Luis Silvestre

Boyan Sirakov

Boundary regularity for viscosity solutions of fully nonlinear elliptic equations

We provide regularity results at the boundary for continuous viscosity solutions to nonconvex fully nonlinear uniformly elliptic equations and inequalities in Euclidian domains. We show that (i) any solution of two sided inequalities with Pucci extremal operators is C 1,α on the boundary; (ii) the solution of the Dirichlet problem for fully nonlinear uniformly elliptic equations is C 2,α on the boundary; (iii) corresponding asymptotic expansions hold. This is an extension to viscosity solutions of the classical Krylov estimates for smooth solutions.

Introduction

In this work we study the boundary regularity of continuous viscosity solutions of fully nonlinear elliptic equations and inequalities such as

(S) F (D 2 u, Du, x) = f (x) (1.1)
in a bounded domain Ω ⊂ R d , with a Dirichlet boundary condition on a part of the boundary ∂Ω. All functions considered in the paper will be assumed continuous in Ω. Standing structure hypotheses on the operator F will be its uniform ellipticity and Lipschitz continuity in the derivatives of u:

(H1) there exist numbers Λ ≥ λ > 0, K ≥ 0, such that for any x ∈ Ω, M, N ∈ § d , p, q ∈ R n ,

M + λ,Λ (M -N ) + K|p -q| ≥ F (M, p, x) -F (N, q, x) ≥ M - λ,Λ (M -N ) -K|p -q|. (1.2)
We denote with M ± λ,Λ (M ) the extremal Pucci operators. We set L := sup Ω f and assume F (0, 0, x) = 0, which amounts to a change of f (x).

We will also consider the larger set of functions which satisfy in the viscosity sense the set of inequalities

(S * ) M + λ,Λ (D 2 u) + K|∇u| ≥ -L M - λ,Λ (D 2 u) -K|∇u| ≤ L in Ω. (1.3)
A natural concept of weak solution for fully nonlinear equations is that of a viscosity solution (standard references on the general theory of viscosity solutions include [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF], [START_REF] Caffarelli | Fully nonlinear elliptic equations[END_REF]). We denote the above problems with (S) and (S * ) in order to use the same notation as in [START_REF] Caffarelli | Fully nonlinear elliptic equations[END_REF].

Viscosity solutions are a priori only continuous functions, so it is clearly a fundamental problem to understand whether and when a viscosity solution has some smoothness. A regularity result starts from a merely continuous solution and shows that the function is in fact more regular (for example, belongs to C α , C 1,α or C 2,α ). This must not be confused with an a priori estimate, in which one assumes from the beginning that the solution is classical, and only proves an estimate on the size of some norm. The a priori estimates are technically easier to prove because one can make computations with derivatives of the solution without worrying about their existence and continuity. A regularity result is practically always accompanied by an a priori estimate, but not necessarily the other way around.

Boundary a priori estimates for solutions to fully nonlinear elliptic equations were first proved by Krylov in [START_REF] Krylov | Boundedly inhomogeneous elliptic and parabolic equations in a domain[END_REF], who thus upgraded his and Evans' interior C 2,α -estimates for convex fully nonlinear equations to global estimates. More references will be given below.

In this paper we prove some boundary regularity results for viscosity solutions, in situations when these solutions do not have the same regularity in the interior of the domain. We stress that all the estimates we prove are known (at least to the experts) if the solution is a priori assumed to be globally smooth. Due to this, one may expect that the corresponding results for viscosity solutions can be obtained by direct extension to viscosity solutions of the known techniques. It turns out however that some difficulties specific to viscosity solutions arise, and workarounds become necessary. These will be discussed in more detail below.

Before stating the main theorems, we make several simple observations on the relation between (S) and (S * ). Obviously if u satisfies (S) then it satisfies (S * ). The converse is true if u is a classical solution of (S * ), in the sense that there exists a linear operator F (depending on u and not necessarily continuous in x) satisfying (H1) such that u is a solution of (S). However, in general viscosity solutions of (S * ) are not solutions of a uniformly elliptic equation in the form (S). An important observation is that under (H1) each partial derivative of a C 1 -smooth solution of F (D 2 u, Du) = 0 is a viscosity solution of (S * ), by the stability properties of viscosity solutions with respect to uniform convergence.

Our first theorem concerns the boundary C 1,α -regularity of solutions of (S * ). In the sequel we assume that 0 ∈ ∂Ω, and denote

Ω + R = Ω ∩ B R , Ω 0 R = ∂Ω ∩ B R , where B R = B R (
0) is the ball centered at 0 with radius R. Theorem 1.1. Suppose (H1) holds, Ω is a C 2 -domain and u is a viscosity solution to (1.3) such that the restriction g = u| ∂Ω ∈ C 1,α (Ω 0 1 ), for some α > 0. Then there exists a function

G ∈ C α (Ω 0 1/2 , R d ), the "gradient" of u on ∂Ω, such that G C α (Ω 0 1/2 ) ≤ CW, (1.4)
and for every x ∈ Ω + 1 and every x 0 ∈ Ω 0 1/2 we have

|u(x) -u(x 0 ) -G(x 0 ) • (x -x 0 )| ≤ CW |x -x 0 | 1+α , (1.5)
where

W := u L ∞ (Ω + 1 ) + L + g C 1+α (Ω 0 1
) . Here α = α(d, λ, Λ) ∈ (0, α); C depends on d, λ, Λ, K, and the maximal curvature of Ω.

The second theorem concerns the boundary C 2,α -regularity of solutions of (S). We need to assume that F is Hölder continuous in x, in the following sense (H2) there exist α, C > 0 such that for all

M ∈ § d , p ∈ R d , x, y ∈ Ω, |F (M, p, x) -F (M, p, y)| ≤ C|x -y| α (|M | + |p|).
Note that (H1)-(H2) imply that the solutions of (S) have Hölder continuous gradients in Ω, see Theorem 1.4 below.

Theorem 1.2. Suppose (H1)-(H2) hold, Ω is a C 2,α -domain, and f ∈ C α (Ω). Let u be a viscosity solution to (1.1) such that the restriction g = u| ∂Ω ∈ C 2,α (Ω 0 1 ). Then there exists a function H ∈ C α (Ω 0 1/2 , R d×d ), the "Hessian" of u on ∂Ω, such that

F (H(x 0 ), Du(x 0 ), x 0 ) = f (x 0 ) for each x 0 ∈ Ω 0 1/2 , H C α (Ω 0 1/2 ) ≤ CW, (1.6)
and for every x ∈ Ω + 1 and every x 0 ∈ Ω 0 1/2 we have

|u(x) -u(x 0 ) -Du(x 0 ) • (x -x 0 ) - 1 2 H(x 0 )(x -x 0 ) • (x -x 0 )| ≤ CW |x -x 0 | 2+α , (1.7)
where

W := u L ∞ (Ω + 1 ) + f C α (Ω + 1 ) + g C 2,α (Ω 0 1
) . Here α = α(d, λ, Λ, α) > 0; C depends on d, λ, Λ, K, α, C and the C 2,α regularity of ∂Ω.

The solutions in the above theorems do not have the same regularity in the interior of the domain as on the boundary. Specifically, solutions of (S * ) are in general only Hölder continuous in Ω and solutions of (S) have only Hölder continuous gradients in Ω; and these cannot be improved, at least if d ≥ 5. Indeed, it was proved by Nadirashvili and Vladut [START_REF] Nadirashvili | Singular solutions of hessian elliptic equations in five dimensions[END_REF] that for each β > 0 there exists a operator F = F (M ) which satisfies (H1) and can even be taken rotationally invariant and smooth, such that F (D 2 u) = 0 has a (1 + β)-homogeneous solution in B 1 . The derivatives of u are then solutions of (S * ) which do not belong to C β (B 1 ).

Note in these counterexamples the singularity of the solution occurs in the center of the ball, i.e. far from the boundary. By combining Theorem 1.2 with a "regularity under smallness" result due to Savin, we can show that solutions of (S) are C 2,α -smooth in a whole neighbourhood of a C 2,α -smooth level set, provided F (M, p, x) is C 1 in the M -variable.

Theorem 1.3. Suppose (H1)-(H2) hold, Ω is a C 2,α domain, and f ∈ C α (Ω). Suppose in addition that F (M, p, x) is continuously differentiable in M . Let u be a viscosity solution to (1.1) such that the restriction g = u| ∂Ω ∈ C 2,α (Ω 0 1 ). Then there exist α, δ > 0 such that

u ∈ C 2,α (Ω δ )
where Ω δ = {x ∈ Ω : dist(x, ∂Ω) < δ}.

Here α = α(d, λ, Λ, α) > 0; δ depends on d, λ, Λ, K, α, C, ∂Ω, and a modulus of continuity of

D M F on B C 0 × Ω, where B C 0 is a ball in § d × R d with radius C 0 depending on d, λ, Λ, K, α, C, ∂Ω.
Acknowledgement. The result of Theorem 1.3 was suggested by Nikolai Nadirashvili to the first author after his talk in a conference in Paris, in June 2011. Another application of Theorem 1.2 is contained in [START_REF] Silvestre | Overdetermined problems for fully nonlinear elliptic equations[END_REF] where we used this theorem to deduce Serrin-like symmetry results for fully nonlinear overdetermined problems, without making regularity assumptions on the solution.

Discussion of difficulties and more context

In general one expects a regularity result to hold whenever an a priori estimate exists. This is in particular the case for global estimates in the presence of a uniqueness result for viscosity solutions, since then one can use the a priori estimate and the standard continuity method to link the fully nonlinear equation to the Laplace equation, and deduce the existence of a solution in the space where the a priori estimate is proven. Sometimes it is also possible to approximate the equation by more treatable equations, but in general it is difficult to approximate a fully nonlinear elliptic PDE with some equation that retains its main properties (for results in that direction we refer to [START_REF] Caffarelli | Smooth approximations of solutions to nonconvex fully nonlinear elliptic equations[END_REF] and [START_REF] Krylov | On regularity properties and approximations of value functions for stochastic differential games in domains[END_REF]).

Furthermore, translating the proof of an estimate from classical to viscosity solutions has some obvious difficulties. Every time a derivative of the solution would be written down and used for an estimate, an alternative argument is needed. In many cases, there is some more or less standard procedure for extending a proof from classical to viscosity solutions. In a few cases however, there are some special difficulties that make this task much more complicated. The most extreme example is probably the uniqueness of solutions to second order elliptic fully nonlinear equations. While the comparison principle is obvious for classical solutions, it is an important result in the theory of viscosity solutions (see [START_REF] Jensen | The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations[END_REF] and [START_REF] Jensen | A uniqueness result for viscosity solutions of second order fully nonlinear partial differential equations[END_REF]). Another fundamental difference is that classical solutions of (S * ) are solutions of (S) for some F , while viscosity solutions are not, in general.

As we noted above, boundary a priori estimates for non-divergence form elliptic operators were first proved by Krylov, in sections 4-5 of [START_REF] Krylov | Boundedly inhomogeneous elliptic and parabolic equations in a domain[END_REF]. It was already observed in that paper that boundary C 2,α -estimates do not require convexity of the operator. A fundamental role in the proof of these estimates is played by an "improvement of oscillation" estimate close to the boundary for solutions of linear equations with zero boundary condition. Shortly after Krylov's work appeared simplifications of the proof of this estimate, due to Safonov (see [START_REF] Safonov | Smoothness near the boundary of solutions of elliptic Bellman equations[END_REF]) and Caffarelli (unpublished work, to our knowledge referred to for the first time in [START_REF] Kazdan | Prescribing the curvature of a riemannian manifold[END_REF]). The most easily accessible source for Krylov's improvement of oscillation estimate is Theorem 9.31 in [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF], where the proof from [START_REF] Kazdan | Prescribing the curvature of a riemannian manifold[END_REF] is given. In that book the result is stated for strong solutions, and only in the setting of a flat boundary and zero boundary data. It turns out that the proof in [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF], as well as the proof in [START_REF] Safonov | Smoothness near the boundary of solutions of elliptic Bellman equations[END_REF], can be extended to viscosity solutions in the S * class in arbitrary domains with zero boundary data. However, a difficulty arises, somewhat unexpectedly, when trying to extend the same result to arbitrary C 1,α -smooth boundary data, due to the lack of "splitting" in the set of solutions of (S * ). Let us describe this interesting open problem. Open problem. Let u be a solution to (S * ). Is it true that u = v + w, where v solves (S * ) and v = 0 on ∂Ω and w is a solution to (S * ) with L = 0 ? More simply, say u is a solution of

M + (D 2 u) ≥ f (x) ≥ M -(D 2 u)
in Ω, is it true that we have the splitting u = v + w where v satisfies the same inequalities and vanishes on ∂Ω, while

M + (D 2 u) ≥ 0 ≥ M -(D 2 u) in Ω?
Note that this statement would provide a direct argument, based on the maximum principle, which reduces a general C 1,α regularity result to one for functions that vanish on the boundary. Such an argument is described for instance in the proof of Proposition 2.2 in [START_REF] Milakis | Regularity for fully nonlinear elliptic equations with neumann boundary data[END_REF].

Note also that the answer to the above question is clearly affirmative if u is a classical solution. Furthermore, using such splitting is not needed if the boundary data is supposed to be C 2 -smooth, since then one can just remove a C 2 -smooth function from the solution and obtain a new solution which vanishes on the boundary. These two remarks probably explain why this open problem has not been observed before.

We circumvent the lack of splitting by using a Caffarelli-type iteration argument, in which the iteration step is insured by the use of an implicit bound provided by global Hölder estimates, see Lemma 3.4 and Theorem 3.5.

Another example of a difficulty exclusive to viscosity solutions appears in the proof of Theorem 1.2. In Lemma 4.1 we prove that if the boundary is flat, a solution to an autonomous fully nonlinear equation which vanishes on the boundary has a second order expansion there, with the corresponding C 2,α -bound. This lemma can be proved by essentially applying the C 1,α estimates of Theorem 1.1 to the normal derivative ∂ d u -a well-known idea (note ∂ d u does not vanish on the boundary). Previously we have to prove that ∂ d u is C 1,α on the boundary. The known way to do that is to apply Theorem 1.1 (with g=0) to the tangential derivatives ∂ i u for i = 1, . . . , d -1. This implies that ∂ i u is C 1,α and in particular ∂ d ∂ i u is C α on the flat boundary. At this point one would want to imply that ∂ d u is C 1,α on the boundary, which is obvious for a classical solution, since

∂ d ∂ i u = ∂ i ∂ d u.
But for viscosity solutions these second derivatives do not have the classical meaning, and cannot be defined in any way for points that are away from the boundary.

