.1. Up to extraction of another subsequence, there exists then a function U ∈ L ∞ (R×T N -1 ) such that U An → U almost everywhere in R×T N -1 as n → +∞. Furthermore, 0 ≤ U ≤ 1 and

U(• + h, •) ≤ U a.e. in R × T N -1 (2.15) for all h ≥ 0. Similarly, ∇ y U ∈ L 2 (R × T N -1 ) ∩ L ∞ (R × T N -1
) by Lemma 2.1. The equation

∆ y U + (γ -α(y)) U x + f (U) = 0 (2.16)
also holds in the sense of distributions in R × T N -1 , by passing to the limit as n → +∞ in (2.13), with A = A n .

Step 2: the limit function U has constant limits as x → ±∞. Take any non-decreasing sequence (x n ) n∈N converging to +∞. The sequences of functions (U ± n ) n∈N defined by

U ± n (x, y) = U(x ± x n , y) for (x, y) ∈ R × T N -1
are monotone and bounded in L ∞ (R×T N -1 ). They converge almost everywhere in R×T N -1 and in L p ((-M, M) × T N -1 ), for all M > 0 and 1 ≤ p < +∞, to two functions

U ± ∞ such that 0 ≤ U ± ∞ ≤ 1 a.e. in R × T N -1 . Furthermore, U ± ∞ (x + h, y) = U ± ∞ (x, y) a.e. in R × T N -1 for all h ∈ R,
due to (2.15) and x n → +∞, and

∇ y U ± ∞ = 0 since ∇ y U ∈ L 2 (R × T N -1
). In other words, the functions U ± ∞ are constant. Since the equation (2.16) is satisfied by each U ± n and thus by the limiting (constant) functions U ± ∞ in the sense of distributions, one concludes that

f (U ± ∞ ) = 0.
(2.17)

By monotonicity in x, the limits U ± ∞ do not depend on the sequence (x n ) n∈N . Therefore, the limits U(±∞, •) exist and are two constants belonging to the set of zeros of f in [0, 1].

Step 3: the limit function U has uniform limits 0 and 1 as x → ±∞. The last thing to do is to prove that U(+∞, •) = 0 and U(-∞, •) = 1 uniformly in y ∈ T N -1 for some limit function U. Recall that the functions U An are translated in such a way that max

y∈T N-1 U An (0, y) = θ. (2.18)
Hence, for all n ∈ N, 0 < U An ≤ θ in [0, +∞) × T N -1 and

∆ y U An + A -2 n U An,xx + (γ An -α(y)) U An,x = 0 in [0, +∞) × T N -1 . (2.19)
On the other hand, for each λ ∈ R, let µ n (λ) denote the principal eigenvalue of the operator ∆ y + A -2 n λ 2 -λ (γ An -α(y)) in T N -1 , that is µ n (λ) is associated with positive eigenfunctions of this operator. Since each function µ n is convex and satisfies µ n (0) = 0, µ ′ n (0) = -γ An < 0 (because α has zero average over T N -1 ) and µ n (λ) ∼ A -2 n λ 2 as λ → +∞, there is a unique λ n > 0 such that µ n (λ n ) = 0. Let ϕ n be an eigenfunction associated with λ = λ n , that is

∆ y ϕ n + A -2 n λ 2 n ϕ n -λ n (γ An -α(y)) ϕ n = 0 in T N -1 , ϕ n > 0 in T N -1 .
Then e -λnx ϕ n (y) solves (2.19). Up to normalization, one can assume that min

T N-1 ϕ n = θ (2.20)
and it then follows from the maximum principle in half-cylinders as in [START_REF] Vega | On the uniqueness of multidimensional traveling fronts of some semilinear equations[END_REF] (see also [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF]) that

U An (x, y) ≤ e -λnx ϕ n (y) for all (x, y) ∈ [0, +∞) × T N -1 .
Furthermore, for each n ∈ N, because of (2.11) and A n ≥ 1, there holds

µ n (λ) ≤ µ(λ) for all λ ≥ 0,
where µ(λ) denotes the principal eigenvalue of the operator

∆ y + λ 2 -λ (γ -α(y)) in T N -1 .
As above, there is a unique λ > 0 such that µ(λ) = 0, and µ(λ) < 0 for all λ ∈ (0, λ). As a consequence, λ n ≥ λ > 0 for all n ∈ N, whence

U An (x, y) ≤ e -λx ϕ n (y) for all (x, y) ∈ [0, +∞) × T N -1 . (2.21) Let y n ∈ T N -1 be such that ϕ n (y n ) = θ.
Up to extraction of a subsequence, one can assume that y n → y ∈ T N -1 as n → +∞. Pick any ε ∈ (0, 2θ) and define

x = -λ -1 ln ε 2θ ≥ 0.
Because of (2.21), there holds U An (x, y n ) ≤ ε/2 for all n ∈ N. But since the sequence ( ∇ y U An L ∞ (R×T N-1 ) ) n∈N is bounded by Lemma 2.1, and since y n → y as n → +∞, there are r > 0 and n 0 ∈ N such that U An (x, y) ≤ ε for all |y -y| ≤ r and n ≥ n 0 .

By monotonicity in x, it follows that U An (x, y) ≤ ε for all x ≥ x, |y -y| ≤ r and n ≥ n 0 . By passing to a limit as n → +∞ as in step 1 (up to extraction of a sequence), the same inequality holds for the limit function U for almost every (x, y) such that x ≥ x and |y -y| ≤ r. Because of (2.15) and step 2, the constant limit U(+∞, •) of U as x → +∞ is then such that 0 ≤ U(+∞, •) ≤ ε.

Since ε > 0 was arbitrarily small, one concludes that U(+∞, •) = 0.

Let us now prove that U(-∞, •) = 1. First, because of (2.18), there is a sequence (y

′ n ) n∈N in T N -1 such that U An (0, y ′ n ) = θ.
Up to extraction of another subsequence, one can assume that y ′ n → y ∈ T N -1 as n → +∞. As in the previous paragraph, given any ε ∈ (0, θ), it follows from Lemma 2.1 that there are r ′ > 0 and n ′ 0 ∈ N such that

U An (0, y) ≥ θ -ε for all |y -y| ≤ r ′ and n ≥ n ′ 0 ,
whence U An (x, y) ≥ θ -ε for all x ≤ 0, |y -y| ≤ r ′ and n ≥ n ′ 0 . Thus U(x, y) ≥ θ -ε for a.e. such (x, y) and the constant function U(-∞, •) satisfies U(-∞, •) ≥ θ -ε for all ε ∈ (0, θ). Therefore, U(-∞, •) ≥ θ. Because of (1.2) and (2.17), the constant U(-∞, •) is either θ or 1.

Assume now that U(-∞, •) = θ. The monotonicity property (2.15) yields 0 ≤ U ≤ θ a.e. in R × T N -1 , whence

∆ y U + (γ -α(y)) U x = 0 in D ′ (R × T N -1 ).
(2.22)

Let φ : [0, 1] → R be a non-increasing C ∞ function such that φ = 1 on [0, 1/3] and φ = 0 on [2/3, 1]. For each k ∈ N, let φ k be the function defined in R × T N -1 by

φ k (x, y) =      0 if |x| ≥ k + 1, 1 if |x| ≤ k, φ(|x| -k) if k < |x| < k + 1.
By testing (2.22) against φ k , one gets that

(0,1)×T N-1 (γ -α(y)) U(x+k, y) φ ′ (x) dx dy - (0,1)×T N-1 (γ -α(y)) U(x-k -1, y) φ ′ (1-x) dx dy = 0.
Since U(+∞, •) = 0, U(-∞, •) = θ and the function α has zero average over T N -1 , it follows from Lebesgue's dominated convergence theorem, by passing to the limit as k → +∞, that θ γ = 0. But θ > 0 and γ > 0 by (2.14), a contradiction. So U(-∞, •) = 1 and the uniformity of the limit in y (as well as of U(+∞, •) = 0) follows from Lemma 2.1 and U being non-increasing in x. The proof of Proposition 2.2 is complete.

