<span class="mathjax-formula">$\mathcal {C}^{1,\beta }$</span> regularity for Dirichlet problems associated to fully nonlinear degenerate elliptic equations

DOI:10.1051/cocv/2014005 www.esaim-cocv.org

C

REGULARITY FOR DIRICHLET PROBLEMS ASSOCIATED TO FULLY NONLINEAR DEGENERATE ELLIPTIC EQUATIONS

I. Birindelli

and F. Demengel

Abstract. We prove H¨older regularity of the gradient, up to the boundary for solutions of some fully-nonlinear, degenerate elliptic equations, with degeneracy coming from the gradient.

Mathematics Subject Classification. 35J25, 35J60, 35P30.

Received March 7, 2013. Revised December 20, 2013.

Published online August 13, 2014.

1. Introduction

In a recent paper Imbert and Silvestre [14] have proved that the solutions of

|∇u|^αF(D²u) =f(x) inΩ⊂IR^N (1.1)

have first derivative which are H¨older continuous in the interior ofΩwhenα≥0,F is uniformly elliptic andf is continuous.

Results concerning regularity of solutions have an intrinsic interest which doesn’t need to be explained. When α= 0, it is known (seee.g.Evans [11], Cabr´e, Caffarelli [7–9]) thatuisC^1,β for someβ∈(0,1). But for solutions of (1.1) withα >−1 the question of the continuity of the gradient was open and it was naturally raised, in [5].

In fact, the question raised concerned precisely regularity near the boundary. Let us recall that context. The values

μ⁺= sup

μ,∃φ >0 inΩ,|∇φ|^αF(D²φ) +μφ^1+α≤0 in Ω , μ⁻= sup

μ,∃ψ <0 inΩ,|∇ψ|^αF(D²ψ) +μ|ψ|^αψ≥0 inΩ

are generalised principal eigenvalues in the sense that there exists a non trivial solution to the Dirichlet problem

|∇φ|^αF(D²φ) +μ^±|φ|^αφ= 0 in Ω, φ= 0 on ∂Ω, with constant sign.

Keywords and phrases.Regularity, fully nonlinear equations, simplicity of the first nonlinear eigenvalue.

1 Dipartimento di Matematica “G. Castelnuovo”, Sapienza Universit`a di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy

2 Laboratoire AGM, Universit´e de Cergy Pontoise, 2 rue Adolphe Chauvin, 95302 Cergy Pontoise, France.

[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2014

I. BIRINDELLI AND F. DEMENGEL

The main scope in [5] was to prove the simplicity of these principal eigenvalues. The main difficulty comes from the fact that the strong comparison principle holds only in open subsets ofΩ where the gradient is away from zero (in the viscosity sense). It is well known that the Hopf Lemma guarantees that this is true on∂Ω. So that the continuity of the gradient up to the boundary would imply that, in a neighbourhood of it, the strong comparison principle holds, which is exactly what is needed to prove that the eigenvalues are simple.

In that same paper, we proved that if α∈(−1,0) the solutions of the Dirichlet problem associated to (1.1) are indeedC^1,β and we raised the problem of whether that regularity would hold also forα≥0 i.e. when the operator is degenerate elliptic.

[14] was a first answer in that direction, very much inspired by that breakthrough, we wanted to complete the work. We want to point out that a difficulty specific to this type of equation is due to the fact that the difference of two solutions is not a sub- or super solution to another equation, which would provide regularity.

Recall thatF is uniformly elliptic if there existsΛ≥λ >0 such that for any symmetric matricesM andN F(M) +M⁻_λ,Λ(N)≤F(M +N)≤F(M) +M⁺_λ,Λ(N) (1.2) here we have denoted the Pucci operators M⁻_λ,Λ(N) = λtr(N⁺) +Λtr(N⁻) and M⁺_λ,Λ(M) = λtr(M⁻) + Λtr(M⁺). In the rest of the paper we shall drop the indicesλand Λof the Pucci operators.

We now state the main result of the paper in its generality.

Theorem 1.1. Suppose thatΩis a boundedC²domain of IR^N andα≥0. Suppose thatF is uniformly elliptic and that h∈ C(Ω). Let f ∈ C(Ω)andϕ∈ C^1,βo(∂Ω). For anyu, viscosity solution of

|∇u|^α(F(D²u) +h(x)· ∇u) =f inΩ

u=ϕ on∂Ω

there existβ =β(λ, Λ,f∞, N, Ω,h∞, β_o)and C=C(β)such that u_C1,β(Ω)≤C

ϕ_C1,βo(∂Ω)+u_∞+f∞^1+α1

.

For radial solutions, and a more general class of operators, this was proved in [6], with the optimal H¨older’s coefficient inf(_1+α¹ , β_o). In a recent paper the result of Imbert and Silvestre was improved by Araujo, Ricarte, Teixeira in [1].