It is worth mentioning that we have an alternative proof of Lemma 4.1 and Theorem 1.2 which only uses Theorem 1.1 in the particular case g = 0. This proof is based on a direct barrier construction, and does not apply Theorem 1.1 to ∂ d u.

Another particularity in the proof of Theorem 1.2 appears in the passage from the specific case considered in Lemma 4.1 to the general equation (1.1). The perturbation argument that we use is based on an approximation lemma, Lemma 4.2, which appears to be new. This lemma says that two solutions of different equations which are close to each other differ by at most a precise algebraic upper bound. This is a version of Lemma 7.9 from [START_REF] Caffarelli | Fully nonlinear elliptic equations[END_REF] which does not require the equation to have C 1,1 estimates.

Finally, let us give some more context on regularity results for viscosity solutions of fully nonlinear equations. Caffarelli proved in his breakthrough paper [START_REF] Caffarelli | Interior a priori estimates for solutions of fully nonlinear equations[END_REF] that the Alexandrov-Bakelman-Pucci and Harnack inequalities are valid for viscosity solutions of F (D 2 u, x) = f (x), and deduced that these solutions are locally in C 1,α (resp. in C 2,α ), in the presence of a priori bounds in C 1,α (resp. C 2,α ) for the solutions of F (D 2 u, 0) = 0. A complete account of the theory of the latter equation is given in the book [START_REF] Caffarelli | Fully nonlinear elliptic equations[END_REF]. For generalizations to equations with measurable coefficients and the so-called L p -viscosity solutions we refer to [START_REF] Świech | W 1,p -interior estimates for solutions of fully nonlinear, uniformly elliptic equations[END_REF], [START_REF] Crandall | L p -theory for fully nonlinear uniformly parabolic equations[END_REF]. Global regularity results and estimates for viscosity solutions can be found in the appendix of [START_REF] Milakis | Regularity for fully nonlinear elliptic equations with neumann boundary data[END_REF] as well as in [START_REF] Winter | W 2,p and W 1,p -estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations[END_REF]. Combining the results from all these works we obtain the following global results, which we state for the reader's convenience and completeness. We recall that the first assumption in Theorem 1.4 (b) is verified if F (M, 0, 0) is convex in M . The convexity assumption can be removed in some cases, see [START_REF] Cabré | Regularity for viscosity solutions of fully nonlinear equations F (D 2 u) = 0. Topol[END_REF], but not in general.

Theorem 1.4 can be compared to Theorems 1.1-1.3 from the introduction. In these theorems we assume much less on the solution but prove only boundary regularity (and, as we already noted, interior regularity does not hold).

We also observe that it is well-known how to put together boundary regularity results such as the ones proved in Theorems 1.1-1.2 and interior regularity results, in order to deduce global statements. A simple procedure of this sort can be found for instance in Propositions 2.3 and 2.4 in [START_REF] Milakis | Regularity for fully nonlinear elliptic equations with neumann boundary data[END_REF].

Preliminaries

In the sequel we denote with

B + 1 the half ball {x = (x , x d ) ∈ R d : |x| < 1 and x d > 0}. The bottom boundary of the half ball is B 0 1 = {x = (x , 0) ∈ R d : |x | < 1}.
We recall that we can always perform a change of variables to flatten the boundary. Indeed, if Ω is a C 2 domain (resp. C 2,α domain) then, for any point x ∈ ∂Ω, there is a C 2 (resp. C 2,α ) diffeomorphism ϕ which maps a neighborhood of x in Ω to the upper half ball B + 1 . The following proposition recalls the equation satisfied by

u • ϕ -1 . Proposition 2.1. 1. If u is a solution to F (D 2 u, Du, x) = 0 in Ω, then v(x) = u(ϕ -1 (x)) is a solution in B + 1 to F (Dϕ t (ϕ -1 (x))D 2 v(x)Dϕ(ϕ -1 (x)) + Dv(x)D 2 ϕ(ϕ -1 x), Dv(x)Dϕ(ϕ -1 (x)), ϕ -1 (x)) = 0
If we denote with F (D 2 v(x), Dv(x), x) the operator in the left-hand side of this equality and F satisfies (H1) and/or (H2), then F satisfies (H1) and/or (H2), with possibly modified constants K, L, C, depending only on the C 2 (resp.

C 2,α ) norm of ϕ. 2. If u is a solution to (1.3), then v(x) = u(ϕ -1 (x)
) is a solution to (1.3), with possibly modified constants K, L, depending only on the C 2 norm of ϕ.

Proof. This follows from a straightforward computation and use of the definition of a viscosity solution.

We observe that gradient terms and explicit x-dependence are unavoidable after the change of variables. That is why it would not simplify the problem to consider equations without gradient terms or independent of x in the theorems in the introduction. Proposition 2.2 (interior Harnack inequality). Let u be a nonnegative solution of (1.3) in B + 1 . Then for each compact subset Σ of B + 1 there exists a constant C depending on d, λ, Λ, K, and Σ such that sup

Σ u ≤ C(inf Σ u + L).
Proof. This is a well-known result, see for instance Theorem 4.3 in [START_REF] Caffarelli | Fully nonlinear elliptic equations[END_REF] or [START_REF] Wang | On the regularity theory of fully nonlinear parabolic equations[END_REF].

In the following we set e = (0, . . . , 0, 1/2).

Proposition 2.3 (Harnack inequality up to the boundary). Let u be a nonnegative solution of (1.3) in B + 1 which vanishes on B 0 1 . Then sup

B + 1/2 u ≤ C(u(e) + L).
The constant C depends on d, λ, Λ, and K.

Proof. This is Theorem 1.3 in [START_REF] Berestycki | Inequalities for second-order elliptic equations with applications to unbounded domains[END_REF]. In that paper only classical solutions and linear equations were considered; however exactly the same proof applies in our situation, since the proof in [START_REF] Berestycki | Inequalities for second-order elliptic equations with applications to unbounded domains[END_REF] uses only the comparison principle. Proof. This is Lemma 2.1 in [START_REF] Berestycki | Inequalities for second-order elliptic equations with applications to unbounded domains[END_REF].

Next, we observe that after flattening the domain we can zoom into a neighborhood of a point on B 0 1 (which we will always assume to be the origin), and assume that the lower order terms in the equation are as small as we like. That is, we can set u r (x) = u(rx) and observe that the function u r satisfies

M + (D 2 u r ) + rK|∇u r | ≥ -r 2 L in B + 1 , M -(D 2 u r ) -rK|∇u r | ≤ r 2 L in B + 1 , (2.1)
which in particular means that we can assume, by fixing some small r, that in (1.3) we have

max{K, L} ≤ ε 0 , (2.2)
for any initially fixed positive constant ε 0 . We insist that (2.2) is generic in a neighborhood of any given point on the boundary.

Proposition 2.5 (Hopf principle). Let u be a nonnegative solution of (1.3) in B + 1 which vanishes on B 0 1 . There exists ε 0 > 0 such that if |K| ≤ ε 0 , then

u(x) ≥ c 0 (u(e) -C 0 L) x d in B + 1/2 .
The constants ε 0 , c 0 and C 0 depend on d, λ, Λ only.

Proof. This proposition is a quantitative version of Hopf's lemma in terms of extremal equations with nontrivial right-hand sides.

All constants c, C with varying indices that appear below depend on d, λ, Λ only. Observe that if u(e) ≤ C 0 L then we have nothing to prove. So in what follows we assume u(e) > C 0 L (the constant C 0 will be determined below).

We can assume ε 0 ≤ 1. From the interior Harnack inequality, Proposition 2.2, we know that for some

c 1 , C 1 > 0 u ≥ c 1 u(e) -C 1 L in B 99/100 ∩ {x d > 1/16}.
We assume C 0 is chosen so that

C 0 > C 1 /c 1 . Fix x ∈ B + 1/4 , x = (x , x d ). Set z 0 = (x , 1/4
). We define the following barrier function

Ψ(y) = (c 1 u(e) -C 1 L) |y -z 0 | -p -(1/4) -p (1/8) -p -(1/4) -p + L λd (|y -z 0 | 2 -1/16),
where p = 2p * and p * = Λ λ (d -1) -1 is the usual power such that the minimal Pucci operator vanishes when evaluated at the Hessian of |y| -p * , y = 0. Then M - λ,Λ (D 2 |y-z 0 | -p ) ≥ c(p) > 0 in B 1/4 (z 0 ) \ B 1/8 (z 0 ), and the function Ψ satisfies the inequalities

M - λ,Λ (D 2 Ψ) ≥ c 2 (c 1 u(e) -C 1 L) + 2L ≥ 2L in B 1/4 (z 0 ) \ B 1/8 (z 0 ), |∇Ψ| ≤ C 2 (c 1 u(e) -C 1 L) + C 3 L in B 1/4 (z 0 ) \ B 1/8 (z 0 ), Ψ ≤ c 1 u(e) -C 1 L ≤ u on ∂B 1/8 (z 0 ), Ψ = 0 ≤ u on ∂B 1/4 (z 0 ).
Therefore if ε 0 is small enough (smaller than c 2 /(2C 2 ), 1/(2C 3 )), in the annulus

B 1/4 (z 0 ) \ B 1/8 (z 0 ) we have 1 2 M - λ,Λ (D 2 Ψ) ≥ K|∇Ψ| and M - λ,Λ (D 2 Ψ) -K|∇Ψ| ≥ 1 2 M - λ,Λ (D 2 Ψ) ≥ L ≥ M - λ,Λ (D 2 u) -K|∇u|.
From the comparison principle u ≥ Ψ in B 1/4 (z 0 ) \ B 1/8 (z 0 ). Finally, observe that

∂Ψ ∂x d (x , 0) ≥ c 4 (c 1 u(e) -C 1 L) -C 4 L ≥ c 4 2 (c 1 u(e) -C 1 L),
where the last inequality is ensured by using L ≤ u(e)/C 0 and by taking C 0 sufficiently large. The proof is thus finished.

Finally, we recall the following Krylov-Safonov global Hölder estimate for solutions of (S * ).

Proposition 2.6 (global Hölder estimate). Let u be a solution of (1.3) in B + 1 . There exist positive constants α 0 , ρ 0 and C depending on d, λ, Λ, K, and L, such that for all ρ ∈ (0, ρ 0 ) and any ball B ρ (x), x ∈ B + 1 , we have osc

Bρ(x)∩B + 1 u ≤ C ρ α 0 + osc B √ ρ (x)∩∂B + 1 u .
Proof. This follows from Theorem 2 in [START_REF] Sirakov | Solvability of uniformly elliptic fully nonlinear PDE[END_REF]. Observe that theorem is stated for solutions of fully nonlinear equations, however its proof is given for solutions of (S * ) (see also the remark on page 603 of that paper).

3 Boundary C 1,α -regularity for the class S * Lemma 3.1. Let u be a solution of (1.3) in B + 1 which vanishes on B 0 1 , and u L ∞ (B + 1 ) ≤ 1. Then we can find A ∈ R (representing the "normal derivative" of u at the origin) such that for all x ∈ B + 1 ,

|u(x) -Ax d | ≤ C(1 + L)|x| 1+α and |A| ≤ C(1 + L). (3.1)
The positive constant α depends on d, λ, Λ only, and C = C(d, λ, Λ, K).

Proof. Replacing u by u/(1 + L), we can assume that L ≤ 1. We will construct an increasing sequence V k and a decreasing sequence U k so that for r k = 2 -k , k ≥ 1, we have

V k x d ≤ u(x) ≤ U k x d in B r k , (3.2)
and also

U k -V k ≤ M r α k , (3.3)
for constants α and M which depend on the right quantities and will be determined later.

The statement of the lemma easily follows from this construction by taking A = lim k→∞ V k = lim k→∞ U k , since for each x ∈ B + 1/2 we can choose k so that r k+1 < |x| ≤ r k , and then (3.2) and (3.3) imply

u(x) -Ax d ≤ (U k -A)x d ≤ M r α k x d = 2 α M r α k+1 x d ≤ 2 α M |x| 1+α , and similarly u(x) -Ax d ≥ -2 α M |x| 1+α . If x ∈ B + 1 \ B + 1/2
, then (3.1) is obvious. As a first step in the construction of the sequences V k and U k , we obtain V 1 and U 1 from the Lipschitz estimate, Proposition 2.4. In this case we can take U 1 = 2 C and V 1 = -2 C, where C is the constant C from Proposition 2.4 (recall |u| ≤ 1 and L ≤ 1). At this point we fix the constant M as follows:

M = 4 Cr -α k 0 ,
where k 0 is fixed so that r k 0 K ≤ ε 0 and r

1/2 k 0 (1 + 2 C) < ε 1 ,
where ε 0 is the constant from Proposition 2.5 and ε 1 ∈ (0, 1) will be chosen below.

Hence we can take Assume that we have constructed all members of the sequences V j and U j up to the level j = k (k ≥ k 0 for the reason explained above). Let us now construct V k+1 and U k+1 .

V 1 = V 2 = • • • = V k 0 and U 1 = U 2 = • • • = U k 0 ,
Since

V 1 ≤ V k ≤ U k ≤ U 1 , we know that |V k | and |U k | are bounded by 2 C. Note that if U k -V k ≤ M r α
k+1 , then we can take V k+1 = V k and U k+1 = U k . So we can assume that we have

U k -V k > M r α k+1
. Set e k = (0, . . . , 0, r k+1 ). From (3.2) we get

V k r k+1 ≤ u(e k ) ≤ U k r k+1 .
We now distinguish two cases, either u(e k ) ≥ r k+1 (V k + U k )/2 or not. Let us first assume the former.