To complete this section, we prove the existence of solutions (γ, U) of (1.7) when θ = 0, for all

γ ≥ γ * ∞ = lim inf A→+∞ γ * A .
Notice also that 0 < γ * ∞ ≤ max

T N-1 α (2.23)
by (1.4) and (2.11).

Proposition 2.3 Assume that θ = 0. Then, for all γ ≥ γ * ∞ , there exists a solution U of (1.7)

such that ∇ y U ∈ L 2 (R × T N -1 ) ∩ L ∞ (R × T N -1 ).
Proof. Fix a real number γ ∈ [γ * ∞ , +∞). Since γ * A is the minimal speed for the solutions of (2.13) for each A > 0, there exist sequences (A n ) n∈N and (γ n ) n∈N of positive real numbers such that

A n → +∞ and γ n → γ as n → +∞, and γ n ≥ γ * An for all n ∈ N. For each n ∈ N, let U An be a solution of (2.13) with A = A n and speed γ n (so hypotheses of Lemma 2.1 are satisfied). Up to a shift in x, one can assume

(0,1)×T N-1 U An (x, y) dx dy = 1 2 . ( 2

.24)

From Lemma 2.1, it follows as in step 1 of the proof of Proposition 2.2 that, up to extraction of a subsequence, the functions U An converge a.e. in R ×T Because of (2.15), one concludes that 0 ≤ U(+∞, •) ≤ 1/2 ≤ U(-∞, •) ≤ 1. Since 0 and 1 are the only zeros of f in [0, 1] we have

N -1 to a function U ∈ L ∞ (R ×T N -1 ) such that 0 ≤ U ≤ 1 a.e. in R × T N -1 and ∇ y U ∈ L 2 (R × T N -1 ) ∩ L ∞ (R × T N -1
U(+∞, •) = 0 and U(-∞, •) = 1,
which completes the proof of Proposition 2.3.

Proof of Theorem 1.1

This section is devoted to the completion of the proof of Theorem 1.1. The existence of a pair (γ, U) solving (1.7) has already been proven in the previous section. First, in Subsection 3.1, we prove its uniqueness, up to shifts in x for U, when the additional hypothesis (1.8) is satisfied. The proof relies on the regularity theory [START_REF] Rothschild | Hypoelliptic differential operators and nilpotent groups[END_REF] and a strong maximum principle [START_REF] Stroock | On the support of diffusion processes with applications to the strong maximum principle[END_REF] for degenerate elliptic and parabolic equations. From this uniqueness property and the Lipschitz continuity of the unique speeds c * (Aq, f ) with respect to the flow Aq, we deduce in Subsection 3.2 the existence of lim A→+∞ c * (Aq, f )/A with or without (1.8).

3.1 Uniqueness of solutions of (1.7) given (1.8) and 0 < θ < 1

Let (γ, U) and (γ ′ , U ′ ) be two solutions of (1.7). We will prove that γ = γ ′ , and U = U ′ after a shift in x, under the additional hypothesis (1.8). First, because of (1.8), it follows from [35, Theorem 18(c)] with X j = ∂ y j (j = 1, . . . , N-1) and X 0 = (γ -α(y))∂ x (resp. X 0 = (γ ′ -α(y))∂ x ) that U and U ′ are actually continuous. In fact, U and U ′ are slightly smoother than f , at least of class C 1,δ (R × T N -1 )), by repeated application of [START_REF] Rothschild | Hypoelliptic differential operators and nilpotent groups[END_REF]Theorem 18(b)].

Next, [START_REF] Rothschild | Hypoelliptic differential operators and nilpotent groups[END_REF]Theorem 18(d)] and the fact that f (U) ∈ L p loc (R × T N -1 ) (and so f (U) is in the space S p 0 (M) from [START_REF] Rothschild | Hypoelliptic differential operators and nilpotent groups[END_REF], for any bounded open

M ⊆ R × T N -1 ) yield U ∈ S p 2 (M) for all p ∈ (1, ∞). This means that ∆ y U, (γ -α(y))U x ∈ L p loc (R × T N -1 ) for any p ∈ (1, ∞), so (1.7) holds in L p loc (R × T N -1 ) for p ∈ (1, ∞). Thus LU ≤ 0 and LU ′ ≤ 0 in L p loc (R × T N -1 ), where L = ∆ y + (γ -α(y))∂ x .
We will now show that this means that the strong maximum principle from [START_REF] Stroock | On the support of diffusion processes with applications to the strong maximum principle[END_REF] applies to -U and -U ′ . In order to show they satisfy the hypotheses in [START_REF] Stroock | On the support of diffusion processes with applications to the strong maximum principle[END_REF], we will need to approximate them by smooth functions. Let η : R → R be a smooth non-negative function supported in [-1, 1], with

η ε,κ (x, y) :=        ε -1 κ 1-N η x ε N -1 j=1 η y j κ κ > 0, ε -1 η x ε δ 0 (y) κ = 0,
where δ 0 is the delta function at 0 ∈ R N -1 . Extend U to a periodic-in-y function on R N and let U ε,κ := U * η ε,κ .

We have

|U ε,κ x | ≤ 10ε -1 since |U| ≤ 1. So if ω ∈ C([0, +∞)) with ω(0) = 0 is such that |α(y) -α(y ′ )| ≤ ω(κ) whenever |y -y ′ | ≤ √ N -1 κ, then for κ > 0 we have LU ε,κ = (LU) * η ε,κ + R N-1 [α(y -ξ) -α(y)]U ε,0 x (x, y -ξ)κ 1-N N -1 j=1 η ξ j κ dξ ≤ 10ε -1 ω(κ).
So if we define Ūε,κ := U ε,κ -5ε -1 ω(κ)y 2 1 , then L Ūε,κ ≤ 0. Since also Ūε,κ ∈ C 2 (R N ) for κ > 0, the last claim of the first paragraph of [START_REF] Stroock | On the support of diffusion processes with applications to the strong maximum principle[END_REF]Section 6] tells us that -Ūε,κ ∈ H - L (R N ), that is, it has the submartingale property relative to L described in that paragraph (with Ūε,κ independent of the variable t). Finally, since U is continuous, Ūε,κ → U ε,0 as κ → 0 and then U ε,0 → U as ε → 0, both locally uniformly in R N . This means that -U ε,0 and then -U have the same submartingale property, hence -U ∈ H - L (R N ) (and also -U ∈ H - L (R × T N -1 )). Thus the strong maximum principle from [START_REF] Stroock | On the support of diffusion processes with applications to the strong maximum principle[END_REF]Theorem 6.1] applies to -U (and to -U ′ ), which we will use in the rest of this section.