The novelty with respect to the paper of Imbert and Silvestre is two folded, on one hand we have added the lower order termh(x)· ∇u|∇u|^α, and on the other hand we go all the way to the boundary.

The proof follows the scheme of the one in [14], but requires new tools. In particular in Section 2, we give some a priori Lipschitz and H¨older estimates in the presence of boundary conditions on one part of the boundary.

These are important because the proof of Theorem1.1requires that sequence of bounded solutions do converge to a solution of a limit equation.

The main tool is an “improvement of flatness lemma”. In the proof of Lemma3.3, we need to use regularity estimates for a limit equation with boundary terms. The novelty is in guaranteeing that even in the presence of the lower order terms and of the boundary term the limit equations are sufficiently “good” and the new terms behave well, see in particular the Claim in the proof of Lemma3.3.

Let us make a final remark:

Following Caffarelli’s famous technique, the results here obtained could be generalised to an operator depending onxi.e. of the form|∇u|^αF(x, D²u); in particular by imposing conditions on

β(x) = sup

M∈S

F(x, M)−F(x,0)

|M| ·

But we would further need to impose conditions that guarantee thatu, a solution of

|∇u|^αF(x, D²u) = 0 inΩ⇒ F(x, D²u) = 0 inΩ.

For the sake of clarity we have chosen not to treat the case where there is a dependence onx.

After this paper was submitted for publication, Silvestre and Sirakov have posted an interesting related paper [18].

2. Local H¨ older and Lipschitz estimates up to the boundary.

Throughout the paper, the notationB_r(x) indicates the euclidian ball of radiusrand centrex, we will write B_r when no ambiguity arises.

It is a classical fact that in order to prove that uis C^1,β at x_o, it is enough to prove that there exists some constantC such that, for allr <1, there existsp_r∈IR^N, such that osc_Br(xo)(u(x)−p_r·x)≤Cr^1+β.

As a consequence,uisC^1,β in some bounded open setBif there exists a constantC_β such that for allx∈B andr <1, there existsp_r,xsuch that

Boscr(x)(u(y)−p_r,x·y)≤C_βr^1+β. This will be used in the whole paper.

We begin by stating the following comparison theorem which will be needed later, proved in [3] under stronger conditions onh, later improved tohcontinuous and bounded in [17].

Theorem 2.1. [3,17]In the hypothesis of Theorem1.1, letuandv be respectively C(Ω)solutions of

|∇u|^α(F(D²u) +h(x)· ∇u)≤f inΩ and

|∇v|^α(F(D²v) +h(x)· ∇v)≥g inΩ with f andg continuous and bounded such that f < g.

If u≥v on ∂Ω thenu≥v inΩ.

In order to prove H¨older and Lipschitz estimates we fix a few notations concerningΩ and F. We suppose, without loss of generality, that at 0∈ ∂Ω, the interior normal ise_N. By the implicit function theorem, there exist a ballB =B_R(0) in IR^N, andD ⊂B_R (0) ball of IR^N−1 anda∈ C²(D), such thata(0) = 0,∇a(0) = 0 and, fory= (y, y_N),

Ω∩B ⊂ {y_N > a(y), y∈D}, and∂Ω∩B={y_N =a(y), y∈D}.

We shall also act as ifF be positively homogeneous of degree 1i.e. such that for anyt >0,F(tM) =tF(M).

Observe though that, if this doesn’t hold, when necessary it is enough to replaceF(M) byG_t(M) =t⁻¹F(tM);

this operator satisfies (1.2) with the same constants asF and the results are unchanged.

In the lemma below we have supposed, for simplicity, thatB is the unit ball centred at the origin.

I. BIRINDELLI AND F. DEMENGEL

Lemma 2.2. Let ϕ∈ C^1,βo. Let a∈ C²(D) such that a(0) = 0 and∇a(0) = 0. Let dbe the distance to the hyper surface{y_N =a(y)}.

Then, for allr <1 and for all γ <1, there exists δ_o depending onf∞,λ,Λ,h∞,Ω,r andLipϕ, such that for all δ < δ_o, if uis a solution of

|∇u|^α(F(D²u) +h(y)· ∇u) =f in B∩ {y_N > a(y)}

u=ϕ on B∩ {yN =a(y)} (2.1)

such that oscu≤1then it satisfies

|u(y, y_N)−ϕ(y)| ≤ 6 δ

d(y)

1 +d(y)^γ inB_r(0)∩ {y_N > a(y)}.

Proof. We write the details of the proof forϕ= 0. Note that thenu∞≤1. The changes to bring in the case ϕ= 0 will be given at the end of the proof, the detailed calculation being left to the reader.

It is sufficient to consider the set whered(y)< δ since the assumptionu∞≤1 implies the result elsewhere.