We introduce the following rescaled function

v k (x) = r -1-α k (u(r k x) -V k r k x d ), x ∈ B 1 , that is, u(x) = V k x d + r 1+α k v k (x/r k ), x ∈ B r k . Since V k x d ≤ u(x) ≤ U k x d in B + r k , we have by (3.3) 0 ≤ v k ≤ U k -V k r α k x d ≤ M x d in B + 1 .
Moreover, v k satisfies the following scaled version of (1.3)

M + (D 2 v k ) + r k K|∇v k | + r 1-α k L(1 + V k ) ≥ 0 in B + 1 , M -(D 2 v k ) -r k K|∇v k | -r 1-α k L(1 + V k ) ≤ 0 in B + 1 , u = 0 on B 0 1 .
The constant α will be chosen small enough, so we can assume α < 1/2. By the choice of k 0 and k > k 0 we have r

1-α k L(1 + V k ) ≤ r 1/2 k 0 (1 + 2 C) ≤ ε 1 < 1,
and r k K ≤ ε 0 . Therefore we can apply Proposition 2.5 and obtain

v k (x) ≥ c 0 (v k (e 0 ) -C 0 ε 1 )x d in B + 1/2 . (3.4) Recalling that r k = 2 -k , u(e k ) ≥ r k+1 (V k + U k )/2 and U k -V k > M r α k+1 , we see that v k (e 0 ) = r -1-α k (u(e k ) -V k r k+1 ) ≥ r -α k 2 U k -V k 2 ≥ M 2 -α-1 ≥ M 4 .
Now we choose ε 1 = M/(8C 0 ), to obtain from (3.4)

v k (x) ≥ c 0 M 8 x d in B + 1/2 .
In terms of the original scale, this means that

u(x) ≥ (V k + c 0 M 8 r α k )x d in B + r k+1 .

By the induction hypothesis

U k -V k ≤ M r α k , hence u(x) ≥ (V k + c 0 8 (U k -V k ))x d in B + r k+1 .
We now choose

U k+1 = U k and V k+1 = V k + c 0 8 (U k -V k ).
Finally, the constant α is chosen so that 2 -α = (1 -c 0 /8). In this way we have

U k+1 -V k+1 = (1 -c 0 /8)(U k -V k ) = 2 -α (U k -V k ) ≤ M (r k /2) α = M r α k+1 ,
and we finish the iterative step.

In the case that u(e k ) < r k+1 (V k + U k )/2 we proceed in a similar way, by using the scaled function

v k (x) = r -1-α k (U k r k x d -u(r k x)), to obtain (3.2) with V k+1 = V k and U k+1 = U k -c 0 8 (U k -V k ). Theorem 3.2. Let u be a solution of (1.3) in B +
1 which vanishes on B 0 1 . Then there exist α > 0 and a function A ∈ C α (B 0 1/2 ) such that for every x 0 ∈ B 0 1/2 , x ∈ B + 1 we have

A C α (B 0 1/2 ) ≤ CW, and |u(x) -A(x 0 )x d | ≤ CW |x -x 0 | 1+α , (3.5)
where W := u L ∞ (B + 1 ) + L. Here α = α(d, λ, Λ), and C = C(d, λ, Λ, K).

Proof. Replacing u by u/W we can assume that u L ∞ (B + 1 ) ≤ 1 and L ≤ 1. For each x 0 ∈ B 0 1/2 , we know that there is a constant A(x 0 ) for which the second inequality in (3.5) holds, and |A(x 0 )| ≤ C. This follows from an application of Lemma 3.1 appropriately translated to x 0 .

We now have to prove that A(x) is Hölder continuous on B 0 1/2 , with bounded norm. Let

x 1 , x 2 ∈ B 0 1/2 and r = 2|x 1 -x 2 |.
We assume without loss of generality that r < 1/4. Fix a point y ∈ B + r (x 1 ) ∩ B + r (x 2 ) with y d > r/4. We have

|u(y) -A(x 1 )y d | ≤ C|x 1 -y| 1+α ≤ Cr 1+α = C|x 1 -x 2 | 1+α , |u(y) -A(x 2 )y d | ≤ C|x 1 -y| 1+α ≤ Cr 1+α = C|x 1 -x 2 | 1+α . Then |A(x 1 ) -A(x 2 )| ≤ 4r -1 |(A(x 1 ) -A(x 2 ))y d | ≤ 4r -1 |A(x 1 )y d -u(y)| + 4r -1 |u(y) -A(x 2 )y d | ≤ C|x 1 -x 2 | α . Remark 3.3. Of course if u ∈ C 1 in a neighbourhood of B 0 1 then A = Du| B 0 1/2 .
Recall however that there exist functions which satisfy (1.3) and are not C 1 in the interior of B + 1 .

In Theorem 3.2 we proved a boundary gradient Hölder estimate for functions which satisfy (S * ), and vanish on the boundary. We will now extend this to arbitrary C 1,α -boundary data.

We first prove the following lemma.

Lemma 3.4. There exists γ 1 > 0 such that for every γ ∈ (0, γ 1 ) we can find δ > 0 such that if u is a viscosity solution of (1.

3) in B + 1 with u L ∞ (B + 1 ) ≤ 1 and u L ∞ (B 0 1 ) ≤ δ, then there exist A ∈ R such that |A| ≤ C 1 and |u(x) -Ax d | ≤ γ 1+α 1 for all x ∈ B + γ . (3.6)
The positive constants γ 1 , α 1 , C 1 are such that α 1 = α 1 (d, λ, Λ), and γ 1 , C 1 depend only on d, λ, Λ, K (but not on γ or δ).

Proof. We can take α 1 to be any positive number smaller than the exponent α from Theorem 3.2. We choose C 1 to be the constant C from that theorem. In Theorem 3.2 we proved that if δ = 0 then we can get a constant A (bounded by

C 1 ) such that |u(x) -Ax d | ≤ C 1 |x| 1+α for all x ∈ B + 1 . In particular, if we choose γ 1 so small that C 1 γ α-α 1 1 < 1/2, we have, if δ = 0, |u(x) -Ax d | ≤ 1 2 γ 1+α 1 for all γ ∈ (0, γ 1 ), x ∈ B + γ . (3.7)
Now, let us assume that the result we want to prove is false for the choice of γ 1 , α 1 and C 1 that we already made. This means that there exists γ ∈ (0, γ 1 ) and sequences u k ∈ C(B + 1 ) and δ k → 0 such that each u k is a solution of (1.3), with

u k L ∞ (B + 1 ) ≤ 1 and u k L ∞ (B 0 1
) ≤ δ k , and for each A ∈ (-C 1 , C 1 ) the second inequality in (3.6) is false for u k in B + γ . We now apply the global estimate contained in Proposition 2.6, and deduce that for each ε > 0 there exist δ > 0 and N such that x, y ∈ B + 3/4 , |x -y| < δ, and k ≥ N imply |u k (x) -u k (y)| < ε. This is enough to apply the Arzela-Ascoli theorem (or more precisely its proof), and conclude that we can extract a subsequence of u k which converges uniformly in B + 3/4 . Let u ∞ be the limit of this subsequence. By the stability properties of viscosity solutions this limit function u ∞ satisfies (1.3) in B + 3/4 and vanishes on B 0 3/4 . Therefore (3.7) holds for u ∞ , there exists a bounded constant

A, |A| ≤ C 1 , such that |u ∞ (x) -Ax d | ≤ 1 2 γ 1+α 1 for all x ∈ B γ . But u k → u ∞ uniformly in B + 2/3 ⊇ B + γ . In particular |u k -u ∞ | ≤ γ 1+α 1 /2 for k sufficiently large. Thus |u k (x) -Ax d | ≤ γ 1+α 1 for all x ∈ B γ ,
and we arrive to a contradiction.

Theorem 3.5. Let u be a viscosity solution to (1.3) in B + 1 such that the restriction of u to the flat boundary g = u| B 0 1 ∈ C 1,α (B 0 1 ), for some α > 0. Then there exists a function

A ∈ C α (B 0 1/2 ) such that for all x = (x , x d ) ∈ B + 1 and all x 0 = (x 0 , 0) ∈ B 0 1/2 , |u(x) -∇ x g(x 0 ) • (x -x 0 ) -A(x 0 )x d | ≤ C( u L ∞ (B + 1 ) + L + g C 1+α (B 0 1 ) )|x -x 0 | 1+α , (3.8) A C α (B 0 1/2 ) ≤ C( u L ∞ (B + 1 ) + L + g C 1+α (B 0 1 ) ). (3.9)
As usual, α = α(d, λ, Λ) ∈ (0, α), and C depends on d, λ, Λ, K.

Proof. Repeating the argument in the proof of Theorem 3.2, we see it is enough to prove the result with x 0 = 0, that is, there exist A ∈ R such that for all x ∈ B + 1 ,

|u(x) -∇ x g(0) • x -Ax d | ≤ C( u L ∞ (B + 1 ) + L + g C 1+α (B 0 1 ) )|x| 1+α , (3.10)
for some universal C, and

|A| ≤ C( u L ∞ (B + 1 ) + L + g C 1+α (B 0 1 )
). Again, we are going to build an iteration process accounting for the difference at diadic scales between u and an approximate solution.

By subtracting a suitable plane at the origin (adding to L the supremum of |u| + |∇g|, if necessary), we can suppose that u(0) = g(0) = 0 and

∇ x g(0) = 0. Set M = 2 u L ∞ (B + 1 ) + 1 δ g C 1,α (B 0 1 ) + L.
Here δ is the constant from Lemma 3.4 with a value of γ which will be specified below.

We will show that there are constants α > 0 (small), γ > 0 (small) and C 1 > 0 (large) to be chosen below (depending only on d, λ, Λ and K), such that we can construct a sequence of real numbers a k with osc

B + γ k (u(x) -a k • x d ) ≤ M r 1+α k (3.11) |a k+1 -a k | ≤ C 1 M r α k , (3.12)
where we have set r k = γ k . We choose a 0 = 0, hence (3.11) holds for k = 0. We will construct the other values of a k inductively. Let us assume that we already have a sequence a k so that (3.11) holds for k = 0, 1, . . . , k 0 ; we have to show that there is a real number a k 0 +1 such that (3.11) holds for k = k 0 + 1.

Let (we write k instead of k 0 )

u k (x) = M -1 r -(1+α) k [u(r k x) -a k r k x d ].
This scaling means precisely that the (3.11) is equivalent to osc B + 1 u k ≤ 1. In addition, it is easy to see that (1.3) transforms into

M + (D 2 u k ) + Kr k |Du k | + KM -1 r 1-α k |a k | + LM -1 r 1-α k ≥ 0 in B + 1 , M -(D 2 u k ) -Kr k |Du k | -KM -1 r 1-α k |a k | -LM -1 r 1-α k ≤ 0 in B + 1 ,
Since M ≥ L and γ < 1, we have that the last term in these inequalities Lγ k(1-α) M -1 ≤ 1. Next, note that

|a k | ≤ k-1 k=0 |a j+1 -a j | ≤ C 1 M ∞ k=0 (γ α ) k = C 1 M 1 -γ α ,
by using a 0 = 0 and the inductive hypothesis |a j+1 -a j | ≤ C 1 M γ kα for all j < k. Hence the third terms in the above differential inequalities satisfy, for all k ≥ 1

KM -1 r 1-α k |a k | ≤ KC 1 γ 1-α 1 -γ α . At this point we choose γ ∈ (0, γ 1 ) such that KC 1 γ 1-α 1 -γ α < 1 (3.13)
and deduce that u k satisfies (1.3) in B + 1 , with L = 2. Further, on the flat boundary we clearly have

u k (x) = M -1 r -(1+α) k g(r k x) on B 0 1 . Since M ≥ g C 1,α /δ and g(0) = |∇g(0)| = 0, this implies u k L ∞ (B 0 1 ) ≤ δ.
Therefore we can apply Lemma 3.4 to u k , and obtain that there are C 1 and α > 0 (this is where C 1 and α are chosen), as well as a constant ãk such that |ã k | ≤ C 1 , and

|u k (x) -ãk x d | ≤ γ 1+α in B + γ . (3.14)
Note that in Lemma 3.4, we can choose γ arbitrarily small without affecting the choice of constants α and C 1 , but modifying δ accordingly. So, we fix γ > 0 and δ > 0 so that both Lemma 3.4 and (3.13) are satisfied. We set

a k+1 = a k + M r α k ãk . Recall that u(x) = a k x d + r 1+α k u k (x/r k ) if x ∈ B r k . Therefore for all x ∈ B r k we have u(x) -a k+1 x d = M r 1+α k (u k (x/r k ) -ãk x d /r k ) .
The last quantity is smaller than M (r k γ) 1+α = M r 1+α k+1 if x ∈ B r k+1 , by (3.14). This finishes the inductive construction.

Let A = lim k→∞ a k . We claim that

|u(x) -Ax d | ≤ CM |x| 1+α . (3.15)
Indeed, from (3.11), (3.12) and

|a k -A| ≤ ∞ j=k |a j -a j+1 | we get osc Br k (u(x) -Ax d ) ≤ osc Br k (u(x) -a k x d ) + γ k |a k -A| (3.16) ≤ M γ k(1+α) + C 1 M γ k ∞ j=k γ αj (3.17) ≤ M γ k(1+α) + C 1 M γ (1+α)k 1 1 -γ α (3.18) ≤ CM γ k•(1+α) (3.19)
is Hölder continuous on B 0 3/4 (these quantities are not even partial derivatives in the classical sense).

We need to justify that (4.2) implies that u d = ∂ d u is C 1,α on B 0 3/4 , and that its tangential derivatives coincide with H di . This is the content of the following claim.

Claim. The restriction of the normal derivative u d to B 0 3/4 is a C 1,α function, and

∂ i ∂ d u = H di on B 0 3/4
, for each i = 1, . . . , d -1. Proof. Without loss of generality, we prove that u d is C 1,α at the origin. Let τ be a tangential unit vector, say τ = e i for some i = 1, . . . , d -1. Given two small positive numbers h and k, we are going to estimate the difference u(ke d + hτ ) -u(ke d ) in two different ways.