We will also need the following. Next test against φ a,a+ε,b,b+1 , where a is the smallest real number such that

U L ∞ ([a,+∞)×T N-1 ) = θ
(recall that U is continuous). Then take ε → 0 and b → +∞ to obtain

T N-1 (γ -α(y))U(a, y)dy = [a,+∞)×T N-1 f (U(x, y))dxdy = 0.
Condition (1.8), continuity of U ≥ 0, and U(a, y a ) = θ for some y a now yield the second inequality in (3.25). Now, since f (0) = f (1) = 0 and

min y∈T N-1 (γ -α(y)) < 0 < max y∈T N-1 (γ -α(y)), (3.26)
it follows from the maximum principle for degenerate parabolic equations in [START_REF] Stroock | On the support of diffusion processes with applications to the strong maximum principle[END_REF] that U cannot attain the value 0 (resp. 1) unless being identically equal to 0 (resp. 1) in R × T N -1 .

(Here we apply [START_REF] Stroock | On the support of diffusion processes with applications to the strong maximum principle[END_REF]Theorem 6.1] to the functions V (t, x, y) = V (x, y) = -U(x, y) ≤ 0 and V (t, x, y) = e -f ′ ∞ t (U(x, y) -1) ≤ 0, and use that (3.26) yields G(t 0 , x 0 , y 0 ) = [t 0 , +∞) × R × T N -1 in that theorem. Note that [START_REF] Stroock | On the support of diffusion processes with applications to the strong maximum principle[END_REF]Theorem 6.1] applies to V just as it does to V , because of its continuity and this time the fact that

V t + L V ≥ 0 in L p loc (R 2 × T N -1 ) for p ∈ (1, ∞).) But U(-∞, •) = 1 and U(+∞, •) = 0, so we have 0 < U < 1 and 0 < U ′ < 1 on R × T N -1 .
Let us assume here that

γ ′ ≤ γ, (3.27)
which we can ensure by possibly swapping U and U ′ . We shall now slide U ′ in the x-direction and compare it with U. We will first prove that

U ≤ U ′ (• -h, •) in R × T N -1
for h large enough, and then decrease h up to a critical value h * so that U and U ′ (• -h * , •) will touch. It will follow from the maximum principle in [START_REF] Stroock | On the support of diffusion processes with applications to the strong maximum principle[END_REF]

that γ = γ ′ and U = U ′ (• -h * , •). Lemma 3.2 There exists h 0 ∈ R such that U(x, y) ≤ U ′ (x -h, y) for all h ≥ h 0 and all (x, y) ∈ R × T N -1 .
Proof. Extend f by 0 on R\[0, 1] (so it is Lipschitz-continuous on R). Let ρ > 0 be such that f ′ ≤ 0 on [1 -ρ, 1] and let M > 0 be such that

U ≤ θ in [M, +∞) × T N -1 and U ′ ≥ 1 -ρ in (-∞, -M] × T N -1 . (3.28)
We will prove the lemma with h 0 = 4M. Pick any h ≥ h 0 and let

ε * = min ε ≥ 0 U(x, y) -ε ≤ U ′ (x -h, y) for all (x, y) ∈ R × T N -1 . (3.29)
We want to show that ε * = 0, so let us assume ε * > 0. Then continuity of U and U ′ , and the limits in (1.7), show that the function

V (x, y) = U(x, y) -ε * -U ′ (x -h, y) ≤ 0
must attain the value 0 at some (x 0 , y 0 ). Moreover, we have

∆ y V (x, y) + (γ -α(y)) V x (x, y) = (γ ′ -γ)U ′ x (x -h, y) + f (U ′ (x -h, y)) -f (U(x, y)). (3.30)
Assume for now that, in the sense of distributions,

(γ ′ -γ)U ′ x ≥ 0 on R × T N -1 . (3.31) If x 0 ≥ 2M, let G = (M, +∞) × T N -1 . Since U ≤ θ on G, we have LV ≥ 0 on G (as a distribution, but then also in L p loc (G) because LU, LU ′ ∈ L p loc (R × T N -1
)). Then [38, Theorem 6.1] (which again applies to V because it is continuous and LV ≥ 0 in L p loc (G)) together with (3.26) shows V = 0 on G, a contradiction with ε * > 0 and the limits in (1.7).

If instead x 0 ≤ 2M and (3.31) is assumed, let G = (-∞, 3M) × T N -1 and define

V (t, x, y) = e -f ′ ∞ t V (x, y) ≤ 0. (3.32) Since U ′ (• -h, •) ≥ 1 -ρ on G, we have f (U ′ (x -h, y)) -f (U(x, y)) ≥ f (U ′ (x -h, y) + ε * ) -f (U(x, y)) ≥ -f ′ ∞ |V (x, y)| there. Thus V t + L V ≥ 0 (3.33) in L p loc (R × G) and [38, Theorem 6.1] again shows V = 0 on R × G, a contradiction.
It follows that ε * = 0, so the lemma is proved under the assumption (3.31) (e.g., if U = U ′ , since then γ = γ ′ ). The validity of (3.31) will be obtained at the end of the proof of the next lemma.

Next let

h * = min h ′ ∈ R U(x, y) ≤ U ′ (x -h, y) for all (x, y) ∈ R × T N -1 and h ≥ h ′ . (3.34)
If we assume (3.31) (which we shall prove to be true always), then Lemma 3.2 and the limits in (1.7) show that h * is well defined. Proof. Assume again (3.31), so h * is well defined, and let

V (x, y) = U(x, y) -U ′ (x -h * , y) ≤ 0.
Then (3.30) and (3.31) show that V from (3.32) satisfies (3.33) on R 2 ×T N -1 . If V (x 0 , y 0 ) = 0 for some (x 0 , y 0 ), then [START_REF] Stroock | On the support of diffusion processes with applications to the strong maximum principle[END_REF]Theorem 6.1] again implies V = 0 on R 2 × T N -1 , as was to be proved.

Let us therefore assume V < 0 on R × T N -1 and pick η ∈ (0, h * ] such that

U(x, y) < U ′ (x -h, y) for all h ∈ [h * -η, h * ] and all (x, y) ∈ [-2M, 2M] × T N -1 .
This can be done because U and U ′ are continuous. For any such h define ε * as in (3.29). It follows that U(x 0 , y 0 ) -ε * -U ′ (x 0 -h, y 0 ) = 0 for some (x 0 , y 0 ) with |x 0 | > 2M. As in the proof of the last lemma, this, U ≤ θ on (M, +∞) × T N -1 , and

U ′ (• -h, •) ≥ 1 -ρ on (-∞, -M) × T N -1 imply U(x, y) -ε * -U ′ (
x -h, y) = 0 on one of these two sets. In either case we must have ε * = 0. This means that h * in (3.34) can be decreased by η > 0, a contradiction with the assumption that it is minimal. Thus we have proved the lemma assuming (3.31). But (3.31) is satisfied if U = U ′ , so the definition of h * , the limits in (1.7), and the lemma yield h * = 0 in this case and U ′

x ≤ 0. Therefore (3.31) holds a priori (after assuming (3.27)), so this and the previous lemma also hold as stated.