We begin by choosing δ < δ₁, such that ond(y)< δ₁ the distance is C² and satisfies|D²d| ≤C₁. We shall also later chooseδ smaller depending of (λ, Λ,f∞,h∞, N).

In order to use the comparison principle we want to constructwa super solution of

|∇w|^α(M⁺_λ,Λ(D²w) +h(y)· ∇w)<−f∞, inB∩ {yN > a(y), d(y)< δ} (2.2) such thatw≥uon∂(B∩ {yN > a(y), d(y)< δ}).

The candidate is

w(y) =

⎧⎪

⎪⎨

⎪⎪

⎩ 2 δ

d(y)

1 +d^γ(y) for|y|< r 2

δ d(y)

1 +d^γ(y)+ 1

(1−r)³(|y| −r)³ for|y| ≥r.

In order to prove the boundary condition, let us observe that, on{d(y) =δ},w≥²_δ1+δ^δγ ≥1≥u,

on{|y|= 1} ∩ {d(y)< δ},w≥_(1−r)¹ 3(1−r)³≥uand finally onB∩ {y_N =a(y)},w≥0 =u.

We need to check thatwis a super solution. For that aim, we compute

∇w=

⎧⎪

⎪⎨

⎪⎪

⎩ 2 δ

1 + (1−γ)d^γ

(1 +d^γ)² ∇d when |y|< r 2

δ

1 + (1−γ)d^γ

(1 +d^γ)² ∇d+ y

|y|

3

(1−r)³(|y| −r)² if |y|> r.

Note that|∇w| ≥ _4δ¹ as soon asδ≤ ^1−r₁₂ . By constructionwisC² and

D²w=−

2γd^γ−1 δ

(1 +γ) + (1−γ)d^γ

(1 +d^γ)³ ∇d⊗ ∇d+2 δ

1 + (1−γ)d^γ

(1 +d^γ)² D²d+H(y) whereH(y) ≤ _(1−r)⁶ 2 +^3(N−1)_r(1−r).

Standard computations that use (1.2) imply that w satisfies (2.2), whenδ <min(δ₁,^1−r₁₂ ) is small enough that the two following inequalities hold

λ(γδ^γ−2) (1 +γ) (1 +δ^γ)³ >2Λ

6

(1−r)² +3(N−1) r(1−r) +2C₁

δ

+4h∞

δ , λ

2^2+2α(γδ^γ−(2+α)) (1 +γ)

(1 +δ^γ)³ >f∞.

By the comparison principle, Theorem2.1,u≤winB∩ {y_N > a(y)} ∩ {d(y)< δ}.

Furthermore the desired lower bound onuis easily deduced by considering−win place ofwin the previous computations and restricting toB_r∩ {yN > a(y)}. This ends the caseϕ≡0.

Whenϕ≡0, let ψbe a solution of

M⁺_λ,Λ(D²ψ) = 0 inB∩ {y_N > a(y)}

ψ=ϕ onB∩ {y_N =a(y)}.

It is well known thatψ isC^1,βo(B∩ {yN ≥a(y)})∩ C²(B∩ {yN > a(y)}). Furthermore, we can chooseψsuch thatψ∞≤ ϕ∞≤1,∇ψ∞≤c∇ϕ∞, for some constantcwhich depends onλ, Λ, N, Ω, see [9].

We now define

w(y) =

⎧⎪

⎪⎨

⎪⎪

⎩ 8 δ

d(y)

1 +d^γ(y)+ψ(y) for|y|< r 8

δ d(y)

1 +d^γ(y)+ 1

(1−r)³(|y| −r)³+ψ(y) for |y|> r.

Computations similar to the caseϕ= 0 imply that

w≥uon∂(B∩ {y_N > a(y)} ∩ {d(y)< δ}).

Furthermore choosingδ small enough, we can ensure that

|∇w|^α(M⁺_λ,Λ(D²w) +h(x)· ∇w)<−f∞.

For the lower bound, we replacewby 2ψ−w. This ends the proof of Lemma2.2.

Using this estimate together with an argument due to Ishii and Lions [16], one finally gets the H¨older regularity of the solution, which can be stated as follows with the same hypothesis ona, andf as above:

Proposition 2.3. Let ϕbe a Lipschitz continuous function. Suppose that usatisfies(2.1).

For allr <1, and for allγ <1,uisγ H¨older continuous onB_r∩ {yN > a(y)}, with some H¨older’s constant depending on(r, λ, Λ, a, N,|f∞,h∞,Lipϕ).

Remark 2.4. In the absence of boundary conditions, the solutions are H¨older continuous insideB_r for any r such thatB_r⊂⊂B. We do not give the proof which follows the lines in the proof below, it is sufficient to cancel in it the dependence onϕ. This will be used in the proof of the interior improvement of flatness lemma with additional lower terms.