On one hand,

u(ke d + hτ ) -u(ke d ) = h u τ (ke d + ξτ )
by the MVT, for some ξ ∈ (0, h),

≤ h kH di (ξτ ) + Ck 1+α using (4.2), ≤ hkH di (0) + Ckh 1+α + Chk 1+α , using that H di is in C α .
On the other hand, we can also estimate that difference by using the mean value theorem with respect to the normal derivative. For some ξ 1 , ξ 2 ∈ (0, k) we have

u(ke d + hτ ) -u(ke d ) = ku d (ξ 1 e d + hτ ) -ku d (ξ 2 e d ) using that u ∈ C 1 and u = 0 on B 0 1 , ≥ ku d (hτ ) -ku d (0) -Ck 1+α using that u d ∈ C α .
Combining the two estimates above, and dividing by k, we obtain

u d (hτ ) -u d (0) ≤ Ck α + hH di (0) + Ch 1+α + Ck α h.
Since the left hand side of this inequality is independent of k, we can let k → 0, to obtain

u d (hτ ) -u d (0) ≤ hH di (0) + Ch 1+α .
The inequality u d (hτ ) -u d (0) ≥ hH di (0) -Ch 1+α follows analogously (switching the inequalities and the sign of all error terms above). Therefore

|u d (hτ ) -u d (0) -hH di (0)| ≤ Ch 1+α .
This means literally that u d ∈ C 1,α (B 0 3/4 ) and ∂ τ u d = H di on B 0 3/4 . The claim is proved. We thus define H id (x 0 ) := H di (x 0 ), for all x 0 ∈ B 0 3/4 , and all i = 1, . . . , d -1. At this point we can finish the construction of H. We define H dd (x 0 ) as the unique real number for which F (H(x 0 ), Du(x 0 )) = 0 (recall F (M, p) is strictly increasing in the matrix M ). Since F is Lipschitz and

H ij ∈ C α for i < d or j < d, it is obvious that H dd ∈ C α (B 0 3/4
). It remains to show (4.1). Without loss of generality, we will show that this inequality is valid for x 0 = 0.

We start by estimating u(x , t) -u(0, t) for any t > 0. In the following repeated indexes denote summation for i = 1, . . . , d -1.

u(x , t) -u(0, t) = x i ∂ i u(ξ, t) by MVT, for some |ξ| < |x |, ≥ x i ∂ i u(ξ, 0) + H id (ξ)x i t -C 1 (t 1+α |x |) ≥ H id (0)x i t -C 1 t|x |(|x | α + t α ),
where we used ∂ i u = 0 on B 0 1 , (4.2) and the Hölder continuity of H on the flat boundary. Now, let us assume in order to arrive to a contradiction that for some r > 0

u(0, r) -u d (0)r - 1 2 H dd r 2 = ±C 0 r 2+α , (4.3)
where C 0 is a large constant to be chosen below. Say we have plus sign in (4.3) (the minus sign is treated analogously). We construct the auxiliary function

w(x) = Du(0) • x + 1 2 H(0)x • x + C 0 r α x 2 d -2C 1 r α |x| 2 = u d (0)x d + H id (0)x i x d + 1 2 H dd (0)x 2 d + (C 0 -2C 1 )r α x 2 d -2C 1 r α |x | 2 ,
where C 1 is the constant from the inequality on u(x , t) -u(0, t), above. Note that by (4.3) u(0, r) -w(0, r) = 2C 1 r 2+α . (4.4)

For r sufficiently small, w(x) is a subsolution of F (D 2 w, Dw) ≥ 0 in the box Q r := [-r, r] d-1 × [0, r], provided C 0 is chosen sufficiently large. This is so because Let k := max{w(0, t) -u(0, t) : t ∈ [0, r]}.

D 2 w = H(0) + 2(C 0 -2C 1 )r α (e d ⊗ e d ) -2C 1 r α (e i ⊗ e i ) Dw = u d (0)e d + O(r) in Q r ,
Note that k ≥ 0 since w(0, 0) = u(0, 0) = 0.

We will now see that w ≤ u + k on the boundary of Q r . Indeed, on the bottom boundary {x d = 0}, we have w ≤ 0 = u. On the top, {x d = r} and |x | ≤ r, we have, by the definition of w, the above estimate on u(x , t) -u(0, t) and (4.4) that

w(x , r) -u(x , r) -k ≤ (w(x , r) -w(0, r)) -(u(x , r) -u(0, r)) + (w(0, r) -u(0, r)) ≤ H id (0)rx i -2C 1 r α |x | 2 + (-H id (0)rx i + C 1 |x |r(r α + |x | α )) -2C 1 r 2+α ≤ C 1 r|x |(r α + |x | α ) -2(|x | 2 + r 2 )r α ≤ 0.
On the side boundary, |x | = r and t ∈ (0, r), we have

w(x , t) -u(x , t) -k = (w(x , t) -w(0, t)) -(u(x , t) -u(0, t)) + (w(0, t) -u(0, t) -k) ≤ H id (0)tx i -2C 1 r α |x | 2 + (-H id (0)tx i + C 1 |x |t(t α + |x | α )) = C 1 -2r 2+α + t 1+α r + tr 1+α ≤ 0,
Now, for a small ε > 0, we consider the sup-convolution

v ε (x) = max y∈B + 3/4 v(y) - 1 ε |x -y| 2 .
From the elementary properties of sup-convolutions (see [START_REF] Jensen | The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations[END_REF], [START_REF] Jensen | A uniqueness result for viscosity solutions of second order fully nonlinear partial differential equations[END_REF], [6, Chapter 5]), we have F (D 2 v ε , Dv ε , 0) ≥ 0 in D in the viscosity sense as long as we can make sure that for any x ∈ D the maximum in the definition of v ε is attained for some y = x * in the interior of

B + 3/4 . This is true if we choose ε = κ 1-α/2 /2. Indeed, recall that |x -x * | ≤ (εosc H v) 1/2 , (4.6)
where H is the set on which the maximum in the definition of v ε is taken (see for instance Lemma 5.2 in [START_REF] Caffarelli | Fully nonlinear elliptic equations[END_REF]). Since v ∈ C α with a norm smaller than 1/2, iterating (4.6) -first with

H = B+ 3/4 , then with H = B+ 3/4 ∩ Bε 1/2 (x), then with H = B+ 3/4 ∩ Bε 1/2+α/4 (x), etc -we get |x -x * | ≤ ε 1/(2-α) , by using 1 2 -α = 1 2 + α 4 ∞ i=0 α 2 i . Moreover, for all x ∈ D, v ε (x) -v(x) ≤ v(x * ) -v(x) ≤ 1/2|x -x * | α ≤ ε α/(2-α) ≤ κ α/2 .
The function v ε is twice differentiable a.e. and semi-convex, with D 2 v ε ≥ - Fix a matrix M such that M ≤ D 2 ϕ(x), M -= (D 2 ϕ(x)) -and F (M, Dϕ(x), 0) = 0. Thus, we have M -≤ 2 ε I and from the ellipticity of F it easily follows that |M | ≤ C ε . Hence

F (D 2 ϕ(x), Dϕ(x), x) ≥ F (M, Dϕ(x), x), = F (M, Dϕ(x), x) -F (M, Dϕ(x), 0), ≥ - C ε κ = -Cκ α/2 .
Therefore, we showed that

F (D 2 v ε , Dv ε , x) ≥ -Cκ α/2 in D, and
v ε ≤ v + κ α/2 ≤ u + Cκ α/2 on ∂D.
From the Alexandrov-Bakelman-Pucci inequality we get that v ≤ v ε ≤ u + Cκ α/2 in D.

The inequality in the opposite direction follows similarly.

Remark 4.3. If we assume that F is convex or concave in the second derivative, then F (D 2 u, Du, 0) = 0 would have C 1,1 solutions and we could prove Lemma 4.2 by using a simpler idea, as in Lemma 7.9 in [START_REF] Caffarelli | Fully nonlinear elliptic equations[END_REF]. This lack of regularity of the solutions is compensated with the use of sup-convolutions. For nonconvex equations, there is a weaker result in [START_REF] Caffarelli | Fully nonlinear elliptic equations[END_REF] (Lemma 8.2) which is proved by compactness and thus does not give an algebraic expression for the upper bound of the difference between the two solutions.

We are now ready to prove Theorem 1.2 by an iterative argument which makes use of Lemma 4.1.

Proof of Theorem 1.2. Without loss of generality, we can assume that the boundary of Ω is flat. Otherwise we can make a change of variables to flatten the boundary which preserves the hypotheses on the equation F . So we assume that u satisfies the equation in B + 1 and equals zero on B 0 1 . The latter is obtained by removing from u a C 2,α -extension of g in Ω. We are going to show that the statement of Theorem 1.2 is valid for x 0 = 0. We can assume without loss of generality that the C α norm of F in B + 1 is less than ε 0 , a constant to be chosen. To achieve the latter, just replace B + 1 by B + r 0 , for some r 0 such that Cr ᾱ 0 < ε 0 , where C is the constant from (H2).

We can also assume that u L ∞ (Ω) + f C α (Ω) = 1 (if not, set a = u L ∞ (Ω) + f C α (Ω) and replace u by u/a and F by (1/a)F (aM, ap, x)). By Theorem 1.4 we know that the gradient Du is Hölder continuous up to the boundary, so we can replace F (M, p, x) by F (M, x) = F (M, Du(x), x) (we will write F instead of F ).

We will construct iteratively two sequences A k ∈ R and H k ∈ § d such that

|A k -A k+1 | ≤ Cr 1+α k , (4.7)
|H k -H k+1 | ≤ Cr α k , (4.8)

|u(x) -A k x d -H k ij x i x j | ≤ r 2+α k , if |x| ≤ r k , (4.9)
where r k = ρ k , for some ρ ∈ (0, 1) to be determined later, depending on the right quantities. Moreover, along the sequence, we have H k ij = 0 for i, j = 1, . . . , d -1. That is, (Hx, x) = H k ij x i x j = 0 when x ∈ B 0 1 . For k = 0 the choice A k = 0 and H k = 0 obviously works. Now we assume we have constructed these sequences up to certain value of k and aim to find A k+1 , H k+1 .

Note that, by the induction hypothesis,

|H k | ≤ k i=1 |H i -H i-1 | ≤ C ∞ i=1 (ρ α ) k ≤ C,
and similarly for A k . Let P k (x) = A k x d + H k ij x i x j and u k be the rescaled function

u k (x) = r -2 k u(r k x) -r -1 k A k x d -H k ij x i x j = r -2 k (u(r k x) -P k (r k x)), x ∈ B + 1 ,
that is, u(x) = P k (x) + r Finally, we know that v k converges uniformly to zero in B + 3/4 , so (4.10) implies that F (H, 0) = f (0), where H = D 2 P . It only remains to show that the symmetric matrix function H(x 0 ) which we thus constructed for all x 0 ∈ B 0 1/2 is Hölder continuous on B 0 1/2 . This is simple to get, since F (H(x 0 ), x 0 ) = f (x 0 ), F (M, x) and f (x) are Hölder continuous in x, F is Lipschitz and uniformly elliptic in M , and H ij (x 0 ) = 0 for i, j = 1, . . . , d -1.

The proof of Theorem 1.2 is finished.

Theorem 1 . 4 .

 14 (a) Assume (H1)-(H2). If u is a viscosity solution of (1.1) in the bounded C 2 -domain Ω, and g = u| ∂Ω ∈ C 1,α (∂Ω) then u ∈ C 1,α (Ω), with a norm bounded by the quantity W from Theorem 1.1 (with Ω + 1 replaced by Ω and Ω 0 1 replaced by ∂Ω). (b) If in addition the equation F (D 2 u, 0, 0) = 0 admits global a priori bounds in C 2,α (Ω), and g = u| ∂Ω ∈ C 2,α (∂Ω) then u ∈ C 2,α (Ω), with a norm bounded by the quantity W from Theorem 1.2 (with Ω + 1 replaced by Ω and Ω 0 1 replaced by ∂Ω).

Proposition 2 . 4 (

 24 Lipschitz estimate). Let u be a solution of (1.3) in B + 1 which vanishes on B 0 1 . Then |u(x)| ≤ C(u(e) + L)x d in B + 1/2 . The constant C depends on d, λ, Λ, and K.

  and we only need to construct V k and U k satisfying (3.2) and (3.3) for k > k 0 .

  and hence (recalling that F (H(0), u d (0)e d ) = F (H(0), Du(0)) = 0)F (D 2 w, Dw) ≥ F (H(0), Du(0)) + 2r α M -((C 0 -2C 1 )(e d ⊗ e d ) -C 1 (e i ⊗ e i )) -Cr ≥ 0, if (C 0 -2C 1 ) > λ(d -1)C 1 /Λ and r ∈ (0, r 0 ), for some sufficiently small r 0 .

  2 ε I and |Dv ε | < C ε . Let ϕ be a C 2 function touching v ε from above at a given point x ∈ D. Then clearly we also have that D 2 ϕ(x) ≥ -2 ε I and |Dϕ(x)| ≤ C ε . By the definition of a viscosity solution F (D 2 ϕ(x), Dϕ(x), 0) ≥ 0.

2+α 2 ,

 2 2 k u k (x/r k ) for x ∈ B r k . Then we have |u k | ≤ r α k in B + 1 (by (4.9)) andF D 2 u k + H k , r k x = f (r k x) in B + 1 . F D 2 v k + H k , 0 = f (0) in B + 3/4 ,(4.10)v k = u k on ∂B + 3/4 .(4.11)We use (H2) and apply Lemma 4.2, to obtain that|u k -v k | ≤ Cε γ 0 r αγ k in B + 3/4 .We take α to be a positive number smaller than αγ. Now, by applying Lemma 4.1 to (4.10) we get that there exists α > 0, Ãk and Hk such that for allx ∈ B + 3/4 |v k (x) -Pk (x)| = |v k (x) -Ãk x d -Hk ij x i x j | ≤ C 1 v k L ∞ |x| 2+ α ≤ 2C 1 r α k |x| 2+ α, where we also used that |v k | ≤ |u k | + |u k -v k | ≤ Cr α k .Here we choose α < α and ρ so that 2C 1 ρ α-α < 1/2, thus|v k (x) -Pk (x)| ≤ r α k ρ for all x ∈ B ρ .We now choose ε 0 so small that|u k -v k | ≤ Cε γ 0 r αγ k ≤ r α k ρ 2+α 2 for all x ∈ B 3/4 .Finally, we defineP k+1 (x) = P k (x) + r 2 k Pk (x/r k ), in other words, A k+1 = A k + r k Ãk and H k+1 = H k + Hk . Then, if |x| ≤ r k+1 = r k /ρ and y = x/r k we have |u(x) -P k+1 (x)| = r 2 k |u k (y) -Pk (y)| ≤ r 2 k (|u k (y) -v k (y)| + |v k (y) -Pk (y)|) ≤ r 2+αk+1 . The conditions (4.7) and (4.8) are clearly satisfied for k + 1 since by Lemma 4.1 and the global C 1,α -estimates we have | Ãk |, | Hk | ≤ C v k L ∞ ≤ Cr α k . This finishes the construction of the sequences A k and H k . Therefore we can define P (x) = lim P k (x) = ∞ k=1 (P k+1 -P k ), since the last series is convergent. In addition, if x ∈ B r k we have, by (4.7) and (4.8), |P (x) -P k (x)| ≤ ∞ j=k |P j+1 (x) -P j (x)| ≤ Cr 2+α k . Thus, for each x ∈ B + 3/4 we can fix k such that r k+1 < |x| ≤ r k and estimate |u(x) -P (x)| ≤ |u(x) -P k (x)| + |P k (x) -P (x)| ≤ Cr 2+α k .