Finally the function U x ≤ 0 (which is of class C δ (R × T N -1 ), as we have showed earlier) is by (1.7) a distributional solution of

LU x = -f ′ (U(x, y)) U x .
As at the beginning of this subsection, it follows from [START_REF] Rothschild | Hypoelliptic differential operators and nilpotent groups[END_REF]Theorem 18(d)] that ∆ y U x , (γα(y))U xx ∈ L p loc (R × T N -1 ) for any p ∈ (1, ∞), so the above equation holds in L p loc (R × T N -1 ) for any p ∈ (1, ∞). Since U x cannot be identically equal to 0 in R × T N -1 , (3.26) and [START_REF] Stroock | On the support of diffusion processes with applications to the strong maximum principle[END_REF]Theorem 6.1] imply that U x < 0 everywhere in R × T N -1 .

3.2 Existence of the limit (1.6) when 0 < θ < 1

The only part of Theorem 1.1 left to prove is the existence of the limit (1.6). If (1.8) holds, then it follows from the proof of Proposition 2.2 and the uniqueness result in Lemma 3.3 that the bounded family (γ * A ) A≥1 has only one limiting value as A → +∞. In other words, the limit

γ * (q, f ) := lim A→+∞ γ * A = lim A→+∞ c * (Aq, f ) A (3.35)
exists. Furthermore, γ * (q, f ) satisfies (3.25) in this case. Now consider general α, pick any ε > 0, let α satisfying (1.8) be such that α ≤ α ≤ α + ε, and let q(x, y) = (α(y), 0, • • • , 0).

Then for any A ∈ R we have

c * (Aq, f ) -Aε ≤ c * (Aq, f ) ≤ c * (Aq, f )
because the speed of traveling fronts for problem (1.3) is 1-Lipschitz with respect to the L ∞ norm of the flow, see [START_REF] Berestycki | Reaction-Diffusion Equations and Propagation Phenomena[END_REF][START_REF] Berestycki | Traveling fronts in cylinders[END_REF]. Since (3.35) shows that c * (Aq, f )/A converges as A → +∞, and since ε > 0 was arbitrary, existence of the limit (1.6) follows. Moreover (3.25) holds for α and γ * (q, f ), so we obtain

T N-1 α(y)dy ≤ γ * (q, f ) ≤ max y∈T N-1 α(y),
with the first inequality strict unless α is constant by [START_REF] Kiselev | Enhancement of the traveling front speeds in reaction-diffusion equations with advection[END_REF]. The proof of Theorem 1.1 is finished.

Remark 3.4 When the flow α satisfies the non-degeneracy condition (1.8), then the solutions (γ, U) of (1.7) are unique, up to shifts in x for U. Without assumption (1.8), this uniqueness property is not clear in the general case, for which the solution U obtained in Proposition 2.2 is just a distributional solution which may not be of class C 1,δ (R × T N -1 ). Furthermore, the strict inequality γ * (q, f ) < max T N-1 α, which holds under assumption (1.8), may not be true in the general case, when α is flat and equal to its maximum on a sufficiently large region.

Remark 3.5 Since the speed of traveling fronts for problem (1.3) is 1-Lipschitz with respect to the L ∞ norm of the flow, it follows from (1.6) that, if q α (x, y) = (α(y), 0, . . . , 0) and q β (x, y) = (β(y), 0, . . . , 0), then

|γ * (q α , f ) -γ * (q β , f )| ≤ α -β L ∞ (T N-1 ) .
4 Proof of Theorem 1.5

The existence result was proved in Section 2 with

γ * (q, f ) := γ * ∞ = lim inf A→∞ γ * A .
It remains to prove (1.6) and that γ * (q, f ) equals the limit in the statement of the theorem (for general α) and the claims in the last sentence (for α satisfying (1.8)). In fact, it will be sufficient to assume (1.8) in the rest of the proof because (1.6) as well as the θ ′ → 0 limit in the general case then follow as in Subsection 3.2. So let us assume (1.8). Recall that f θ ′ (u) := f (u)χ(u/θ ′ ) for θ ′ ∈ (0, 1/4] and let f θ ′ be the function which equals f θ ′ on [0, 1] and equals 0 on [-θ ′ , 0). For A ≥ 1 and θ ′ ∈ (0, 1/4], let γ * A,θ ′ (resp. γ * A,θ ′ ) be the unique speed for which there is a (unique up to shifts) solution U A,θ ′ (resp. U A,θ ′ ) of (2.13) with f θ ′ (resp. f θ ′ ) in place of f , but with U A,θ ′ instead satisfying -θ ′ < U A,θ ′ < 1 and U A,θ ′ (+∞, •) ≡ -θ ′ . As in Theorem 1.1, γ * (q, f θ ′ ) = lim A→∞ γ * A,θ ′ exists and is the unique speed for which there is a (unique up to shifts) solution U θ ′ of (1.7) with f θ ′ in place of f , and with -θ ′ < U θ ′ < 1 and

U θ ′ (+∞, •) ≡ -θ ′ . It follows from [5] that γ * A,θ ′ < γ * A,θ ′ < γ * A for all A ≥ 1 and θ ′ ∈ (0, 1/4], so γ * (q, f θ ′ ) ≤ γ * (q, f θ ′ ) ≤ γ * (q, f ).
Since the maps θ ′ → γ * A,θ ′ and θ ′ → γ * A,θ ′ are decreasing on (0, 1/4] by [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF], the limits lim θ ′ →0 γ * (q, f θ ′ ) and lim θ ′ →0 γ * (q, f θ ′ ) exist in R, and

γ := lim θ ′ →0 γ * (q, f θ ′ ) ≤ γ := lim θ ′ →0 γ * (q, f θ ′ ) ≤ γ * (q, f ) (= lim inf A→∞ γ * A ). ( 4

.36)

To obtain (1.6) and γ * (q, f ) = lim A→∞ γ * A = lim θ ′ →0 γ * (q, f θ ′ ), we thus only need to prove lim sup

A→∞ γ * A ≤ γ. (4.37)
We first note that since front speed is monotone in f and any f is dominated by some KPP g (i.e., with g(u) ≤ g ′ (0)u), (1.8) and (1.9) immediately imply lim sup

A→∞ γ * A < max T n-1 α. (4.38)
On the other hand, since α is not constant, Theorem 1.1 yields γ * (q, f θ ′ ) > T N-1 α(y)dy for any θ ′ ∈ (0, 1/4], whence

* A,θ ′ (because T N-1 α(y)dy < γ * A,θ ′ < max T N-1 α + A -1 c * (0, f ) by [5]) imply that the quantities U A,θ ′ x L 1 (R×T N-1 ) (= 1) and ∇ y U A,θ ′ L 2 (R×T N-1 ) are bounded independently of A ≥ 1 and θ ′ ∈ (0, 1/4]. Furthermore, since sup θ ′ ∈(0,1/4] f θ ′ C 1 ([-θ ′ ,1]) < +∞, it follows from [6] that the quantities ∇ y U A,θ ′ L ∞ (R×T N-1
) are bounded independently of A ≥ 1 and θ ′ ∈ (0, 1/4]. Therefore, from the characterization of U θ ′ in Theorem 1.1, there holds sup

θ ′ ∈(0,1/4] U θ ′ x L 1 (R×T N-1 ) + ∇ y U θ ′ L 2 (R×T N-1 ) + ∇ y U θ ′ L ∞ (R×T N-1 ) < +∞. (4.40)
Up to shifts, one can assume without loss of generality that max T N-1 U θ ′ (0, •) = 1/2 and by Sobolev embedding one can choose a sequence θ n → 0 such that U θn (x, y) → U (x, y) ∈ [0, 1] for almost every (x, y) ∈ R × T N -1 . The function U is a distributional solution of (1.7), with γ in place of γ, such that U (• + h, •) ≤ U a.e. in R × T N -1 for all h ≥ 0 and ∇ y U ∈ L 2 (R × T N -1 ) ∩ L ∞ (R × T N -1 ). Furthermore, (4.40), U θ ′ x ≤ 0, and max T N-1 U θ ′ (0, •) = 1/2 imply that ± U L ∞ ((±ε,+∞)×T N-1 ) ≤ ±1/2 for any ε > 0. The x → ±∞ limits U ± (y) of U are uniform since ∇ y U ∈ L ∞ (R × T N -1 ) and they must be 0 and 1 because they solve ∆ y U ± + f ( U ± ) = 0 with f > 0 on (0, 1).