In the following proof we shall use directly the definition of viscosity solutions, so, in particular, in order to fix the notations we state the definition of semi-jets:

Definition 2.5. LetS²ⁿ denote the symmetric 2n×2nmatrices. For any continuous function gwe define the intrinsic semi-jets by:

J^2,+g(x) =

(p, X)∈IR^N ×S^N, g(x+h)≤g(x) +p·h+1

2Xh, h+o(h²)∀h∈IR^N

, J^2,−g(x) =

(p, X)∈IR^N ×S^N, g(x+h)≥g(x) +p·h+1

2Xh, h+o(h²)∀h∈IR^N

. and the definition of the closed semi jets

I. BIRINDELLI AND F. DEMENGEL

Definition 2.6.

J^2,+g(x) ={(p, X)∈IR^N ×S^N, ∃x_n, ∃(p_n, X_n)∈J^2,+g(x_n), g(x_n)→g(x),and (p_n, X_n)→(p, X)} and analogous definition forJ^2,−g(x)

Proof of Proposition 2.3. The proof relies on arguments similar to those in [2,3,14]. Let 1> r₁> r. Without loss of generality we can suppose that oscu≤1. Letx_o∈B_r∩ {y_N > a(y)} andΦbe defined as

Φ(x, y) =u(x)−u(y)−M|x−y|^γ−L|x−x_o|²−L|y−x_o|². The scope is to prove that forLandM independent ofx_o, chosen large enough,

Φ(x, y)≤0 on (B_r1∩ {yN > a(y)})². (2.3) This will imply thatuisγ-H¨older continuous onB_r∩ {y_N > a(y)} by takingx=x_o, and letting x_o vary.

To prove (2.3), we begin by observing that the inequality holds on the boundary. First we treat the points wherey_N =a(y). According to Lemma2.2there existsM_o such that forx∈B_r1∩ {yN > a(y)},

|u(x)−ϕ(x)| ≤M_od(x, ∂Ω).

Then, using|x−y| ≤ |x−y|, one has

|u(x, x_N)−u(y, a(y))| ≤ |u(x, x_N)−u(x, a(x))|+|u(x, a(x))−u(y, a(y))|

≤M_od(x, ∂Ω) + Lip_ϕ|x−y|

≤M_o|x−(y, a(y))|+ Lip_ϕ|x−(y, a(y))|.

So, ifM is chosen greater thanM_o+ Lipϕ, then we have obtained thatΦ(x, y)≤0 on (B_r1∩ {yN =a(y)})². In order to satisfy the required estimate on the rest of the boundary, it is enough to chooseL >_(r ⁴

1−r)² and to recall that the oscillation ofuis bounded by 1.

In the sequel we will choose L large and M > _γ(1−γ)^4L2N . Suppose by contradiction that Φ(x, y)>0 for some (x, y)∈B_r1∩ {y_N > a(y)}. Then there exists (¯x,y) such that¯

Φ(¯x,y) = sup¯

Br1

(Φ(x, y))>0.

Clearly ¯x= ¯y. Furthermore the hypothesis on Lforces ¯xand ¯yto be inBr1+r

2 ∩ {yN > a(y)}. Then, as in [2], for all small >0 depending on the norm ofQ:=D²(M|x−y|^γ), using Ishii’s Lemma [13], there existX and Y such that

(γM(¯x−y)|¯¯ x−y|¯^γ−2+ 2L(¯x−x_o), X)∈J^2,+u(¯x) (γM(¯x−y)|¯¯ x−y|¯^γ−2−2L(y−x_o),−Y)∈J^2,−u(¯y)

with

X 0 0 Y

≤

Q −Q

−Q Q

+ (2L+) I0

0I

.

In the sequel, since we assumed that _M^L ≤ ^C_L one also has ^L+_M ≤ ^C_L and then we dropfor simplicity.

Let us denote q_x =γM(¯x−y)¯ |x¯−y¯|^γ−2+ 2L(¯x−x_o), and q_y =γM(¯x−y)¯ |x¯−y¯|^γ−2−2L(¯y−x_o). Since γ <1, as soon as 2L(r+r₁)≤^γ₂M r^γ−1₁ , we get that 2L|¯x−x_o| ≤^γM₂ |x¯−y|¯^γ−1i.e.

2γM|¯x−y|¯^γ−1≥(|q_x|, |q_y|)≥1

2γM|¯x−y|¯^γ−1.

Since|q_x−q_y| ≤4L, by the mean value theorem and using a constantκ < α ifα <1 andκ= 1 ifα≥1:

||qx|^α− |qy|^α| ≤α|qx−q_y|^κ2^|α−1||γM|¯x−y|¯^γ−1|^1−κ|γM|¯x−y|¯^γ−1|^α−1

≤C(M γ|¯x−y|¯^γ−1)^α−κL^κ=o

M|¯x−y|¯^(γ−1)_α .