Acknowledgments

Luis Silvestre was partially supported by NSF grants DMS-1254332 and DMS-1065979.

We easily obtain (3.15) by taking k such that γ k+1 < |x| ≤ γ k and appying the last inequality. Theorem 1.1 is a direct consequence of Theorem 3.5, taking G(x 0 ) = (D x g(x 0 ), A(x 0 )).

4 C 2,α regularity on the boundary for fully nonlinear equations

We will first prove Theorem 1.2 in the particular case of an autonomous equation and a solution which vanishes on a flat part of the boundary. The general case will be obtained later by an iterative perturbative procedure.

Lemma 4.1. Let u be a viscosity solution to the equation

and F satisfies (H1). Then there is a Hölder continuous function

In addition

. Here α = α(d, λ, Λ) and C depends on d, λ, Λ, and K.

Proof. Recall that u ∈ C 1,α (B 0 3/4 ) for some α > 0, see Theorem 1.4. Without loss of generality, we assume that u L ∞ (B + 1 ) = 1 (if not, set a = u L ∞ (B + 1 ) and replace u by u/a and F (M, p) by (1/a)F (aM, ap)).

For each i ∈ {1, . . . , d-1}, by (H1) the incremental quotient v h (x) = 1 h (u(x+he i )-u(x)) satisfies in B + 1-h the inequalities of Theorem 1.1, with L = 0. Since viscosity solutions are stable with respect to uniform convergence, the partial derivative u i = ∂ i u is also a solution of the same inequalities. Since u ≡ 0 on B 0 1 and B 0 1 is flat, we have ∂ i u ≡ 0 on B 0 1 . Thus, by applying Theorem 1.1 to ∂ i u, for each i = 1, . . . , d -1 and x 0 ∈ B 0 3/4 we obtain a quantity G i (x 0 ) which is a Hölder continuous function on B 0 3/4 , and

We now define

Note that by definition

By the comparison principle, w ≤ u + k everywhere in the box Q r . Now, if k > 0, this means that w(0, t) = u(0, t) + k for some t ∈ (0, r) -a contradiction with the strong maximum principle. On the other hand, if k = 0, we get a contradiction with the Hopf lemma, since ∂ d (u -w) = 0 at the origin. Thus, by translating the origin along B 0 2/3 , we have proved that for any

We now use that Du = u d ∈ C 1,α (B 0 3/4 ) and H ∈ C α (B 0 3/4 ), which implies

Then (4.1) with x 0 = 0 follows from plugging the last two inequalities into (4.5). Lemma 4.1 is proved.

On order to extend Lemma 4.1 to the general equation (1.1) we will use the following approximation result. Lemma 4.2. Assume (H1). Let u be a solution to

and u L ∞ (B + 1 ) ≤ 1. Let v be a solution to

Assume also that for some κ > 0

Then there exist γ = γ(d, λ, Λ) > 0, and C depending on d, λ, Λ, K, such that

Proof. By Proposition 2.6 the functions u and v are in C α (B + 3/4 ) for some α > 0, with C α -norms bounded by C u L ∞ (B + 1 ) ≤ C. We choose γ = α/2. Without restricting the generality (replacing, if necessary, B + 1 and B + 3/4 by B + r 0 and B + 3r 0 /4 , for some fixed r 0 depending only on d, λ, Λ, K), we can assume that v C α (B + 3/4 ) ≤ 1/2. By the Hölder regularity we clearly have |u -v| ≤ Cκ α/2 in B + 3/4 \ B + 3/4-κ 1/2 , and also in

We are left to prove the estimate in the remaining part of B + 3/4 , which we will call

Proof of Theorem 1.3. As in the previous proof, we can assume that g = 0, the boundary is flat, and we can write F (M, x) instead of F (M, p, x). From Theorem 1.2 we know that at any point x 0 ∈ ∂Ω there exists a second order polynomial P = P x 0 , which is Hölder continuous in x 0 and such that |u(x) -P (x)| ≤ C|x -x 0 | 2+α for some α > 0. Let x ∈ Ω δ . From the definition of Ω δ , there exists a point x 0 ∈ ∂Ω such that |x -

In [START_REF] Savin | Small perturbation solutions for elliptic equations[END_REF], Ovidiu Savin proved that solutions with sufficiently small oscillation are C 2,αsmooth. We will use the extensions of this result given in [1, Proposition 4.1] and [9, Theorem 1.2], which say that if F (M, x) is C 1 in M and |u(x) -P (x)| ≤ εr 2 in B r (x) for sufficiently small ε > 0, then u ∈ C 2,α (B r/2 (x)) (we replace u by u -P and F (M, x) by F (M + D 2 P, x)). This smallness assumption is satisfied if we choose δ such that C 1 δ α < ε. Hence u is C 2,α -smooth in the interior of Ω δ .

To put together this interior regularity result with the boundary result from Theorem 1.2 we repeat the proof of Proposition 2.4 in [START_REF] Milakis | Regularity for fully nonlinear elliptic equations with neumann boundary data[END_REF]. This proves that u ∈ C 2,α in a neighborhood of any point in Ω δ . The rest follows by an easy covering argument.


Boyan Sirakov
email: bsirakov@mat.puc-rio.br

Non uniqueness for the Dirichlet problem for fully nonlinear elliptic operators and the Ambrosetti-Prodi phenomenon

We study the uniformly elliptic fully nonlinear PDE F (D 2 u, Du, u, x) = f (x) in Ω, where F is a convex positively 1-homogeneous operator and Ω ⊂ R N is a regular bounded domain. We prove non-existence and multiplicity results for the Dirichlet problem, when the two principal eigenvalues of F are of different sign. Our results extend to more general cases, for instance, when F is not convex, and explain in a new light the classical results of Ambrosetti-Prodi type in elliptic PDE.

Introduction and Main Results

This paper is devoted to the study of the existence and the uniqueness of solutions of the Dirichlet boundary value problem

H(D 2 u, Du, u, x) = f (x) in Ω u = 0 on ∂Ω, (1)
where Ω ⊂ R N is a bounded domain with C 1,1 -boundary, f ∈ L ∞ (Ω), and H(M, p, u, x) is an uniformly elliptic fully nonlinear operator, globally Lipschitz in (M, p) and locally Lipschitz in u. A particular type of operators to which our results apply are Isaacs and Hamilton-Jacobi-Bellman operators. Boundary value problems of this type have been very extensively studied in the framework of classical, strong and viscosity solutions, see for example [START_REF] Lions | Résolution analytique des problèmes de Bellman-Dirichlet[END_REF], [START_REF] Evans | Classical solutions of fully nonlinear, convex, second-order elliptic equations[END_REF], [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF], [START_REF] Crandall | User's guide to viscosity solutions of second-order partial differential equations[END_REF], [START_REF] Caffarelli | On viscosity solutions of fully nonlinear equations with measurable ingredients[END_REF], [START_REF] Crandall | Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations[END_REF]. Most work on fully nonlinear problems concerns proper operators, that is, the case when H is nonincreasing in u.

Recently nonproper problems of type [START_REF] Amann | On the number of solutions of nonlinear equations in ordered Banach spaces[END_REF] have been studied in [START_REF] Quaas | Existence results for nonproper elliptic equations involving the Pucci operator[END_REF] and [START_REF] Quaas | Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators[END_REF], see also the references in these papers. The present work continues a study started in [START_REF] Quaas | Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators[END_REF]. For all M ∈ S N (R), p ∈ R N , define the extremal operators L -, L + by

L -(M, p) = M - λ,Λ (M ) -γ |p|, L + (M, p) = M + λ,Λ (M ) + γ |p|,
for some positive constants λ, Λ, γ. Here M + , M -denote the Pucci operators M + λ,Λ (M ) = sup A∈A tr(AM ), M - λ,Λ (M ) = inf A∈A tr(AM ), where A ⊂ S N denotes the set of matrices whose eigenvalues lie in the interval [λ, Λ].

We suppose that H in (1) satisfies the following hypothesis : for all M ∈ S N (R), p ∈ R N , u ∈ R, x ∈ Ω, and for some constants A 0 , c, δ, F (M, p, u, x) -A 0 ≤ H(M, p, u, x) ≤ L + (M, p)

+ c|u| + A 0 , (2)
where F (M, p, u, x) is some nonlinear operator, such that

          
L -(M, p) -δ|u| ≤ F (M, p, u, x) ≤ L + (M, p) + δ|u| F (tM, tp, tu, x) = tF (M, p, u, x) for t ≥ 0 F is convex in (M, p, u), F (M, 0, 0, x) ∈ C(S N (R) × Ω, R).

(

) 3
We assume that H is Lipschitz continuous and uniformly elliptic, in the following sense : for each R ∈ R there exists c R ∈ R such that for all

M, N ∈ S N (R), p, q ∈ R N , x ∈ Ω, u, v ∈ [-R, R], H(M, p, u, x) -H(N, q, v, x) ≥ L -(M -N, p -q) -c R |u -v| H(M, p, u, x) -H(N, q, v, x) ≤ L + (M -N, p -q) + c R |u -v|. (4)
Note that (3) implies ( 4) with H = F and c R = δ, see [START_REF] Quaas | Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators[END_REF] (or inequalities [START_REF] Arcoya | The Ambrosetti-Prodi problem for the p-Laplacian[END_REF] below). Our final standing assumption is that the proper operator

H v [u] := H(D 2 u, Du, v(x), x) -u (5)
satisfies the comparison principle for each v ∈ C(Ω), in the sense that if

H v [u 1 ] ≥ f ≥ H v [u 2 ] in Ω, and v 1 = v 2 = 0 on ∂Ω then v 1 ≤ v 2 in Ω.
This is satisfied for instance when H is Hölder continuous in x with a sufficiently large Hölder constant or when H is convex in M and H(M, 0, 0, x) is uniformly continuous. Many other conditions which ensure uniqueness for proper equations can be found in [START_REF] Crandall | User's guide to viscosity solutions of second-order partial differential equations[END_REF], [START_REF] Ishii | Viscosity solutions of fully nonlinear second-order elliptic partial differential equations[END_REF], [START_REF] Caffarelli | On viscosity solutions of fully nonlinear equations with measurable ingredients[END_REF], [START_REF] Jensen | Uniqueness and existence of maximal and minimal solutions of fully nonlinear PDE[END_REF].

For instance, F can be a Hamilton-Jacobi-Bellman (HJB) operator, that is, a supremum of linear second order operators with bounded coefficients and continuous second order coefficients -see [START_REF] Quaas | Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators[END_REF] for examples and discussions. HJB operators are basic in control theory. On the other hand, H can be an Isaacs operator, that is, a sup-inf of linear operators (these operators are essential in game theory). The Dirichlet problem for such operators has been widely studied in the proper case, and still many open question subsist, see the references above. Of course H can be a semilinear or quasilinear operator satisfying the hypotheses we made.

It was shown in [START_REF] Lions | Bifurcation and optimal stochastic control[END_REF], [START_REF] Berestycki | The principal eigenvalue and maximum principle for second order elliptic operators in general domains[END_REF], [START_REF] Quaas | Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators[END_REF] (see also [START_REF] Birindelli | Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators[END_REF], [START_REF] Armstrong | Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations[END_REF] for related results) that under hypothesis (3) F has two principal eigenvalues λ + 1 (F, Ω) ≤ λ - 1 (F, Ω), which correspond to a positive and a negative eigenfunction, such that (1) with H = F has a unique solution for all

f if λ + 1 > 0, while if λ - 1 > 0 ≥ λ + 1
then (1) has a solution for f ≥ 0 but (1) does not have solutions for f ≤ 0, f ≡ 0. The question of uniqueness in the last case was left open, since λ - 1 > 0 alone does not imply a comparison principle. It is this question that we address in the present article. We will show that uniqueness fails when only one of the two eigenvalues is positive.

We will use the following decomposition of the right-hand side f (x) in ( 1) [START_REF] Quaas | Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators[END_REF]. Since F 0 is proper, we have λ + 0 = λ + 1 (F 0 , Ω) > 0, see [START_REF] Quaas | Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators[END_REF]. Whenever we speak of a solution of (1) we shall mean a function in C(Ω) which satisfies [START_REF] Amann | On the number of solutions of nonlinear equations in ordered Banach spaces[END_REF] in the L N -viscosity sense. See [START_REF] Caffarelli | On viscosity solutions of fully nonlinear equations with measurable ingredients[END_REF] for definitions and properties of these solutions. Note that u ∈ W 2,N loc (Ω) ∩ C(Ω) satisfies (1) almost everywhere in Ω if and only if it is a L N -viscosity solution of (1).

f (x) = -tφ(x) + h(x), where t ∈ R, φ = ϕ + 1 (F 0 , Ω) is the first positive eigenfunction of the operator F 0 (M, p, x) = F (M, p, 0, x), normalized so that max Ω φ = 1. The existence of φ ∈ W 2,p loc (Ω) ∩ C(Ω), p < ∞, φ > 0 in Ω, F 0 (D 2 φ, Dφ, x) = -λ + 0 φ in Ω was shown in
Here is our main result.