Since f ∈ C 1,1 , it follows from [START_REF] Rothschild | Hypoelliptic differential operators and nilpotent groups[END_REF]Theorem 18(c,b)] that U is actually a classical solution of class C 2 (R × T N -1 ), with uniformly bounded first and second derivatives (this is true for any solution, proving the first claim in the last sentence of the Theorem). We now let V := -U x ≥ 0 and obtain from (1.7) for U , ∆ y V + ( γ -α(y)) V x = -f ′ ( U (x, y))V. (here B s (x, y) is the ball of radius s centered at (x, y)). Since V ≥ 0 satisfies (4.41) and (4.42), the Harnack inequality from [START_REF] Hamel | The Harnack inequality for a class of degenerate elliptic operators[END_REF] holds for V , so there is C ′ s > 0 such that V L ∞ ([x-s,x+s]×T N-1 ) ≤ C ′ s V (x, y) for all (x, y) ∈ R × T N -1 .

Thus for any fixed s > 0 and C := C s C ′ s > 0, ∆ y w + A -2 w xx + f (w) -w x + α(y) , as follows from the same arguments as the ones used in [START_REF] Hamel | Formules min-max pour les vitesses d'ondes progressives multidimensionnelles[END_REF] in the case of infinite cylinders with bounded cross sections and Neumann boundary conditions, this implies γ * A ≤ γ +CA -2 . It follows that (4.37) holds, and thus also (1.6) and

γ * (q, f ) = lim θ ′ →0 γ * (q, f θ ′ ) = lim θ ′ →0 γ * (q, f θ ′ ).
Finally, assume that U ′ is any solution of (1.7) with γ ′ < γ * (q, f ) and let U := U θn for θ n as above and small enough so that γ := γ * (q, f θn ) > γ ′ . Then the argument after (3.25) in Subsection 3.1 applied to U, U ′ and with θ replaced by 0 in (3.28) yields a contradiction in the same way as before, with (3.30) and (3.31) replaced by ∆ y V (x, y) + (γ ′ -α(y)) V x (x, y) = (γ ′ -γ)U x (x -h, y) + f (U ′ (x -h, y)) -f (U(x, y))

and by (γ ′ -γ)U x ≥ 0, and considering the operator L ′ from the right-hand side of the above equation instead of L. (Note that the upper bound in (3.25) holds for γ ′ because it holds for γ and the lower bound in (3.25) follows from the first part of the proof of Lemma 3.1 which only uses that f ≥ ≡ 0 on [0, 1], so (3.26) actually makes the proof even easier.) Thus we must have γ ′ ≥ γ * (q, f ) and the proof of Theorem 1.5 is finished.

1 - 1

 11 η(ζ)dζ = 1 and η ′ ∞ ≤ 10. For ε > 0 and κ ≥ 0 define the mollifiers

Lemma 3 . 1

 31 If(1.8) holds and (γ, U) is a solution of (1.7), thenT N-1 α(y)dy < γ < max y∈T N-1 α(y).(3.25)Proof. For a 1 < a 2 ≤ b 1 < b 2 , let φ a 1 ,a 2 ,b 1 ,b 2 : R → [0, 1] be a smooth compactly supported test function, equal to 0 on (-∞, a 1 ] ∪ [b 2 , +∞), to 1 on [a 2 , b 1 ], increasing on [a 1 , a 2 ], and decreasing on [b 1 , b 2 ]. Test the PDE (1.7) against φ a,a+1,b,b+1 (extended onto R × T N -1 so it is independent of y) and take -a, b → +∞ to obtain T N-1 (γ -α(y))dy = R×T N-1 f (U(x, y))dxdy > 0 (the last inequality is due to U being continuous). The first inequality in (3.25) now follows.

Lemma 3 . 3

 33 We have γ = γ ′ and U(x, y) = U ′ (x -h * , y) for all (x, y) ∈ R × T N -1 .

1 γ- 1 γ

 11 -α(y) < 0 < max y∈T N-α(y) .(4.42)Since f ′ ( U (x, y)) is bounded and Lipschitz, we obtain from [35, Theorem 18(c)] and subsequent repeated application of[START_REF] Rothschild | Hypoelliptic differential operators and nilpotent groups[END_REF] Theorem 18(b)] that for each s > 0 there isC s > 0 such that |V x (x, y)| ≤ C s V L ∞ (Bs(x,y)) for all (x, y) ∈ R × T N -1

1 .

 1 |V x (x, y)| ≤ CV (x, y) for all (x, y) ∈ R × T N -But this means that∆ y U + A -2 U xx + ( γ + CA -2 -α(y)) U x + f ( U) ≤ 0.On the other hand, as for the function U in Theorem 1.1 under assumption (1.8),[START_REF] Stroock | On the support of diffusion processes with applications to the strong maximum principle[END_REF] Theorem 6.1] and (4.42) imply that U x < 0 everywhere in R × T N -1 . Sinceγ * A = min w∈C 2 (R×T N-1 ), wx<0, w(-∞,•)=1, w(+∞,•)=0 sup (x,y)∈R×T N-1

  holds for γ ′ . Moreover,lim x→+∞ U(x, y) ≡ -θ n < 0 ≡ lim x→+∞ U ′ (x, y)


François Hamel

Andrej Zlatoš

The Harnack inequality for a class of degenerate elliptic operators

We prove a Harnack inequality for distributional solutions to a type of degenerate elliptic PDEs in N dimensions. The differential operators in question are related to the Kolmogorov operator, made up of the Laplacian in the last N -1 variables, a first-order term corresponding to a shear flow in the direction of the first variable, and a bounded measurable potential term. The first-order coefficient is a smooth function of the last N -1 variables and its derivatives up to certain order do not vanish simultaneously at any point, making the operators in question hypoelliptic.

Introduction

We prove a Harnack inequality for distributional solutions to the degenerate elliptic PDE ∆ y u + β(y)u x + γ(x, y)u = 0 (1.1) in cylindrical domains in R N with axes in the direction of the first variable x. Here γ is bounded measurable and β is a smooth function such that the operator

L = N -1 n=1 X 2 n + X 0 := N -1 n=1 (∂ yn ) 2 + β(y)∂ x = ∆ y + β(y)∂ x (1.2)
satisfies Hörmander's hypoellipticity condition. That is, vector fields {X n } N -1 n=0 and their commutators up to certain order span the whole tangent space R N at each (x, y). Moreover, β changes sign so that L is not parabolic, since then the "elliptic" Harnack inequality (1.4) below would not hold in general. These conditions on β are equivalent to hypothesis (1.3) below and our result is then as follows: Remark. We note that ∆ y could be replaced by any x-independent, uniformly elliptic in y operator on D, but for the sake of simplicity we state the theorem with ∆ y instead.

