A Neumann eigenvalue problem for fully nonlinear operators

In this paper we study the asymptotic behavior of the principal eigenvalues associated to the Pucci operator in bounded domain $\Omega$ with Neumann/Robin boundary condition i.e. $\partial_n u=\alpha u$ when $\alpha$ tends to infinity. This study requires Lipschitz estimates up to the boundary that are interesting in their own rights.


Introduction
In this introduction and in the rest of the paper we quote some works of Louis Nirenberg that are used explicitly in order to give the right definitions and to prove the results; but the influence of his research, here and in all the papers both the authors have written, goes well beyond the citations. His mathematical ideas have been very important for us, specially for the first named author, but his teaching of how to approach mathematical problems has been as important. We are happy to have this opportunity to thank him for his generosity.
In this paper, for Ω a C 2 bounded domain of R n and for any α > 0, we consider the eigenvalue problem: It is useless to emphasize the importance of the concept of eigenvalue for the understanding of the structural properties of the solutions both for linear and non linear equations. The pioneering work of Berestycki, Nirenberg and Varadhan [4] has open the way to enlarge this fundamental concept to non linear operators. Indeed, even if they treat linear equations, their theory is very well adapted to fully nonlinear operators and viscosity solutions being based primarily on the use of the maximum principle. This has been done by many and in many different contests, let us mention the works of Armstrong, Busca, Demengel, Juutinen, Ishii, Quaas, Sirakov, Yoshimura and the authors of this note ( [1,5,6,11,12,16,17]). It should be mentioned that P.-L. Lions in [13], with a completely different approach, first introduces what he called demi-eigenvalues. Indeed when the operator is not odd with respect to the Hessian (as is the case of the Pucci operators), eigenvalues corresponding to positive eigenfunctions or to negative eigenfunctions may not coincide and one could interpret these two eigenvalues as a "splitting" of the eigenvalue.
The eigenvalue problem for Robin boundary conditions associated with a fully-nonlinear operator was already treated in [16]. The novelty here is that we consider α > 0 which is the "wrong sign" in the sense that the boundary conditions are not "proper", see e.g. [8]. The boundary source and the reaction-diffusion equation are somehow in competition.
In analogy to [4] we define the eigenvalues in the following way: λ + α := sup{λ ∈ R | ∃ v > 0 on Ω bounded viscosity supersolution of The first step is to prove that there exists u + α > 0 and u − α < 0 solutions of (1) when respectively λ = λ + α and λ = λ − α (Proposition 4.1). We shall also prove that below these eigenvalues there are solutions of the equation with a forcing term f (x) as long as the f has the right sign, i.e. f ≤ 0 below λ + α and f ≥ 0 below λ − α . We are mainly interested in the asymptotic behavior with respect to α of the eigenvalues. When α → 0, λ + α and λ − α tend to 0 which is the principal eigenvalue of the pure Neumann boundary problem But our main goal is to study the behavior when α → +∞, this is done in our main Theorem 1.1. The following limits hold: Interestingly this asymptotic behavior emphasizes the "splitting" of the eigenvalue. In the linear case, i.e. when a = A = 1 and the Pucci operator is nothing else but the Laplacian, this problem was treated in [14] by Lou and Zhu with a variational approach. Very recently Daners and Kennedy [9] have proved that this asymptotic behavior is valid for the whole spectrum.
We also prove that for any K ⊂⊂ Ω, the normalized eigenfunctions u + α and u − α satisfy u + α L ∞ (K) → 0 and u − α L ∞ (K) → 0 as α → +∞. So that the eigenfunctions tend to concentrate on the point of the boundary where they reach the sup or the inf.
The idea of the proof of Theorem 1.1 which somehow follows the line adopted in [14], is the following: first we establish that u + α reaches its maximum on the boundary and then we perform a blow up around this point.
Then a key tool will be a Liouville result in the half space (Theorem 5.1). Precisely we prove that for γ > A (respectively γ > a) there are no bounded subsolutions (respectively supersolutions) of M + a,A (D 2 u) − γu = 0 in R n + , − ∂u ∂xn = u on ∂R n . that are positive (repectively negative) somewhere. In [14] the analogous result for the Laplacian is proved using the construction of sub and super solutions in the flavor of what is done in [3]. Let us mention here that it would be interesting to extend the results of Berestycki, Caffarelli, Nirenberg [3] in half spaces, to this class of fully-nonlinear operators and to these boundary conditions.
Lipschitz estimates up to the boundary will be required in the proofs of both the existence results and the asymptotic behavior. These estimates which are interesting in their own right, are established here using an argument inspired by [10] (see also Barles and Da Lio [2] and Milakis and Silvestre [15] ).
In the whole paper the fully-nonlinear operator considered is the Pucci operator M + a,A , but, mutatis mutandis, parallel results can be stated for the Pucci operator M − a,A defined by M − a,A (M) = inf 0<aI≤σ≤AI tr(σM).