Theorem 1 Suppose F and H verify (2)-( 5), and

λ + 1 (F, Ω) < 0 < λ - 1 (F, Ω). ( 6
)
Then for each h ∈ L ∞ (Ω) there exists a number t * (h) ∈ R such that:

(1) if t < t * (h) then (1) has at least two solutions ;

(2) if t = t * (h) then (1) has at least one solution ;

(3) if t > t * (h) then (1) has no solutions. The map

h → t * (h) is continuous from L ∞ (Ω) to R. Remark 1. If H(M, p, u, x) is convex in M then the solutions obtained in Theorem 1 belong to W 2,p loc (Ω) ∩ C(Ω)
, for all p < ∞. The acknowledged reader may have noticed that the conclusion in Theorem 1 is similar to results obtained in the framework of the so-called Ambrosetti-Prodi problem, classical in the theory of semilinear elliptic PDE's. We shall quote here the original work [START_REF] Ambrosetti | On the inversion of some differentiable mappings with singularities between Banach spaces[END_REF], as well as the subsequent developments [START_REF] Berger | On the solutions of a nonlinear Dirichlet problem[END_REF], [START_REF] Kazdan | Remarks on some quasilinear elliptic equations[END_REF], [START_REF] Dancer | On the ranges of certain weakly nonlinear elliptic partial differential equations[END_REF], [START_REF] Hess | On a nonlinear elliptic boundary value problem of the Ambrosetti-Prodi type[END_REF], [START_REF] De Figueiredo | A variational approach to superlinear elliptic problems[END_REF], [START_REF] Rynne | Half-eigenvalues of elliptic operators[END_REF], [START_REF] Chang | Ambrosetti-Prodi type results in elliptic systems[END_REF], [START_REF] De Figueiredo | On the Ambrosetti-Prodi problem for non-variational elliptic systems[END_REF]. Quasilinear operators were recently considered in [START_REF] Arcoya | Quasilinear elliptic problems interacting with its asymptotic spectrum[END_REF], [START_REF] Arcoya | The Ambrosetti-Prodi problem for the p-Laplacian[END_REF]. Here is the most typical Ambrosetti-Prodi type result : given the operator H L (M, p, u, x) = tr(A(x)M ) + b(x).p + g(x, u), if g is a Lipschitz function such that g(x, u) ≥ c 1 u + -c 2 u --c 0 , and if c 1 > λ 1 > c 2 , where λ 1 is the usual first eigenvalue of the linear operator L(M, p, x) = tr(A(x)M ) + b(x).p, then the same conclusion as in Theorem 1 holds for [START_REF] Amann | On the number of solutions of nonlinear equations in ordered Banach spaces[END_REF] with H = H L . Actually, this statement is nothing but Theorem 1 applied to H = H L and F = F L , where

F L (M, p, u, x) = tr(A(x)M ) + b(x).p + c 1 u + -c 2 u -.
Here u + = max{u, 0}, u -= -min{u, 0}, A ∈ C(Ω) is a positive definite matrix, b is a bounded vector, and

c 1 > c 2 . Then λ + 1 (F L , Ω) (resp. λ - 1 (F L , Ω)) is obviously equal to λ 1 -c 1 (resp. λ 1 -c 2 ).
In other words, and this is the second main conclusion of the paper, the Ambrosetti-Prodi phenomenon turns out to be due to nonuniqueness of solutions of the Dirichlet problem for a convex nonlinear operator with one positive and one negative principal eigenvalue. Remark 2. Many of the quoted papers on the Ambrosetti-Prodi problem contain results also for systems of equations or for the case when g(x, u) in H L does not have a linear but rather a power growth in u. Such extensions for fully nonlinear equations and systems of type (1) might be the subject of a future work. Remark 3. It is only a matter of technicalities to show the results extend to the case when h(x) in (1) and A 0 in (2) belong to L p (Ω), p > N .

The next section contains the proof of Theorem 1. Its overall scheme (that is, the statements of the steps of the proof) is similar to the classical one used to prove the Ambrosetti-Prodi type results quoted above. It combines Perron's method with a priori bounds and degree theory, see the next section for more details. Of course, the proofs of some steps are rather different, and require a specific nonlinear approach. We find it quite remarkable how naturally the theory of viscosity solutions and eigenvalues for fully nonlinear operators permit to carry out these proofs. We begin the next section by an overview. Acknowledgement. The author is indebted to F. Demengel for a number of remarks which lead to a substantial improvement of the paper. Remark. The results of this paper first appeared as a preprint in 2008. This preprint was later used a basis for the more detailed study of resonance problems in [START_REF] Felmer | Resonance phenomena for stochastic control equations[END_REF], as well as for the result of the same type in [START_REF] Demengel | Non uniqueness of solutions for Dirichlet problem related to fully nonlinear singular or degenerate operator[END_REF] on more general fully nonlinear operators modeled on |∇u| γ M + λ,Λ (D 2 u). Other related results can be found in [START_REF] Armstrong | The Dirichlet problem for the Bellman equation at resonance[END_REF], [START_REF] Felmer | Landesman-Lazer type results for second order Hamilton-Jacobi-Bellman equations[END_REF].

Proof of Theorem 1

From now on h ∈ L ∞ (Ω) will be fixed and we shall refer to (1) as problem (P t ) or (P t,h ), when we need to stress the dependence on t or h.

We first give the plan of the proof of Theorem 1.

1. prove an a priori upper bound on t, such that (P t ) has a solution ;

2. prove an a priori bound on u, for t ≥ -C ;

3. prove subsolutions of (P t ) exist for all t, supersolutions exist for sufficiently small t, deduce by Perron's method that solutions of (P t ) exist for t ∈ (-∞, t * ) ;

4. prove for each t ∈ (-∞, t * ) there exists a subsolution of (P t ) which is smaller than all solutions of (P t ) ;

use fixed point theorems and degree theory to conclude ;

Let us review the main points and the difficulties in the proofs. Steps 1 and 2 above are rather classical for operators in divergence form, that is, for cases when (1) has an equivalent formulation in terms of integrals. Then one can prove Step 1 by testing the equation with the first eigenfunction of F 0 and after that carry out a contradiction (blow-up) argument to obtain the statement in Step 2. This is not possible for operators in non-divergence form. Recently a different method was developed in [START_REF] De Figueiredo | On the Ambrosetti-Prodi problem for non-variational elliptic systems[END_REF], for the semilinear operators F L , H L , which gives a simultaneous proof of Steps 1 and 2, and which applies to operators with power growth in u. The proof in [START_REF] De Figueiredo | On the Ambrosetti-Prodi problem for non-variational elliptic systems[END_REF] depends on the linearity of L = F 0 . We will show here that it is actually the nonlinear structure of F and H, as described in our hypotheses, which provides for such a method to be applicable.

Further, Step 3 above is proved with the help of an one-sided Alexandrov-Bakelman-Pucci (ABP) inequality combined with an existence result, both obtained in [START_REF] Quaas | Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators[END_REF], for operators with only one positive principal eigenvalue, which we recall below.

Another important difference with the semilinear case appears in proving Step 4. If F = F L then it is automatic that the restriction of F L to the cone {(M, p, u, x) : u ≤ 0} satisfies a comparison principle in this cone (since F L is linear and coercive there). In the nonlinear case this is not clear ; however we manage to prove that subsolutions can be chosen to satisfy properties which permit to us to use a more restrictive comparison result, which we establish, based on the fraction rather than the difference between the two functions that we compare -see Lemma 2.5 and the comments there.

Finally, the multiplicity result (Step 5) relies on an argument which uses the properties of the Leray-Schauder degree of compact maps.

We next list several preliminary results, mostly from [START_REF] Quaas | Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators[END_REF]. It was shown in [START_REF] Quaas | Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators[END_REF] that hypothesis (3) implies

F (M -N, p -q, u -v, x) ≥ F (M, p, u, x) -F (N, q, v, x) F (M + N, p + q, u + v, x) ≤ F (M, p, u, x) + F (N, q, v, x), (7)
for all M, N ∈ S N (R), p, q ∈ R N , u, v ∈ R, x ∈ Ω.
We recall that the principal eigenvalues of F are defined by

λ + 1 (F, Ω) = sup {λ ∈ R | ∃ψ > 0 in Ω, F (D 2 ψ, Dψ, ψ, x) + λψ ≤ 0 in Ω}, λ - 1 (F, Ω) = sup {λ ∈ R | ∃ψ < 0 in Ω, F (D 2 ψ, Dψ, ψ, x) + λψ ≥ 0 in Ω}.
In the sequel we shall need the following one-sided ABP estimate, obtained in [START_REF] Quaas | Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators[END_REF]. A complete version of the Alexandrov-Bakelman-Pucci inequality for proper operators can be found in [START_REF] Caffarelli | On viscosity solutions of fully nonlinear equations with measurable ingredients[END_REF] (an ABP inequality for the Pucci operator was first proved in [START_REF] Caffarelli | Interior estimates for fully nonlinear elliptic equations[END_REF]). We recall that λ + 1 , λ - 1 are bounded above and below by constants which depend only on N, λ, Λ, γ, δ, Ω, and that both principal eigenvalues of any proper operator are positive, see [START_REF] Quaas | Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators[END_REF].

Theorem 2 ( [START_REF] Quaas | Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators[END_REF]) Suppose the operator F satisfies [START_REF] Ambrosetti | On the inversion of some differentiable mappings with singularities between Banach spaces[END_REF].

I. If λ - 1 (F, Ω) > 0 then for any u ∈ C(Ω), f ∈ L N (Ω), the inequality F (D 2 u, Du, u, x) ≤ f implies sup Ω u -≤ C(sup ∂Ω u -+ f + L N (Ω) ),
where C depends on Ω, N, λ, Λ, γ, δ.

II. In addition, if λ + 1 (F, Ω) > 0 then F (D 2 u, Du, u, x) ≥ f implies sup Ω u ≤ C(sup ∂Ω u + + f - L N (Ω) ). Set E p = W 2,p loc (Ω) ∩ C(Ω), p ≥ N .
We shall use the following existence result.

Theorem 3 ( [START_REF] Quaas | Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators[END_REF]) Suppose the operator F satisfies (3).

I. If λ - 1 (F, Ω) > 0 then for any f ∈ L p (Ω), p ≥ N , such that f ≥ 0 in Ω, there exists a solution u ∈ E p of F (D 2 u, Du, u, x) = f in Ω, u = 0 on ∂Ω, such that u ≤ 0 in Ω.
II. In addition, if λ + 1 (F, Ω) > 0 then for any f ∈ L p (Ω), p ≥ N , there exists a unique solution u ∈ E p of F (D 2 u, Du, u, x) = f in Ω, u = 0 on ∂Ω.

We will actually need to apply parts II in the above two theorems only to the proper operator F 0 (M, p, x) = F (M, p, 0, x).

We now move to the proof of Theorem 1. First we will show that solutions of (P t ) admit an a priori bound, which is uniform in t ∈ (m, ∞), for each m ∈ R. In the sequel C will denote a constant which may change from line to line and which depends on N, λ, Λ, γ, δ, A 0 , c, Ω, and h L ∞ (Ω) .

The next proposition realizes Steps 1 and 2 (see the beginning of this section) of the proof of Theorem 1.

Proposition 2.1 For each m 0 ∈ R + there exists a constant C such that for any t ≥ -m 0 and any solution u of (P t ) we have

u L ∞ (Ω) ≤ C and t ≤ C.
In particular, there do not exist solutions of (P t ) for large t.

Proof. We divide the proof in three steps. Claim 1. For each m 0 ∈ R + there exists a constant C = C(m 0 ) such that for any t ≥ -m 0 and any solution u of (P t ) with this t we have

u - L ∞ (Ω) ≤ C.
Proof. This is an immediate consequence of (2), [START_REF] Armstrong | The Dirichlet problem for the Bellman equation at resonance[END_REF], and Theorem 2 I with f replaced by m 0 φ + h.

Claim 2. There exists a constant C such that for solution u of (P t ) we have

t ≤ C(1 + u L ∞ (Ω) ).
Proof. By (2) and the definition of φ we have

F (D 2 u, Du, u, x) - t λ + 0 F (D 2 φ, Dφ, 0, x) ≤ h(x) + A 0 . (8)
By ( 7) and (3) we have (recall we have set

F 0 (M, p, x) = F (M, p, 0, x)) F (M, p, u, x) ≥ F (M, p, 0, x) -F (0, 0, -u, x) ≥ F 0 (M, p, x) -δ|u|.
Hence, by [START_REF] Arcoya | The Ambrosetti-Prodi problem for the p-Laplacian[END_REF], [START_REF] Bardi | On the strong maximum principle for fully nonlinear degenerate elliptic equations[END_REF], and the homogeneity of F

-F 0 D 2 -u + t λ + 0 φ , D -u + t λ + 0 φ , x ≤ h(x) + A 0 + δ|u|.
Then the second part of Theorem 2 implies that for all x ∈ Ω -u(x)

+ t λ + 0 φ(x) ≤ C h(x) + A 0 + δ|u| L ∞ (Ω) .
Taking x such that φ(x) = max Ω φ = 1 finishes the proof of Claim 2.

Conclusion.

Suppose the a priori bound on u in the statement of Proposition 2.1 is false, that is, there exist sequences

{t n }, {u n } such that t n ≥ -m 0 , u n L ∞ (Ω) → ∞, and
H(D 2 u n , Du n , u n , x) = -t n φ + h.
By ( 2), (3) and Claim 2 we have

L -(D 2 u n , Du n ) ≤ δ u n L ∞ (Ω) + m 0 + A 0 + h L + (D 2 u n , Du n ) ≥ -C(1 + u n L ∞ (Ω) ) + h. Hence, setting v n = u n / u n (so that v n L ∞ (Ω) = 1), L -(D 2 v n , Dv n ) ≤ C and L + (D 2 v n , Dv n ) ≥ -C.
We now use the following result from the general theory of viscosity solutions of fully nonlinear PDE (it is a particular case, for instance, of Proposition 4.2 in [START_REF] Crandall | Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations[END_REF]).

Proposition 2.2 For any given

M ∈ R the set of functions u ∈ C(Ω) such that u L ∞ (Ω) ≤ M, L -(D 2 u, Du) ≤ M, and L + (D 2 u, Du) ≥ -M is precompact in C(Ω).
Hence a subsequence of {v n } converges uniformly to a function v in Ω. Note that v ≥ 0 in Ω, by Claim 1, and v L ∞ (Ω) = 1.

Again by ( 2)

F (D 2 u n , Du n , u n , x) ≤ m 0 + A 0 + h, so, by the homogeneity of F F (D 2 v n , Dv n , v n , x) ≤ o(1)
.

By viscosity solutions theory (see Theorem 3.8 in [START_REF] Caffarelli | On viscosity solutions of fully nonlinear equations with measurable ingredients[END_REF]) we can pass to the limit in this inequality, obtaining

F (D 2 v, Dv, v, x) ≤ 0. ( 9
)
We recall the following strong maximum principle (Hopf lemma), a consequence from the results in [START_REF] Bardi | On the strong maximum principle for fully nonlinear degenerate elliptic equations[END_REF].

lim inf t 0 w(x 0 + tν) -w(x 0 ) t > 0,
where ν is the interior normal to ∂O at x 0 .