This result is motivated by its application in our work [START_REF] Hamel | Speed-up of combustion fronts in shear flows[END_REF] on large amplitude A → ∞ asymptotics of traveling fronts in the x-direction, and their speeds, for the reaction-advectiondiffusion equation

v t + A α(y) v x = ∆ x,y v + f (v) (1.5)
on R N +1 , with the first order term representing a shear flow in the x-direction and f a nonnegative reaction function vanishing at 0 and 1. The front speeds in question are proved to satisfy lim A→∞ c * (Aα, f )/A = κ(α, f ) for some constant κ(α, f ) ≥ 0, so after substituting the front ansatz v(t, x, y) = u(xc * (Aα, f )t, y) into (1.5) and scaling by A in the x variable, one formally recovers (1.1) in the limit A → ∞, with β(y) := κ(α, f )α(y) and γ(x, y) := -f (u(x, y))/u(x, y).

The study of hypoelliptic operators of the form

L = M n=1 X 2 n + X 0
(where X n are first order differential operators with smooth coefficients), possibly with an additional potential term, has been systematically pursued since Hörmander's fundamental paper [START_REF] Hörmander | Hypoelliptic second order differential equations[END_REF]. Although various regularity and maximum principle results have been obtained soon thereafter (see, e.g., [START_REF] Bony | Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés[END_REF][START_REF] Fefferman | Subelliptic eigenvalue problems[END_REF][START_REF] Folland | Subelliptic estimates and function spaces on nilpotent Lie groups[END_REF][START_REF] Nagel | Balls and metrics defined by vector fields. I. Basic properties[END_REF][START_REF] Rothschild | Hypoelliptic differential operators and nilpotent groups[END_REF][START_REF] Stroock | On the support of diffusion processes with applications to the strong maximum principle[END_REF]), Harnack inequalities and related heat kernel estimates for such operators have initially been proved only in the case when the tangent space at each point is spanned by the fields {X n } M n=1 and their commutators, sometimes with X 0 either zero or a linear combination of {X n } M n=1 [START_REF] Bony | Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés[END_REF][START_REF] Jerison | The Poincaré inequality for vector fields satisfying Hörmander's condition[END_REF][START_REF] Jerison | Estimates for the heat kernel for a sum of squares of vector fields[END_REF][START_REF] Kusuoka | Applications of the Malliavin calculus III[END_REF][START_REF] Kusuoka | Long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator[END_REF]. More recently, Harnack inequalities have been obtained without this assumption for certain special classes of operators, not including (1.1) with general β, γ. Specifically, some operators with constant and linear coefficients, such as the Kolmogorov operator L = ∂ 2 yy + y∂ x -∂ t , were considered in [START_REF] Garofalo | Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type[END_REF][START_REF] Pascucci | On the Harnack inequality for a class of hypoelliptic evolution equations[END_REF], and cases of more general coefficients satisfying somewhat rigid structural assumptions (see hypothesis [H.1] in [START_REF] Pascucci | Harnack inequalities and Gaussian estimates for a class of hypoelliptic operators[END_REF]) were studied in [START_REF] Kogoj | An invariant Harnack inequality for a class of hypoelliptic ultraparabolic equations[END_REF][START_REF] Pascucci | Harnack inequalities and Gaussian estimates for a class of hypoelliptic operators[END_REF] and with a potential term in [START_REF] Polidoro | Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term[END_REF]. The domains involved in the obtained inequalities have to

1 For ζ = (ζ 1 , . . . , ζ N -1 ) ∈ N N -1 , we let |ζ| = ζ 1 + • • • + ζ N -1 and D ζ β(y) = ∂ |ζ| β ∂y ζ 1 1 •••∂y ζ N -1 N -1 (y).
depend on the metrics associated to the operators rather than the Euclidian metric, as shows a counter-example to a Harnack inequality in [START_REF] Garofalo | Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type[END_REF]. This is related to the need for the signchanging assumption on β here. We also note that the operators considered in these papers involve the term ∂ t and appropriate "parabolic-type" Harnack inequalities are obtained, but corresponding "elliptic" inequalities follow from these.

It was a mild surprise to us that we were not able to find in the literature a sufficiently general result which would include our case (1.1). It appears that Harnack inequalities and heat kernel estimates become much more involved when the field X 0 is required for Hörmander's condition to be satisfied. One hint in this direction is the fact that the signchanging hypothesis on β is necessary for (1.4) to hold, so hypoellipticity of L is in itself not a sufficient condition.

We therefore believe that our method of proof of Theorem 1.1 in the next section is itself also a valuable contribution to the problem of quantitative estimates for hypoelliptic operators. The proof is based on the Feynman-Kac formula for the stochastic process associated with the operator L, and uses the independence of L, and thus also of the stochastic process, on x. It is not immediately obvious whether this requirement can be lifted and replaced, for instance, by some assumption on the relation of the stochastic processes associated to L and starting from two different points which can be connected by a path with tangent vector X 0 at each point. We leave this as an open problem.

We thank Luis Caffarelli, Nicola Garofalo, Nicolai Nadirashvili, Brian Street, and Daniel Stroock for useful discussions and pointers to references. FH is indebted to the Alexander von Humboldt Foundation for its support. His work was also supported by the French Agence Nationale de la Recherche through the project PREFERED. AZ was supported in part by NSF grants DMS-1113017 and DMS-1056327, and by an Alfred P. Sloan Research Fellowship. Part of this work was carried out during visits by FH to the Departments of Mathematics of the Universities of Chicago and Wisconsin and by AZ to the Faculté des Sciences et Techniques, Aix-Marseille Université, the hospitality of which is gratefuly acknowledged.

Proof of Theorem 1.1

Without loss we can assume inf D ′ β < 0 < sup D ′ β and D ′ connected, after possibly enlarging D ′ . We will also assume a = -5,

a ′ = 0, b ′ = 1, b = 6, D = B 3 (0), D ′ = B 1 (0)
, and γ ∞ ≤ 1, with C then only depending on β, because the general case is analogous. We also note that [START_REF] Rothschild | Hypoelliptic differential operators and nilpotent groups[END_REF]Theorem 18(c)] and boundedness of u show that u is actually continuous.

We first claim that for each d > 0 there is

C d,β ≥ 1 such that sup [0,1]×A d u ≤ C d,β inf [0,1]×B 1 (0) u, (2.6)
with

A d := A + d ∪ A - d and A ± d := {y ∈ B 1 (0) ± β(y) > d}.
Clearly it suffices to show this for all small enough d such that A ± d = ∅, which we shall assume. To this end, note that parabolic regularity theory with x as the time variable, applied on

[-1, 5] × {y ∈ B 2 (0) -β(y) > d/2}, yields sup [0,1]×A - d u ≤ C ′ d,β inf [2,5]×A - d u, (2.7)
where C ′ d,β > 0 depends only on d and β. Similarly, we obtain sup

[3,4]×A + d u ≤ C ′ d,β inf [-1,2]×A + d u, (2.8)
Next, consider the stochastic process (X x,y t , Y x,y t ) starting at (x, y) ∈ R × B 2 (0) and satisfying the stochastic differential equation (dX x,y t , dY x,y t ) = (β(Y x,y t )dt, √ 2 dB t ), (X x,y 0 , Y x,y 0 ) = (x, y).