Preliminary results
Let us recall the definition of viscosity sub and supersolution of the Neumann problem associated to a general elliptic operator F : Ω × R × R n × S(n) → R. Here S(n) is the space of symmetric matrices on R n , equipped with the usual ordering. We denote by USC(Ω) (resp., LSC(Ω)) the set of upper (resp., lower) semicontinuous functions on Ω. Let f : Ω → R, g : ∂Ω × R → R.
Definition 2.1. A function u ∈ USC(Ω) (resp., u ∈ LSC(Ω) ) is called viscosity subsolution (resp., supersolution) of if the following conditions hold (i) For every x 0 ∈ Ω, for any ϕ ∈ C 2 (Ω), such that u − ϕ has a local maximum (resp., minimum) at x 0 then (ii) For every x 0 ∈ ∂Ω, for any ϕ ∈ C 2 (Ω), such that u − ϕ has a local maximum (resp., minimum) at x 0 then A viscosity solution is a continuous function which is both a subsolution and a supersolution.
One of the motivation for these relaxed boundary conditions is the stability under uniform convergence. Actually, if the domain Ω satisfies the exterior sphere condition and F is uniformly elliptic, viscosity subsolutions (resp., supersolutions) satisfy in the viscosity sense ∂u ∂ − → n ≤ (resp., ≥ )g(x, u) for any x ∈ ∂Ω, see e.g. Proposition 2.1 in [16]. We assume throughout the paper that Ω is a bounded domain of R n of class C 2 . Assume that c and f are continuous on Ω. Let u ∈ USC(Ω) and v ∈ LSC(Ω) be respectively a sub and a supersolution of If u ≤ v on Ω then either u < v on Ω or u ≡ v on Ω.

Lipschitz estimates
In this section we shall prove a local Lipschitz regularity result for solutions of the Neumann problem associated to general uniformly elliptic operators, that we will use in the next sections. Let us consider the Neumann problem where the operator F is supposed to be continuous on Ω × R × R n × S(n) and satisfying the following assumptions: Proposition 3.1. Assume that (F1) and (F2) hold. Let f : Ω → R be bounded, g : ∂Ω → R be Lipschitz continuous. Let u ∈ C(Ω) be a viscosity solution of (5), then, for any x 0 ∈ Ω and for any ρ > 0, there exists K > 0 such that where M ≤ C(|u| L ∞ (B 3ρ (x 0 )∩Ω) + |g| L ∞ (∂Ω) + 1) and C depends on a, A, C 1 , n and Ω.
Proof of Proposition 3.1 We follow the proof of Proposition III.1 of [10], that we modify taking test functions which depend on the distance function and that are suitable for the Neumann boundary conditions. Moreover, as in [2], we are going to use a regularization of g. In order to do so, it is convenient to introduce the following classical lemma.
for some positive constant C 0 ≤ C|g| C 0,1 (R n ) , with C depending only on ρ and n.
We first extend g to a C 0,1 function of R n and we still denote by g this extension. Then, we consider the function g associated to g as in Lemma 3.3.
Since Ω is a domain of class C 2 , it satisfies the uniform exterior sphere condition, i.e., there exists r > 0 such that B(x + r − → n (x), r) ∩ Ω = ∅ for any x ∈ ∂Ω. From this property it follows that |y − x| 2 for x ∈ ∂Ω and y ∈ Ω.
Moreover, the C 2 -regularity of Ω implies the existence of a neighborhood of ∂Ω in Ω on which the distance from the boundary is of class C 2 . We still denote by d a C 2 extension of the distance function to the whole Ω. Without loss of generality we can assume that |Dd(x)| ≤ 1 on Ω.
We are going to show that u is Lipschitz continuous on B Ω (x 0 , ρ). For this aim, let us introduce the functions where L is a fixed number greater than 1 r with r the radius in (10), K and M are positive constants to be chosen later and δ is a small parameter. We also use the notation MK|x|.
We define We fix M > 1 and j > 0 such that , and we claim that taking K large enough, for any small δ one has (13) u To show (13) we suppose by contradiction that the maximum of where The point (x, y) belongs to int(∆ K ) ∩ B Ω (x 0 , ρ) 2 . Indeed, if |x − y| = 1 4K , by (12) and (11), we have On the other hand, if |x − x 0 | = ρ, then Since x = y we can compute the derivatives of ϕ at (x, y) obtaining
Here and henceforth C denotes various positive constants independent of K, b, c, f, g and u.
By Lemma 3.3 Hence, using (10), for MK > 16rC 0 e Ld 0 (3rL−2) , since x = y and L > 1 Then, by definition of sub and supersolution in Ω 2 . Then applying Theorem 3.2 in [8], for every ǫ > 0 there exist X, Y ∈ S(n) such that (−D y ϕ(x, y), Y + C 0 δD 2 d(y)) ∈ J 2,− u(y) and Now we want to estimate the matrix on the right-hand side of the last inequality. Using Lemma 3.3, it is easy to check that Next, let us estimate D 2 Ψ 1 (x, y).
We set A 1 := Φ(x − y)D 2 (e −L(d(x)+d(y)) ), Indeed for ξ, η ∈ R n we compute Now we consider A 3 . The matrix D 2 (Φ(x − y)) has the form and the Hessian matrix of Φ(x) is , then we have the following estimates where I 2n := I 0 0 I . Then using (18), (19), (20), (23) and observing that from (17) we can conclude that The last inequality can be rewritten as follows with X = X − (MO(K) + CC 0 )I and Y = Y + (MO(K) + CC 0 )I. Now we want to get a good estimate for tr( X − Y ), as in [10]. For that aim let We have to compute tr(P B). From (21), observing that the matrix Then, since trP = 1 and 4K|x − y| ≤ 1, we have This gives The Lemma III.I in [10] ensures the existence of a universal constant C depending only on n such that Thanks to the above estimates we can conclude that Now, using assumptions (F1) and (F2) concerning F , the definition of X and Y and the fact that u is sub and supersolution we compute From these inequalities, using (15), (26) and (25), for K > K, where K is a constant depending only on a, A, C 1 , n and Ω, we get Then, since we have chosen M > 1, for K > K we obtain , and this is a contradiction for K large enough. This implies that there exists K satisfying (28), such that (13) holds true. Next, choosing x = x 0 , (13) gives Repeating the proof in B Ω (x, 2ρ) for any x ∈ B Ω (x 0 , ρ), we finally find the u satisfies (6) and (7).