We now treat the terms concerning the second order derivative. The previous inequalities can also be written

as

X−2LI 0 0 Y −2LI

≤

Q −Q

−Q Q

.

We prove, in what follows, that L =o(|tr(X +Y)|), that there exist constantsc and C such that |X|,|Y| ≤ C|tr(X+Y)|and that

cM|¯x−y|¯^γ−2≤ |tr(X+Y)| ≤CM|¯x−y|¯^γ−2. Indeed, let

P := (¯x−y¯⊗x¯−y)¯

|¯x−y|¯² ≤I.

Using 4L−(X+Y)≥0, (I−P)≥0 and the properties of the symmetric matrices, one has tr(X+Y −4L)≤tr(P(X+Y −4L)).

Remarking in addition that X +Y −4L ≤ 4Q, one sees that tr(X +Y −4L) ≤ 4tr(P Q). But tr(P Q) = γM(γ−1)|¯x−y|¯^γ−2<0, hence

|tr(X+Y −4L)| ≥4γM(1−γ)|¯x−y|¯^γ−2. (2.4) Furthermore, by Lemma III.1 of [16], there exists a universal constantCsuch that

|X|,|Y|,|X−2L|,|Y −2L| ≤C(|tr(X+Y −4L)|+|Q|¹²|tr(X+Y −4L)|¹²)

≤C|tr(X+Y −4L)|

≤C|tr(X+Y)|,

since|Q|and|tr(X+Y −4L)|are of the same order, and _M^L =o(1). This will yield the required estimates.

For some positive constants c₂, c₃, sinceuis both a sub- and a super- solution of (2.1), using the uniform ellipticity ofF and the assumptions onh:

f(¯x)≤ |q_x|^α(F(X) +h(¯x)·q_x)

≤ |qy|^α(F(X) +h(¯x)·q_x)

+o(M γ|¯x−y|¯^γ−1)^α(Λ|X|+h∞

γM|¯x−y|¯^γ−1+ 2L)

≤ |q_y|^α(F(−Y) +h(¯y)·q_x+ 4h_∞L) + +|qy|^αtr(X+Y)Λ+o

M^1+α|¯x−y|¯^{(γ−1)α+γ−2}

≤M^αc₂|x¯−y¯|^(γ−1)αtr(X+Y) +o

M^1+α|x¯−y¯|^{(γ−1)α+γ−2} +f(¯y)

≤ −c₃M^1+α|¯x−y|¯^{(γ−1)α+γ−2}+f(¯y).

This is clearly false as soon asL(and thenM) is large enough and it ends the proof.

Compactness near the boundary is a natural consequence of Proposition2.3.

I. BIRINDELLI AND F. DEMENGEL

Corollary 2.7. Suppose that(u_n)is a bounded sequence of continuous functions which satisfy |∇un|^α(F(D²u_n) +h(y)· ∇un) =f_n inB∩ {yN > a(y)}

u_n=ϕ onB∩ {y_N =a(y)}

and suppose that(f_n)converges simply to some continuous functionf. Then for allr <1, one can extract from (u_n)a subsequence which converges uniformly, onB_r∩ {y_N > a(y)}, to a solution of

|∇u|^α(F(D²u) +h(y)· ∇u) =f in B∩ {y_N > a(y)}

u=ϕ onB∩ {y_N =a(y)}.

Remark 2.8. In the absence of boundary conditions, the analogous result holds, in the sense that one can extract from (u_n)_n a subsequence which converges uniformly on everyB_r⊂⊂Bto ua solution of

|∇u|^α(F(D²u) +h(y)· ∇u) =f in B.

When we shall treat, in the improvement of flatness lemma up to the boundary, the case where the boundary is locally straight, we shall need the following Lipschitz estimate’s near the boundary for some different but related equation.

Proposition 2.9. (Lipschitz estimates for large ps) Let ϕ be a Lipschitz continuous function. Suppose that h_N ≡0. Assume thatusolves

|peN+∇u|^α(F(D²u) +h(y)· ∇u) =f in B₁(x)∩ {yN >0}

u=ϕ on B₁(x)∩ {y_N = 0}

withosc_{B1(x)∩{yN>0}}u≤1andfL^∞(B1(x)∩{yN>0})≤_o<1. Then, for allr <1, there existsb_o depending on (λ, Λ, N, α, r, _o,Lipϕ), such that if|p|> _b¹

o,uis Lipschitz continuous inB_r(x)∩ {y_N >0}with some Lipschitz constant depending on (λ, Λ, N, α, r, _o,Lipϕ).

Remark 2.10. In the absence of boundary conditions, the solutions in some ball B are Lipschitz in B_r, for anyr such thatB_r⊂⊂B with Lipschitz constant independent ofp.

This will be used in the proof of the interior improvement of flatness lemma with lower order terms.