Here t is a new time variable and B t is a normalized Brownian motion on R N -1 with B 0 = 0 (defined on a probability space (Ω, B, P)). We then have

(X x,y t , Y x,y t ) = (X 0,y t + x, √ 2B t + y).
(2.9)

for any (x, y) ∈ R × B 2 (0), in particular, Y x,y t is independent from x. For any y ∈ B 2 (0) we also define the stopping time

τ = τ y := inf t > 0 Y x,y t / ∈ B 2 (0) .
If t ∧ τ := min{t, τ }, then by the Feynman-Kac formula, γ ∞ ≤ 1, and the parabolic comparison principle, we have for each t ≥ 0 and (x,

y) ∈ R × B 2 (0), e -t E(u(X x,y t∧τ , Y x,y t∧τ )) ≤ u(x, y) ≤ e t E(u(X x,y t∧τ , Y x,y t∧τ )). (2.10)
(The Feynman-Kac formula is usually stated for C 2 functions so we provide a proof of (2.10) in Lemma 2.1 below.) Here

E(u(X x,y t∧τ , Y x,y t∧τ )) = Ω u(X x,y t∧τ (ω), Y x,y t∧τ (ω))dP(ω) = R×B 2 (0)
u(x ′ , y ′ )dµ x,y t (x ′ , y ′ ), (2.11) with the probability measure µ x,y t on R × B 2 (0) such that µ x,y t (A) = P((X x,y t∧τ , Y x,y t∧τ ) ∈ A) for Borel sets A ⊆ R × B 2 (0). Notice that µ x,y t is supported on [xβ ∞ t, x + β ∞ t] × B 2 (0) and µ x,y t (R × ∂B 2 (0)) = P(τ y ≤ t). By (2.9), translation in x equally translates the µ x,y t , and the (x-independent) measure on B 2 (0) given by ν y t (A) = µ x,y t (R × A) is just the law of √ 2B t∧τy + y, the Brownian motion on B 2 (0) starting at y, with stopping time τ y , and with time scaled by a factor of two. In particular for each t > 0 there is h t > 0 such that for any y 1 , y 2 ∈ B 1 (0) and any Borel sets

A 1 ⊆ B 1 (0) and A 2 ⊆ B 2 (0), h t ν y 1 t (A 2 ) ≤ ν y 2 t (A 2 ) ≤ h -1 t ν y 1 t (A 2 ) and h t |A 1 | ≤ ν y 1 t (A 1 ) ≤ h -1 t |A 1 |. (2.12)
From this it follows for t :

= β -1 ∞ that inf [0,1]×B 1 (0) u ≥ C ′′ d,β inf [-1,2]×A + d u (
sup [0,1]×B 1 (0) v ≤ C z,β inf [0,1]×B 1 (0) u (2.15)
holds for some C z,β ≥ 1. Indeed, it follows from (2.9), (2.10), (2.11) that for each (x, y) ∈ R × B 2 (0),

e -t R×B 2 (0) u(x ′ , y ′ )dµ x,y;z t (x ′ , y ′ ) ≤ v(x, y) ≤ e t R×B 2 (0) u(x ′ , y ′ )dµ x,y;z t (x ′ , y ′ ),
where µ x,y;z t

(x ′ , y ′ ) = µ x,y t (x ′ , y ′ ) * (χ [-z,z] (x ′ )dx ′ δ 0 (y ′ )).
The above claims about µ x,y t and the definition of ν y t imply that

µ x,y;z t (x ′ , y ′ ) ≤ κ x;z t (x ′ ) × ν y t (y ′ ) ≤ M m=-M µ x+2mz,y;z t (x ′ , y ′ ),
where κ x;z t is the measure on R with κ x;z t (B) = |B∩[x-zβ ∞ t, x+z+ β ∞ t]| for any Borel set B ⊆ R, and M is such that (2M +1)z ≥ 2(z+ β ∞ t), for instance, M := ⌈1/2+ β ∞ t/z⌉. This and the first claim in (2.12) means that

v(x, y 1 ) ≤ e 2t h -2 t M m=-M v(x + 2mz, y 2 ) (2.16)
for any x ∈ R, y 1 , y 2 ∈ B 1 (0) and t > 0. Now we take any x ∈ [0, 1], y 1 ∈ B 1 (0), and y 2 ∈ A d for some fixed d > 0 such that

A ± d = ∅. Pick t := (2 β ∞ ) -1 z and M = 1 to obtain using (2.16), v(x, y 1 ) ≤ e 2t h -2 t 3z -3z u(x + s, y 2 )ds ≤ e 2t h -2 t 2 -1 u(x ′ , y 2 )dx ′ .
Since (2.6) and its shifts in x give for c = -1, 0, 1, sup

[c-1,c]×A d u ≤ C d,β inf {c}×A d u ≤ C d,β sup [c,c+1]×A d u, sup [c-1,c]×A d u ≤ C d,β inf {c-1}×A d u ≤ C d,β sup [c-2,c-1]×A d u,
we obtain (2.6) with [0, 1] and C d,β replaced by [-1, 2] and C 3 d,β . This proves (2.15). Similarly, (2.15) with [-1, 0] and [START_REF] Ben Arous | The Poisson kernel for certain degenerate elliptic operators[END_REF][START_REF] Bony | Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés[END_REF] in place of [0, 1], together with (2.6), yield sup

[-1,2]×B 1 (0) v ≤ C z,β C d,β inf [0,1]×B 1 (0) u.
In a similar way one can also obtain sup

[-1,2]×B 2 (0) v ≤ C z,d,β inf [0,1]×B 2 (0) u.
(2.17)

for some C z,d,β > 0 (recall that B 2 (0) ⊂⊂ D = B 3 (0)).

We will now need to use (1.3) to finish the proof. This assumption makes the differential operator on the left-hand side of (1.1) hypoelliptic in the sense of Hörmander. It follows that for t > 0, the measure µ x,y t is absolutely continuous when restricted to R × B 2 (0) and also to R × ∂B 2 (0) (as an (N -1)-dimensional measure in the latter case), with densities p t (x, y, •, •), q t (x, y, •, •) ≥ 0 such that p t (x, y, x ′ , y ′ ) = p t (0, y, x ′x, y ′ ), q t (x, y, x ′ , y ′ ) = q t (0, y, x ′x, y ′ ), and p t , q t are bounded functions when restricted to y ∈ B 1 (0) (with y ′ ∈ B 2 (0) for p t and y ′ ∈ ∂B 2 (0) for q t ). For p t this follows from the same claim for the corresponding measure µ x,y t on R N given by (2.11) with t in place of t ∧ τ and β smoothly extended to a periodic function on R N -1 (whose density is smooth in all arguments, [START_REF] Ichihara | A classification of the second order degenerate elliptic operators and its probabilistic characterization[END_REF]Theorem 3]). This is because µ x,y t (A) ≥ µ x,y t (A) for any Borel set A ⊆ R × B 2 (0). For q t this would follow from the same claim for the corresponding escape measure µ x,y τ on R × ∂B 2 (0) given by (2.11) with τ = τ y in place of t ∧ τ . We know of such a result for bounded domains only [1, Corollary 2.11] but since µ x,y t is supported on a bounded cylinder, it applies in our case as well. Specifically, take any a -<β ∞ t and a + > β ∞ t. There is a convex open domain G with a smooth boundary whose intersection with [a -, a + ] × R N -1 is [a -, a + ] × B 2 (0), and the intersection with [(-∞, a -) ∪ (a + , ∞)] × R N -1 are two smooth "slanted" conical caps G ± ⊆ R × B 2 (0) over the (N -1)-dimensional balls {a ± } × B 2 (0) with the two (rounded) tips at points with y ′ coordinates y ′ ± such that ±β(y ′ ± ) > 0 and sufficiently long so that for any (x ′ , y ′ ) ∈ ∂G ± ∩ ∂G, the unit outer normal vector n(x ′ , y ′ ) to ∂G ± at (x ′ , y ′ ) satisfies