Proof of Corollary 3.2 Let us define
Then, v is a solution of where F (x, r, p, X) = M + a,A {X − α(Dd ⊗ p + p ⊗ Dd) + α 2 r(Dd ⊗ Dd) − αrD 2 d} + λr. It is easy to check that F satisfies assumptions (F1) and (F2) with C 1 = 0, and where C depends on a, A, n and Ω. Then, by Proposition 3.1, the Lipschitz constant of v on Ω ρ is bounded from above by M v K v , where M v ≤ C(|e αd(x) u| L ∞ (Ω 3ρ ) + 1) and K v satisfies (9). Hence, for any x, y ∈ Ω ρ , we have and this concludes the proof.

Properties of the principal eigenvalues
Proof. We follow the arguments of [5]. To show the existence of positive eigenfunctions, the first step is to prove that if f is a continuous function such that f ≤ 0, f ≡ 0, then for any λ < λ + α there exists a positive solution of Observe that v ≡ 1 is a positive subsolution of (4) for λ ≥ 0. This implies, by Proposition 2.2, that if λ < λ + α then λ < 0. Let (v n ) n be the sequence defined by v 1 = 0 and v n+1 be the solution of − αrD 2 d} and c = C(α 2 + α). By comparison, the sequence is positive and increasing. Let (u n ) n be the sequence defined by u n (x) := e −αd(x) v n (x), then u n+1 is solution of We claim that (u n ) n is bounded. Suppose that it is not, then defining w n := un |un|∞ one gets that w n+1 is a solution of By Corollary 3.2, (w n ) n converges along a subsequence to a positive function w which satisfies where k := lim sup n→+∞ |un|∞ |u n+1 |∞ ≤ 1. This contradicts the Maximum Principle, Proposition 2.2. Then (u n ) n is bounded and letting n go to infinity, by the compactness result, the sequence converges uniformly to a function u which is a solution of (31). Moreover, u is positive by the Strong Comparison Principle, Theorem 2.1.
We are now in position to construct a sequence (u n ) n of positive solutions of where (λ n ) n is an increasing sequence which converges to λ + α . The sequence (u n ) n is unbounded, otherwise one would contradict the definition of λ + α (see Theorem 8 of [5]). Then, up to subsequence, |u n | ∞ → +∞ as n → +∞ and defining φ n := un |un|∞ one gets that φ n satisfies M + a,A (D 2 φ n ) + λ n φ n = − 1 |un|∞ in Ω, ∂φn ∂ − → n = αφ n on ∂Ω.

Liouville type results
For γ > 0 let us introduce the system Theorem 5.1. If γ > A, any bounded subsolution of (35) is non-positive in R n + . If γ > a, any bounded supersolution of (35) is non-negative in R n + . Hence, if γ > A there are no, non trivial bounded solutions of (35).
Both inequalities contradict the definition of sub and supersolution, therefore x, y ∈ R n + .
6. Asymptotic behavior and Proof of Theorem 1.1 We start by the following simple result: Proof. By Proposition 4.5, λ + α increases to some value λ 0 ≤ 0. On the other hand, the sequence of normalized solutions (u + α ) α , by the Lipschitz estimates Corollary 3.2, converges to u 0 a positive solution of M + a,A (D 2 u) + λ 0 u = 0 in Ω, ∂u ∂ − → n = 0 on ∂Ω, which satisfies |u 0 | = 1. Recall that 0 is the principal eigenvalue for the Neumann problem. If λ 0 < 0, the Maximum Principle below the first eigenvalue, i.e. Proposition 2.2, implies that u 0 ≤ 0 a contradiction.