Therefore v > 0 in Ω. Using [START_REF] Berestycki | The principal eigenvalue and maximum principle for second order elliptic operators in general domains[END_REF], the existence of such function contradicts the definition of λ + 1 (F, Ω) and the hypothesis λ + 1 (F, Ω) < 0. Hence u L ∞ (Ω) is bounded, and, by Claim 2, t is bounded as well.

We turn to existence of subsolutions and supersolutions of (P t ). We shall need the following boundary Lipschitz estimate for fully nonlinear equations (for a proof see Proposition 4.9 in [START_REF] Quaas | Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators[END_REF]). Proposition 2.4 Suppose H satisfies (4) and Ω satisfies an uniform exterior sphere condition. Suppose u ∈ C(Ω) satisfies H(D 2 u, Du, u, x) = h, u = 0 on ∂Ω, where h ∈ L ∞ (Ω). Then there exists a constant k depending on N, λ, Λ, γ, δ, diam(Ω), u L ∞ (Ω) , h L ∞ (Ω) , and the radius of the exterior spheres, such that for each

x 0 ∈ ∂Ω |u(x)| ≤ k|x -x 0 | for each x ∈ Ω.
First we deal with the existence of supersolutions.

Lemma 2.1 There exists t 0 ∈ R, depending on the constants in ( 2)-( 4) and on h L ∞ (Ω) , such that for each t ≤ t 0 there exists a supersolution u of (P t ),

such that u ≥ 0 in Ω, u ∈ E p , p < ∞.
Proof. Let u be the unique solution of the Dirichlet problem (see Theorem 3 above, or Corollary 3.10 in [START_REF] Caffarelli | On viscosity solutions of fully nonlinear equations with measurable ingredients[END_REF])

L + (D 2 u, Du) = -h -(x) in Ω u = 0 on ∂Ω, (10)
The ABP inequality shows that u ≥ 0 in Ω, u L ∞ (Ω) ≤ C and u satisfies the boundary inequality in Proposition 2.4. On the other hand, the Hopf lemma and the inequality F 0 (D 2 φ, Dφ, x) ≤ 0 imply that there exists a constant α > 0 such that for all x 0 ∈ ∂Ω

lim inf t 0 φ(x 0 + tν) -φ(x 0 ) t ≥ α,
where ν is the inner normal to ∂Ω. Therefore there exists t 0 < 0 such that -t 0 φ ≥ δu in Ω. Hence by [START_REF] Amann | Fixed point equations and elliptic eigenvalue problems in ordered Banach spaces[END_REF] we have H(D 2 u, Du, u, x) ≤ -tφ + h, for all t ≤ t 0 , which is the required result.

The next lemma concerns the existence of subsolutions.

Lemma 2.2 For any t ∈ R there exists a subsolution u ≤ 0 in Ω, u ∈ E p , p < ∞, of (P t ). In addition, given a compact interval I ⊂ R, u can be chosen so that u ≤ u in Ω, for all solutions u of (P t ), t ∈ I.

The difficulty in Lemma 2.2 is in the second statement. As a step in its proof, we will obtain the following uniform boundary Hopf Lemma, which is of independent interest.

Lemma 2.3

Assume Ω satisfies an uniform interior sphere condition. Suppose F satisfies (3), λ - 1 (F, Ω) > 0, and f ≡ 0, 0 ≤ f ≤ M in Ω. Then there exists α 0 > 0 depending only on λ, Λ, ν, δ, Ω, and M , such that for any solution of F (D 2 u, Du, u, x) = f in Ω, u ≤ 0 in Ω, u = 0 on ∂Ω, and all x 0 ∈ ∂Ω we have

V x 0 (u) := lim inf t 0 u(x 0 ) -u(x 0 + tν) t ≥ α 0 .
Proof. Suppose the lemma is false, that is, there is a sequence of solutions u n ≤ 0 in Ω and points x n ∈ ∂Ω (we can suppose

x n → x ∈ ∂Ω) such that V xn (u n ) → 0. Note that u n L ∞ (Ω) ≤ C, by Theorem 2. From (3) we have L -(D 2 u n , Du n ) ≤ C and L + (D 2 u n , Du n ) ≥ -C.
By Proposition 2.2 a subsequence of {u n } converges uniformly to a function u in Ω, and F (D 2 u, Du, u, x) = f in Ω. Note that, by the strong maximum principle, u n < 0 and u < 0 in Ω (since f ≡ 0 excludes u n ≡ 0 or u ≡ 0).

By (3) and properties of Pucci operators (M

-(M ) = -M + (-M )), the positive functions v n = -u n satisfy M - λ,Λ (D 2 v n ) -ν|Dv n | -δv n ≤ 0 (11)
in Ω. Let ρ be the radius of the interior spheres. Fix p ∈ ∂Ω and let B ρ ⊂ Ω be a ball tangent to ∂Ω at p. Introduce the (standard) barrier function, defined in B ρ ,

z(r) = e -βr 2 -e -βρ 2 ,
where r is the distance to the center of B ρ and β is a positive constant yet to be chosen. We recall the following fact. Using this lemma and the fact that

M - λ,Λ (M ) = λ {e i >0} e i + Λ {e i <0} e i , M + λ,Λ (M ) = Λ {e i >0} e i + λ {e i <0} e i ,
where e i denote the eigenvalues of M , an elementary computation shows that

M - λ,Λ (D 2 z) -ν|Dz| -δz ≥ 0 (12)
in the annulus

B ρ \ B ρ/2 , if β = β(ρ) is chosen sufficiently large. Let the point q n ∈ ∂B ρ/2 be such that v n (q n ) = min ∂B ρ/2 v n and set σ n = v n (q n ) e -β(ρ/2) 2 -e -βρ 2 . Then σ n z ≤ v n on ∂(B ρ \ B ρ/2
) and, by the comparison principle for proper operators (see [START_REF] Caffarelli | On viscosity solutions of fully nonlinear equations with measurable ingredients[END_REF] or [START_REF] Quaas | Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators[END_REF], note that the operator which appears in [START_REF] Birindelli | Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators[END_REF], [START_REF] Busca | Harnack type estimates for nonlinear elliptic systems and applications[END_REF] is proper),

σ n z ≤ v n in B ρ \ B ρ/2 . Hence σ n ∂z ∂ν (p) ≤ -V p (v n ) = V p (u n ),
which implies min

∂B ρ/2 v n ≤ a 0 V p (u n )
for some a 0 > 0, which depends on the appropriate quantities, and for all p ∈ ∂Ω. Therefore, there exists a sequence of points y n ∈ Ω such that dist(y n , ∂Ω) ≥ ρ/2 and v n (y n ) → 0. Hence there exists a point y ∈ Ω such that v(y) = 0, a contradiction.

Proof of Lemma 2.2. Set

M = A 0 + sup t∈I -tφ + h L ∞ (Ω)
. By Theorem 3, there exists a solution u < 0 in Ω of F (D 2 u, Du, u, x) = M in Ω, u = 0 on ∂Ω. Hence u is a subsolution of (P t ) for t ∈ I, by [START_REF] Amann | Fixed point equations and elliptic eigenvalue problems in ordered Banach spaces[END_REF].

Next, note that if u is a solution of (P t ) for some t ∈ I, then both functions ψ = u and ψ = 0 are solutions of the inequality

F (D 2 ψ, Dψ, ψ, x) ≤ F (D 2 u, Du, u, x).
Since -u -= min{u, 0} and the minimum of two viscosity supersolutions is a viscosity supersolution, we have

F (D 2 (-u -), D(-u -), -u -, x) ≤ F (D 2 u, Du, u, x).
Observe we cannot directly infer from this inequality that u ≤ -u -≤ u since F does not satisfy a comparison principle (λ + 1 (F, Ω) < 0). However, as we will show now, we can gain enough information on these functions in order to prove the inequality by considering their quotient instead of their difference.

By Proposition 2.4 and Lemma 2.3 we can fix k sufficiently large so that for any solution u of (P t ), t ∈ I, and any x 0 ∈ ∂Ω we have lim sup

t 0 -u -(x 0 + tν) ku(x 0 + tν) ≤ 1 4
Note that ku is a subsolution of (P t ) for k ≥ 1 and t ∈ I, by ( 2) and ( 3). Fix a solution u of (P t ), t ∈ I. Then there exists d > 0 sufficiently small, so that, setting Ω d = {x ∈ Ω : dist(x, ∂Ω) > d}, we have

0 < w := -u - ku ≤ 1 2 in Ω \ Ω d .
The proof of Lemma 2.2 is finished with the help of the following comparison result.

Lemma 2.5 Suppose v 1 , v 2 are such that v 1 ≤ 0, v 2 < 0 in Ω, v 2 ∈ E p , p < ∞, F (D 2 v 1 , Dv 1 , v 1 , x) ≤ F (D 2 v 2 , Dv 2 , v 2 , x) in Ω, ( 13
) 0 < F (D 2 v 2 , Dv 2 , v 2 , x) in Ω, (14)
and, for some d > 0,

w := v 1 v 2 < 1 2 in Ω \ Ω d . Then v 1 > v 2 in Ω.
Remark. Considering the quotient rather than the difference of two functions can often be a successful technique when proving comparison results for a nonlinear operator. For fully nonlinear equations this has been used, in a different setting, for instance in [START_REF] Birindelli | Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators[END_REF].

Proof of Lemma 2.5. For any two vectors p, q ∈ R N we denote the symmetric tensorial product by p ⊗ q = 1 2 (p i q j + p j q i ) N i,j=1 ∈ S N . By replacing v 1 by wv 2 in (13) and by using [START_REF] Arcoya | The Ambrosetti-Prodi problem for the p-Laplacian[END_REF] and the homogeneity of F we get

wF (Dv 2 , Dv 2 , v 2 , x) + v 2 F (D 2 w + 2 Dv 2 v 2 ⊗ Dw, Dw, 0, x) (15)
= F (wDv 2 , wDv 2 , wv 2 , x) -

F (-v 2 D 2 w -2Dv 2 ⊗ Dw, -v 2 Dw, 0, x) ≤ F (D 2 v 1 , Dv 1 , v 1 , x) ≤ F (D 2 v 2 , Dv 2 , v 2 , x),
where we have used the equality

D 2 (u 1 u 2 ) = u 1 Du 2 + 2Du 1 ⊗ Du 2 + u 2 Du 1 ,
valid for u 1 , u 2 ∈ E p . In case u 1 is only continuous, we use test functions in E p to prove (15) -this is very standard, so we shall omit it. We obtain from ( 15)

F (D 2 (w -1), D(w -1), x) + c(x)(w -1) ≥ 0 in Ω d/2 , (16)
where we have set

F (M, p, x) = F (M + 2b(x) ⊗ p, p, 0, x), b(x) = Dv 2 (x) v 2 (x) ∈ L ∞ (Ω d/2 ), c(x) = F (D 2 v 2 (x), Dv 2 (x), v 2 (x), x) v 2 (x) < 0,
by [START_REF] Caffarelli | On viscosity solutions of fully nonlinear equations with measurable ingredients[END_REF]. Note that w-1 < 0 in a neighbourhood of ∂Ω d/2 . Then the existence of a point in Ω d/2 at which w-1 attains a positive maximum would contradict (16) -just test [START_REF] Chang | Ambrosetti-Prodi type results in elliptic systems[END_REF] with a constant function. So w-1 ≤ 0. Finally, w-1 < 0 is a consequence of the strong maximum principle.

The following existence result is an easy consequence from the previous lemmas.

Proposition 2.5 There exists a number t * such that problem (P t ) has a solution for t ≤ t * and does not have a solution for t > t * .

Proof. We use the following lemma which is based on the Perron's method. This type of result in the viscosity setting goes back at least to Ishii [START_REF] Ishii | On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs[END_REF] recall we assumed in the introduction that the operator H v [u] satisfies the comparison principle. Lemma 2.6 Suppose u 0 ∈ E N is a subsolution and v 0 ∈ E N is a supersolution of H(D 2 u, Du, u, x) = f , where f ∈ L ∞ (Ω), H satisfies (4). Suppose in addition that u 0 ≤ v 0 in Ω, u 0 ≤ 0 on ∂Ω, and v 0 ≥ 0 on ∂Ω. Then there exists a solution u of

H(D 2 u, Du, u, x) = f in Ω u = 0 on ∂Ω.
For a proof of this lemma see for example Lemma 4. Now we can move to the realization of Step 5 of the proof of Theorem 1. The argument which follows is inspired by a classical reasoning of Amann [START_REF] Amann | On the number of solutions of nonlinear equations in ordered Banach spaces[END_REF], [START_REF] Amann | Fixed point equations and elliptic eigenvalue problems in ordered Banach spaces[END_REF]. We refer for instance to [START_REF] Chang | Methods in nonlinear analysis[END_REF] for a systematic treatment of existence results based on degree theory.

In what follows we shall use the following global C 1,α -estimate, proved in [START_REF] Trudinger | On regularity and existence of viscosity solutions of nonlinear second order, elliptic equations[END_REF], [START_REF] Świech | W 1,p -estimates for solutions of fully nonlinear uniformly elliptic equations[END_REF], [START_REF] Winter | W 2,p and W 1,p -estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations, to appear in[END_REF].