This Proposition is a consequence of the following

Lemma 2.11. Suppose that ϕ is Lipschitz continuous and that h_N ≡ 0. For all γ < 1, for all r < 1, there existsδ=δ(f_∞, λ, Λ, r,Lipϕ), such that for b < ^δ₄, any solutionuof

|e_N+b∇u|^α(F(D²u) +h(y)· ∇u) =f inB₁(x)∩ {y_N >0}

u=ϕ onB₁(x)∩ {y_N = 0}.

such that osc(u)≤1, satisfies|u(y, y_N)−ϕ(y)| ≤ ²_δ1+y^yNγ

N inB_r(x)∩ {yN >0}.

Proof of Lemma2.11.Suppose for simplicity thatϕ= 0. Ifb= 0 the result is known by properties of solutions ofF(D²u) +h(x)· ∇u=f which are zero on the boundary. So we assume in what follows thatb= 0.

We proceed as in Lemma2.2, replacing the distance ofy to the boundary byy_N; so we consider

w(y) =

⎧⎪

⎪⎨

⎪⎪

⎩ 2 δ

y_N

1 +y_N^γ fory_N < δ,|y|< r 2

δ y_N

1 +y_N^γ + 1

(1−r)³(|y| −r)³ fory_N < δ,|y|> r.

Similarly to the proof of Lemma 2.2, it is sufficient to consider the set where y_N < δ, since the assumption oscu≤1 implies u_∞≤1, so the result holds elsewhere. Furthermore we only prove thatu≤w, the desired lower bound can be obtained by considering−win place ofw.

In order forwto satisfy

|e_N +b∇w|^α(M⁺_λ,Λ(D²w) +h(y)· ∇w)≤ −f_∞, in B, it is sufficient to chooseδ such that

1 2

_α

λγδ^γ−2(1−γ) 1

(1 +δ^γ)³ >f∞+ 2Λ 6

(1−r)² +3(N−1)

r(1−r) +4h∞

δ

, b < ^δ₄, and recall that|∇w| ≤ ²_δ. Furthermorew≥uon∂(B∩ {0< y_N < δ}).

Hence we can use the comparison principle in Theorem2.1, for ˜w(x) =x_N +bw(x) and ˜u(x) =x_N +bu(x), with b = 0, (recall that h_N ≡0 )which implies that u≤w in B∩ {yN >0}. Finally the desired estimate is obtained in{|y|< r, y_N >0}.

In the case ϕ≡0, we take the function was in the proof of Lemma 2.2, with dreplaced by y_N. Requiring

sufficient restriction on the smallness ofδgive the result.

We are now ready to give the

Proof of Proposition 2.9. The proof proceeds as in [14] so we just detail the differences. Recall that u is a solution of

|eN +bDu|^α(F(D²u) +h(y)· ∇u) = ˜f withb=¹_p and ˜f =|p|^−αf.

Letr < r₁<1 and let x_o∈B_r1(x)∩ {yN >0},L₂= _(r ⁴

1−r)²,

ψ(z, y) =u(z)−u(y)−L₁ω(|z−y|)−L₂|z−x_o|²−L₂|y−x_o|² whereω(s) =s−ω_os³² ifs≤s_o=

2 3ωo

₂

andω(s) =ω(s_o) ifs≥s_o. We also requireL₁>3(²_δ + Lip_ϕ).

If we prove thatψ(z, y)≤0 inB_r1(x), sinceL₁ is independent ofx_o, by choosingz=x_o one gets u(x_o)−u(y)≤L₁|xo−y|+L₂|xo−y|²;

next choosingy=x_ofor allz∈B_r1,

u(z)−u(x_o)≤L₁|xo−z|+L₂|xo−z|².

Finally, for (x, y)∈B_r(x),|u(x)−u(y)| ≤L₁|x−y|+L₂|x−y|², which implies the desired result.

We begin to observe that if the supremum is achieved in (¯x,y)¯ ∈B_r(x) then, with our choice ofL₁, neither

¯

xnor ¯y can belong to the part{z_N = 0}according to Lemma2.11.

The rest of the proof is as in [2] and [14], (see also the Proof of Prop.2.3) as long as we chooseδsmall enough in order that

1 2

_α

λ(γδ^γ−2) (1 +γ)

2(1 +δ^γ)³ >f∞+ 2Λ 6

(1−r)² +3(N−1)

r(1−r) +4h∞

δ

and such that b < ^δ₄. Let us note that this implies that b is small enough depending on

(λ, Λ, N, α, r, _o,h_∞,Lipϕ).