|n(x ′ , y ′ ) • (1, 0, • • • , 0)| ≤ 1 2 ( β -1 ∞ + 1) whenever ± β(y ′ ) ≤ 0.
Then G satisfies the hypotheses of [1, Corollary 2.11] (it satisfies the escape condition and all points of ∂G are τ ′ -regular). It follows that the escape measure µ x,y τ has a density q τ (x, y, •, •) which is a continuous function of (x, y, x ′ , y ′ ) ∈ G×∂ * G, where ∂ * G is the set of "good" points of ∂G, that is, all (x ′ , y ′ ) ∈ ∂G except of the two cone tips, where n(x ′ , y ′ ) = (±1, 0, • • • , 0). Thus q τ is bounded on S := {0} ×B 1 (0) ×(a -, a + ) ×∂B 2 (0). Since {X 0,y s } s≤t∧τ almost surely stays in (a -, a + ), we obtain q t ≤ q τ on S and q t = 0 on [{0} × B 1 (0) × R × ∂B 2 (0)] \ S. Finally, q t (x, y, x ′ , y ′ ) = q t (0, y, x ′ -x, y ′ ) shows that q t is bounded on R×B 1 (0)×R×∂B 2 (0). Let d > 0 be such that A ± d = ∅, let z := 1/3, t := β -1 ∞ , and

C t := max{ sup R×B 1 (0)×R×B 2 (0) p t , sup R×B 1 (0)×R×∂B 2 (0) q t } < ∞. Then p t (x, y, x ′ , y ′ ), q t (x, y, x ′ , y ′ ) ≤ C t χ [x-1,x+1] (x ′ ) because the measure µ x,y t is supported on [x -1, x + 1] × B 2 (
0), so we obtain from (2.10) and (2.11)

sup [0,1]×B 1 (0) u ≤ C t e t [-1,2]×B 2 (0) u(x ′ , y ′ )dx ′ dy ′ + C t e t [-1,2]×∂B 2 (0) u(x ′ , y ′ )dx ′ dy ′ ≤ 10C t e t sup [-1,2]×B 2 (0) v ≤ 10C t C z,d,β e t inf [0,1]×B 1 (0) u by using [-1, 2] = [-1, -1/3] ∪ [-1/3, 1/3] ∪ [1/3, 1] ∪ [1, 5/3] ∪ [4/3, 2
] and (2.17). This is (1.4), so the theorem will be proved once we establish (2.10).

Lemma 2.1 If u, β, γ, X x,y t , Y x,y t , τ y are as in the proof of Theorem 1.1 (in particular, γ ∞ ≤ 1), then (2.10) holds for (x, y) ∈ R × B 2 (0).

Proof. Let Z x,y t = t so that dZ x,y t = dt and K := ∆ y + β(y)∂ x + ∂ z is the generator of the process (X x,y t , Y x,y t , Z x,y t ). If we let v(x, y, z) := e z u(x, y), then Kv ≥ 0 on R × B 3 (0) × R in the sense of distributions, that is, We next apply Dynkin's formula [START_REF]Stochastic Differential Equations[END_REF]Theorem 7.4.1] to the smooth function v ε , the process (X x,y t , Y x,y t , Z x,y t ), and stopping time t ∧ τ (with τ = τ y ), to obtain E [v ε (X x,y t∧τ , Y x,y t∧τ , Z x,y t∧τ )] = v ε (x, y, 0) + E t∧τ 0 (Kv ε )(X x,y s , Y x,y s , Z x,y s )ds ≥ v ε (x, y, 0) -2 N +2 εe t+ε u ∞ t.

R×B 3 (0)×R vK * φ dxdydz ≥ 0 for any φ ∈ C ∞ 0 (R × B 3 (0) × R), with K * := ∆ y -β(y)∂ x -∂ z the adjoint of K. For any ε > 0 let δ ε ∈ (0, 1/2 √ N -
Since v ε → v uniformly on [xβ ∞ t, x + β ∞ t] × B 2 (0) × [0, t] as ε → 0 (by continuity of v) and Z x,y t∧τ ≤ t, it follows that e t E [u(X x,y t∧τ , Y x,y t∧τ )] ≥ E [v(X x,y t∧τ , Y x,y t∧τ , Z x,y t∧τ )] ≥ u(x, y). This is the second inequality in (2.10). The first inequality is obtained in the same way, this time with v(x, y, z) := e -z u(x, y), so that Kv ≤ 0 on R × B 3 (0) × R and (Kv ε )(x, y, z) ≤ 2 N +2 εe -z+ε u ∞ .

Theorem 1 . 1 1 ( 1 . 3 )

 11113 Let D ⊆ R N -1 be open connected and u : (a, b) × D → [0, ∞) a bounded distributional solution of (1.1) with γ bounded measurable and β satisfying for some r ∈ N, β ∈ C ∞ (D), inf D β < 0 < sup D β, and 0≤|ζ|≤r |D ζ β(y)| > 0 for all y ∈ D. Then for each [a ′ , b ′ ] ⊆ (a, b) and bounded open D ′ with D ′ ⊆ D, there is C > 0, depending only on D, D ′ , β and an upper bound on (a ′a) -1 , (bb ′ ) -1 , b ′a ′ , and γ ∞ , such that sup (a ′ ,b ′ )×D ′ u ≤ C inf (a ′ ,b ′ )×D ′ u. (1.4)

1 ) 1 - 1

 111 be such that |β(y)β(y ′ )| ≤ ε 2 whenever y, y ′ ∈ B 5/2 (0) and |yy ′ | ≤ √ N -1 δ ε . Let η : R → [0, 1] be a smooth non-negative function supported in [-1, 1], with η(x ′ )dx ′ = 1 and η ′ ∞ ≤ 2. For ε > 0 define the mollifierη ε (x, y, z) := ε -2 δ 1-N and let v ε := v * η ε and φ ε;x,y,z (x ′ , y ′ , z ′ ) := η ε (xx ′ , yy ′ , zz ′ ). For ε ∈ (0, 1) the smooth function v ε then satisfies on R × B 2 (0) × R (Kv ε )(x, y, z) = R×B 3 (0)×R vK * φ ε;x,y,z dx ′ dy ′ dz ′ + R×B 3 (0)×R v(x ′ , y ′ , z ′ )[β(y ′ )β(y)]φ ε;x,y,z x ′ (x ′ , y ′ , z ′ ) dx ′ dy ′ dz ′ .The first integral is non-negative. The integrand in the second vanishes when |x ′ -x| > ε or |y ′ ny n | > δ ε for some n or |z ′ -z| > ε, and |φ ε;x,y,zx ′ (x ′ , y ′ , z ′ )| ≤ 2ε -3 δ 1-N ε , so we have (Kv ε )(x, y, z) ≥ -2 N +2 εe z+ε u ∞ .