Theorem 4 Suppose H satisfies (4), Ω is a C 1,1 -domain and u is a solution of (1). Then there exists α, C 0 > 0 depending on N, λ, Λ, γ, δ, Ω, such that u ∈ C 1,α (Ω), and

u C 1,α (Ω) ≤ C 0 u L ∞ (Ω) + f L ∞ (Ω) .
Let t 1 be such that there exists a solution u for (P t 1 ). Fix t < t 1 . Then u is a strict supersolution of (P t ). By Lemma 2.2 there is a subsolution u of (P t ) such that u < u in Ω. By the choice of u, u and Hopf's lemma, we can also ensure that ∂u ∂ν < ∂u ∂ν on ∂Ω. Let c R 0 is the constant from hypothesis (4), with

R 0 = max{ u L ∞ (Ω) , u L ∞ (Ω) }.
For any v ∈ C(Ω) we define H v (M, p, x) = H(M, p, v(x), x). For each v ∈ C(Ω) we denote with u = K t (v) the solution of the Dirichlet problem

H v (D 2 u, Du, x) -c R 0 u = f (x) -c R 0 v in Ω u = 0 on ∂Ω. (17)
This problem has a unique solution, by hypothesis (5), Theorem 3, and Perron's method. By the ABP inequality K t maps bounded sets in C(Ω) into bounded sets in C(Ω). Hence, by Proposition 2.2 and Theorem 4 (recall C 1,α (Ω) → C 1 (Ω) is compact) the map K t sends bounded sets in C 1 (Ω) into precompact sets in C 1 (Ω), that is, K t : C 1 (Ω) → C 1 (Ω) is a compact map. Note that solutions of (1) are fixed points of K t and vice versa. Define To prove this claim it is sufficient to show that if u ≤ v ≤ u in Ω then u < K t (v) < u in Ω and ∂u ∂ν < ∂Kt(v) ∂ν < ∂u ∂ν on ∂Ω. So let v ∈ C(Ω) be such that u ≤ v ≤ u and set u = K t (v). Then we have, by ( 4 This implies, again by (4),

O = {v ∈ C 1 (Ω) : u < v < u
L + (D 2 (u -u), D(u -u)) -c R (u -u) > 0
in Ω, and u -u = 0 on ∂Ω. It follows from the maximum principle for proper operators (or from Theorem 2) and from the strong maximum principle that u < u in Ω and ∂u ∂ν < ∂u ∂ν on ∂Ω. In the same way we obtain the inequality for u.

To finish the proof of our main theorem we shall use the following lemma, concerning the Leray-Schauder degree of the compact map I -K t . It is wellknown how to prove this type of result, we give a proof for completeness. Hence v n is a positive supersolution of (P t * n +ε,hn ) which implies that this problem has a solution as well (we know subsolutions always exist). This is a contradiction with the definition of t * n .

Proposition 2 . 3 (

 23 [START_REF] Bardi | On the strong maximum principle for fully nonlinear degenerate elliptic equations[END_REF]) Let O ⊂ R N be a regular domain and let γ ∈ R,δ ≤ 0. Suppose w ∈ C(O) is a viscosity solution of M - λ,Λ (D 2 w) -γ|Dw| -δw ≤ 0 in O,and w ≥ 0 in O. Then either w ≡ 0 in O or w > 0 in O and at any point x 0 ∈ ∂O at which w(x 0 ) = 0 we have

Lemma 2 . 4

 24 Suppose u ∈ C 2 (B) is a radial function, defined on a ball B, say u(x) = g(|x|). Then the matrix D 2 u(x) has g (|x|) as a simple eigenvalue, and |x| -1 g (|x|) as an eigenvalue of multiplicity N -1.

  ),H(D 2 u, Du, u(x), x) = H(D 2 u, Du, u(x), x) + c R u -c R u ≥ H(D 2 u, Du, v(x), x) + c R v -c R u = f (x) + c R u -c R u > H(D 2 u, Du, u(x), x) + c R (u -u).

Lemma 2 . 7

 27 For any t 0 ∈ (-∞, t * ) there exist R 1 , R 2 ∈ R such that R 1 < R 2 and deg(I -K t 0 , O ∩ B R 1 , 0) = 1 and deg(I -K t 0 , B R 2 , 0) = 0, (18)whereB R = {u ∈ C 1 (Ω) : u C 1 (Ω) < R}.By the ABP inequality and the boundary estimate (Theorem 2 and Proposition 2.4), we have w n ⇒ 0 and c R |w n | ≤ εφ in Ω for large n, where c R is the constant from (4), with R = u L ∞ (Ω) + 1.Set v n = u + w n . Then, by (4), if n is sufficiently large,H(D 2 v n , Dv n , v n , x) ≤ H(D 2 v n , Dv n , v n , x) -H(D 2 u, Du, u, x) -(a + 3ε)φ + h ≤ L + (D 2 w n , Dw n ) + c R w n -(t * n + 2ε)φ + h ≤ -(t * n + ε)φ + h n ,

  It follows from Lemmas 2.6 and 2.2 that if for some t problem (P t ) has supersolution then it has a solution. It is obvious that if u is a supersolution for (P t 0 ) then it is also a supersolution for all (P t ), t < t 0 . By Lemma 2.1 t * is well defined and by Proposition 2.1 t

	3 in [37].
	Next, set
	t

* = sup{t ∈ R : (P t ) has a supersolution }. * is finite. The existence of solution for t = t * follows from a passage to the limit t n → t * , thanks to Proposition 2.2 and Theorem 3.8 in

[START_REF] Caffarelli | On viscosity solutions of fully nonlinear equations with measurable ingredients[END_REF]

.

  Note the setO is open in C 1 (Ω). Claim. K t (O) ⊂ O. In particular, K t (O) ∩ ∂O = ∅.

	in Ω and	∂u ∂ν	<	∂v ∂ν	<	∂u ∂ν	on ∂Ω}.

Proof. Let R be an upper bound (given by Theorem 4) for C 1 (Ω)-norms of solutions of [START_REF] Crandall | Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations[END_REF]

To prove the first equality in [START_REF] Crandall | L p theory for fully nonlinear uniformly parabolic equations[END_REF], fix w ∈ O ∩ B R 1 and consider the compact homotopy

By the choice of R 1 and the claim above we have (I -

By combining Proposition 2.1 with Theorem 4 we see that for each m 0 there exists an uniform bound C(m 0 ) for the C 1 (Ω)-norms of the solutions of (P t ) with t ≥ m 0 . Then we take

But the last degree is zero, since K t 1 has no fixed points at all, by Proposition 2.5. This proves the second equality in [START_REF] Crandall | L p theory for fully nonlinear uniformly parabolic equations[END_REF].

So, to complete the proof of the multiplicity result in Theorem 1 we can use the excision property of the degree together with Lemma 2.7, which leads to deg(I -K t 0 , B R 2 \ (O ∩ B R 1 ), 0) = -1, hence problem (1) (i.e. problem (P t 0 )) has a second solution in B R 2 \ (O ∩ B R 1 ), apart from the solution in O ∩ B R 1 , given by Proposition 2.5.

Finally, let us show the mapping

for large n, since any solution of (1) with h replaced by -h L ∞ (Ω) -1 is a supersolution of (P t * n ,hn ). So t * n is bounded. Take a subsequence of t * n and let a be the limit of some subsequence of this subsequence (which we denote by t * n again). Let u n be a solution of (P t * n ,hn ) (we already know such a solution exists). By Proposition 2.1 {u n } is bounded in L ∞ (Ω). Hence, by the equation satisfied by u n , (4) and Proposition 2.2, some subsequence of u n converges to a solution of (P a,h ). Hence a ≤ t * . Suppose a < a+3ε < t * , for some ε > 0. Let u be a positive supersolution of (P a+3ε,h ) -we already know such supersolutions exist. Let w n be the solution of the Dirichlet problem

w n = 0 on ∂Ω.


L Dupaigne
email: louis.dupaigne@math.cnrs.fr

A Farina

B Sirakov

Regularity of the extremal solutions for the Liouville system

In this short note, we study the smoothness of the extremal solutions to the following system of equations:

(1)

     -∆u = µe v
in Ω,

-∆v = λe u in Ω, u = v = 0 on ∂Ω,
where λ, µ > 0 are parameters and Ω is a smoothly bounded domain of R N , N ≥ 1. As shown by M. Montenegro (see [START_REF] Montenegro | Minimal solutions for a class of elliptic systems[END_REF]), there exists a limiting curve Υ in the first quadrant of the (λ, µ)-plane serving as borderline for existence of classical solutions of [START_REF] Cowan | Regularity of the extremal solutions in a Gelfand systems problem[END_REF]. He also proved the existence of a weak solution u * for every (λ * , µ * ) on the curve Υ and left open the question of its regularity. Following standard terminology (see e.g. the books [START_REF] Dupaigne | Stable solutions of elliptic partial differential equations[END_REF], [START_REF] Esposito | Mathematical analysis of partial differential equations modeling electrostatic MEMS[END_REF] for an introduction to this vast subject), u * is called an extremal solution. Our result is the following.

Theorem 1 Let 1 ≤ N ≤ 9. Then, extremal solutions to (1) are smooth.

Remark 2 C. Cowan ([1]

) recently obtained the same result under the further assumption that (N -2)/8 < λ/µ < 8/(N -2).

Any extremal solution u * is obtained as the increasing pointwise limit of a sequence of regular solutions (u n ) associated to parameters (λ n , µ n ) = (1 -1/n)(λ * , µ * ). In addition, see [START_REF] Montenegro | Minimal solutions for a class of elliptic systems[END_REF], u n is stable in the sense that the principal eigenvalue of the linearized operator associated to (1) is nonnegative. In other words, there exist λ 1 ≥ 0 and two positive functions

ϕ 1 , ψ 1 ∈ C 2 (Ω) such that (2)      -∆ϕ 1 -g (v)ψ 1 = λ 1 ϕ 1 in Ω, -∆ψ 1 -f (u)ϕ 1 = λ 1 ψ 1 in Ω. ϕ 1 = ψ 1 = 0 on ∂Ω, dfs2.tex November 27, 2012 2
where, in the context of (1), g(v) = µe v and f (u) = λe u . This motivates the following useful inequality. Let f, g denote two nondecreasing C 1 functions and consider the more general system (3)

     -∆u = g(v)
in Ω,

-∆v = f (u) in Ω, u = v = 0 on ∂Ω. Lemma 3 Let N ≥ 1 and let (u, v) ∈ C 2 (Ω) 2 denote a stable solution of (3).
Then, for all ϕ ∈ C 1 c (Ω), there holds

(4) Ω f (u)g (v)ϕ 2 dx ≤ Ω |∇ϕ| 2 dx
Remark 4 As we just learnt, the same inequality has been obtained independently by C. Cowan and N. Ghoussoub. See [START_REF] Cowan | Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains[END_REF].

Proof. Since (u, v) is stable, there exist λ 1 ≥ 0 and two positive functions

ϕ 1 , ψ 1 ∈ C 2 (Ω) solving (2). Given ϕ ∈ C 1 c (Ω)
, multiply the first equation in (2) by ϕ 2 /ϕ 1 and integrate. Then, ( 5)

Ω g (v) ψ 1 ϕ 1 ϕ 2 dx ≤ Ω ϕ 2 ϕ 1 (-∆ϕ 1 ) = - Ω |∇ϕ 1 | 2 ϕ ϕ 1 2 + 2 Ω ϕ ϕ 1 ∇ϕ∇ϕ 1 = - Ω ϕ ϕ 1 ∇ϕ 1 -∇ϕ 2 + Ω |∇ϕ| 2 ≤ Ω |∇ϕ| 2 .
Working similarly with the second equation, we also have

(6) Ω f (u) ϕ 1 ψ 1 ϕ 2 dx ≤ Ω |∇ϕ| 2 dx
(4) then follows by combining the Cauchy-Schwarz inequality and ( 5)- [START_REF] Montenegro | Minimal solutions for a class of elliptic systems[END_REF].

Thanks to the inequality (4), we obtain the following estimate.

Lemma 5 Let N ≥ 1. There exists a universal constant C > 0 such that any stable solution of (1) satisfies

(7) e u+v dx ≤ C |Ω| λ µ + µ λ .
Proof. Multiply the second equation in ( 1) by e v -1 and integrate.

λ .

Ω e u+v dx ≥ λ Ω e u (e v -1)dx = Ω ∇v∇(e v -1) dx (8) = 4 Ω ∇(e v/2 -1)
Applying once more (14), we obtain the desired estimate.

We can now prove Theorem 1.

Step 1. Case 1 ≤ N ≤ 3. It is enough to treat the case N = 3, the cases N = 1, 2 being easier. By ( 8) and ( 7), e v/2 -1 is bounded in H 1 0 (Ω) (with a uniform bound with respect to λ and µ). By the Sobolev embedding, it follows that e v is bounded in

L N N -2 (Ω). By (8) and elliptic regularity, u is bounded in W 2, N N -2 . For N = 3, N N -2 > N 2
. By Sobolev's embedding, we deduce that u is bounded, and so must be v. This implies the desired conclusion for the corresponding extremal solution.

Step 2. General case. We adapt a method introduced in [START_REF] Dupaigne | The Gel'fand problem for the biharmonic operator[END_REF]. Fix α > 1/2 and multiply the first equation in ( 1) by e αu -1. Integrating over Ω, we obtain Using (7), we deduce that ( 21)

Ω e α+1 2 u e v 2 dx ≤ ε 2 µ λ Ω e αu e v dx + 1 2ε λ µ C |Ω| λ µ + µ λ .
where C is the universal constant of lemma 3.

λµ X ≤ α 4 + ε µ X 2α-1 2α Y 1 2α + C λ ε |Ω| λ µ + µ λ .
By symmetry, we also have

λµ Y ≤ α 4 + ε λ Y 2α-1 2α X 1 2α + C µ ε |Ω| λ µ + µ λ .
Multiplying these inequalities, we deduce that

1 - α 4 + ε 2 X Y ≤ C 1 λ µ + µ λ 2 1 + X 2α-1 2α Y 1 2α + Y 2α-1 2α X 1 2α
.

where C 1 > 0 depends on Ω, α, only. Hence, for every α < 4, either X or Y must be bounded (with a uniform bound with respect to λ and µ). Without loss of generality, λ ≥ µ and by the maximum principle, v ≥ u. It follows that e u is bounded in L p (Ω) for every p = α + 1 < 5. Using standard elliptic regularity, the result follows.

Remark. O. Goubet informed us that J. Dávila and him obtained a simpler proof of Theorem 1.

3 Using ( 4 )Ω e u+v dx ≥ 4 λµ Ω e u+v 2 (e v/ 2 - 1 2 e 1 √ 2 2 eΩ e u+v dx 2 + 2 Ω e u+v 2 dx 2 ,e v dx ≥ Ω e u+v dx 2 .

 34221212222222 with test function ϕ = e v/2 -1, it follows thatλ v dx -8 λµ Ω e u+v 2 e v/2 dx. By Young's inequality, e v/2 = v dx.Using Young's inequality, the left-hand side in the above inequality is bounded above by(13) 2λµ C(λ + µ)where C is a universal constant. In addition, by the Cauchy-Schwarz inequality, Plugging (14) in (13) and remembering that (13) is an upper bound of the left-hand side in (12)

e αu e v dx + 2 λµ Ω e α+1 2 u e v 2 dx

 2 -1) e v dx = α Ω e αu |∇u| 2 dx = Let us estimate the terms on the right-hand side. By Hölder's inequality,