I. BIRINDELLI AND F. DEMENGEL

As a corollary of this Lemma one has the following compactness result

Corollary 2.12. Letϕbe a Lipschitz continuous function. Suppose that(u_n)is a sequence of uniformly bounded continuous viscosity solutions of

|e_N +b_n∇u_n|^α

F(D²u_n) +h· ∇u_n

=f_n in B∩ {y_N > a(y)},

u_n=ϕ on B∩ {y_N =a(y)}.

where b_n ≤ b_o, b_o is given above in Proposition 2.9. Suppose that f_n converges simply to some function f in Ω. Then for all r < 1, one can extract from (u_n, b_n) a subsequence which converges uniformly on B_r∩ {yN > a(y)} ×IR, and the limit(u,¯b)satisfies

|eN+ ¯b∇u|^α

F(D²u) +h· ∇u

=f inB∩ {yN > a(y)},

u=ϕ onB∩ {yN =a(y)}.

Remark 2.13. In the absence of boundary conditions, the conclusion is that the sequence (u_n) contains a subsequence which converges locally uniformly and up to a constant toward a solution of

|e_N + ¯b∇u|^α

F(D²u) +h· ∇u

=f in B.

3. Proof of Theorem 1.1.

In fact Theorem1.1 is an immediate consequence of the following local result up to the boundary, together with some argument of finite covering:

Theorem 3.1. Suppose thatF,handf are as in Theorem1.1andϕis a function inC^1,βo. LetB be a ball in IR^N and letabe a C² function defined on IR^N−1 witha(0) = 0,∇a(0) = 0. There existsβ such that for anyu solution of |∇u|^α(F(D²u) +h(x)· ∇u) =f in B∩ {x_N > a(x)}

u=ϕ onB∩ {xN =a(x)},

uisC^1,β( ˜B∩ {xN > a(x)})for any B˜ ⊂⊂B.

Theorem3.1is provedviathe following two “improvement of flatness” lemma and their consequences.

Lemma 3.2. There exist_o∈]0,1[andρ∈]0,1[depending on(α,h∞, λ, Λ, N)such that for anyp∈IR^N and for any viscosity solution uof

|p+∇u|^α

F(D²u) +h(y)·(∇u+p)

=f inB₁ such that osc_B1u≤1 andf_L^∞_(B1)≤_o, there existsp ∈IR^N such that

oscBρ(u−p·x)≤ 1 2ρ.

and

Lemma 3.3. For any a ∈ C², such that a(0) = 0 and∇a(0) = 0, there exist _o >0 and ρ which depend on (α, λ, Λ, N,D²a∞,h∞,ϕ_C1,βo)such that for anyp∈IR^N andua viscosity solution of

|p+∇u|^α(F(D²u) +h(y)·(∇u+p)) =f in B∩ {y_N > a(y)}

u+p·y=ϕ on B∩ {y_N =a(y)},

the following holds: for all x ∈ B such that B₁(x) ⊂ B, if osc_{B1(x)∩{yN>a(y})}u ≤ 1, and f_L^∞_{(B1(x)∩{yN>a(y)})}≤_o then there existsq_x,ρ∈IR^N such that

Bρ(x)∩{yoscN>a(y)}(u(y)−q_x,ρ·y)≤ρ 2·

Suppose that these Lemmata have been proved and let us derive the following one.

Lemma 3.4. Suppose that ρ and _o ∈ [0,1] and B are as in Lemma 3.3 and suppose that u is a viscosity

solution of

|∇u|^α(F(D²u) +h(y)· ∇u) =f in B∩ {y_N > a(y)}

u=ϕ onB∩ {y_N =a(y)} (3.1)

with oscu ≤ 1 and f_∞ ≤ _o, then, there exists β ∈]0,1[, such that for all k and for all x ∈ B such that B₁(x)⊂B, there existsp_k ∈IR^N for which

B_rk(x)∩{yoscN>a(y)}(u(y)−p_k·y)≤r_k^1+β wherer_k =ρ^k.

Remark 3.5. Of course, in the absence of boundary condition we obtain the interior regularity: Suppose that ρand_o∈[0,1] are as in Lemma3.2and suppose thatuis a viscosity solution of

|∇u|^α(F(D²u) +h(·)· ∇u) =f in B₁

with oscu≤1 andf∞≤_o, then there existsβ ∈]0,1[ such that for allk there existsp_k∈IR^N such that

Bosc_rk(u(x)−p_k·x)≤r_k^1+β. Proof of Lemma 3.4. As in [14] we use a recursive argument.

We first remark that one can assume thatϕ(x) =∂_iϕ(x) = 0 fori= 1, . . . , N−1.

Indeed, letube a solution of (3.1). Thenv(y) :=u(y)−u(x)− ∇ϕ(x)·(y−x) satisfies |∇v+q|^α(F(D²v) +h(y)·(∇v+q)) =f inB₁(x)∩ {yN > a(y)}

v(y, a(y)) =φ(y) onB₁(x)∩ {y_N =a(y)}

where q = (∇ϕ(x),0) and φ(y) = ϕ(y)−ϕ(x)− ∇ϕ(x)·(y−x) which satisfies φ(x) = ∂_iφ(x) = 0 for i= 1, . . . , N−1.

So the result obtained forvwould easily transfer to u.

Chooseβ small enough in order thatρ^β>¹₂, and definer_k =ρ^k.

We can start the recursive argument. Fork= 0, takingp_o= 0 yields the desired inequality. Suppose thatp_k exists we now constructp_k+1.

Letϕ_k(y) =r_k^−1−βϕ(x+r_k(y−x)), which satisfies, forβ < β_o, with the above choice ofϕ, ϕk_C^1,β_(B1(x))≤ ϕ_C^1,β.

We consider

u_k(y) =r^−1−β_k (u(r_k(y−x) +x)−p_k·(r_k(y−x) +x)). u_k is well defined onB₁(x)∩ {yN > a_k(y)}, wherea_k(y) =x_N(1−_r¹_k) +^a(rk(y−x_r ^)+x)

k .

It is immediate to see thatu_k is a solution of

|p_kr^−β_k +∇u_k|^α(F(D²u_k) +h_k·(p_kr^−β_k +∇u_k)) =f_k inB₁(x)∩ {y_N > a_k(y)}

u_k+r^−β_k p_k·y=r^−β_k (r_k−1)p_k·x+ϕ_k(y) onB₁(x)∩ {yN =a_k(y)}

withf_k(y) =r^1−β(1+α)_k f(r_k(y−x) +x) andh_k(y) =r_kh(r_k(y−x) +x).

Observe that ify∈B₁(x)∩ {y_N =a_k(y)}, then

a(r_k(y−x) +x)−x_N r_k

≤1

I. BIRINDELLI AND F. DEMENGEL

but this implies, using the mean value’s theorem and|y−x|<1, that

|x_N−a(x)| ≤r_k(1 +|∇a|).

So ifxis not on the boundary i.e. {xN > a(x)} then, fork sufficiently large,B₁(x)⊂ {yN > a_k(y)}, and in that case we don’t have to worry about the boundary terms.

A direct computation gives that,|∇a_k(y)|=|∇a(r_k(y−x) +x)|, while,D²a_k(y) =r_kD²a(r_k(y−x) +x), and henceD²a_k∞=r_kD²a_∞≤ D²a_∞.

Furthermore, as long asβ < _1+α¹ ,

B1(x)∩{yoscN>ak(y)}u_k ≤1, fk_L^∞_{(B1(x)∩{yN>ak(y)})}≤_o andhk∞≤ h∞. Hence, using Lemma3.3with obvious changes, there existsq_k+1∈IR^N such that

Bρ(x)∩{yoscN>ak(y)}(u_k(y)−q_k+1·y)≤ ρ 2. Defining p_k+1=p_k+q_k+1r^β_k, with the assumptions onβ andρ, one gets:

B_rk+1(x)∩{yoscN>a(y)}(u(y)−p_k+1·y)≤ ρ

2r^1+β_k ≤r^1+β_k+1,

since the oscillation is invariant by translation. This is the desired conclusion.

There remains to prove the improvement of flatness lemmata. We start by the interior case with lower order terms.

Proof of Lemma 3.2.Suppose by contradiction that there exist a sequence of functions (f_n)_n whose norm goes to zero, a sequence of (p_n)_n∈IR^N and a sequence of functions (u_n)_n with oscu_n≤1, solutions of

|p_n+∇u_n|^α

F(D²u_n) +h(y)·(∇u_n+p_n)

=f_n, (3.2)

such that, for allq∈IR^N, and anyρ∈(0,1), oscBρ

(u_n(y)−q·y)> ρ

2· (3.3)

Let us suppose first that (p_n)_nis bounded then, up to subsequences, it converges top_∞. Consideringv_n(y) = u_n(y) +p_n·y and using the compactness Remark2.8, we can extract form (v_n)_n a subsequence converging to a limitv_∞, which satisfies

|∇v∞|^α(F(D²v_∞) +h(x)· ∇v∞) = 0.

Remark next that the solutions of such an equation are solutions of

F(D²v_∞) +h(x)· ∇v_∞= 0 (3.4)

as it is the case forh= 0 (see [14]). But, passing to the limit in (3.3) gives that osc_Bρ(v_∞−(q−p_∞)·x)> ^ρ₂. This contradicts the regularity results known for solutions of equation (3.4), (see [20]) and it ends the case where the sequence (p_n)_n is bounded.

In the case where (p_n)_n is unbounded, take a subsequence such that _|p^pn

n| converges to somep_∞. Claim.There existq_∞∈IR^N and a subsequenceσ(n) such that for anyr <1,

n→∞lim h(y)·p_σ(n)=h(y)·q_∞ uniformly inB_r.

We postpone the proof of that claim and we end the Proof of Lemma3.2.