Existence and uniqueness of Lipschitz continuous graphs with prescribed Levi curvature

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Existence and uniqueness of Lipschitz continuous graphs with prescribed Levi curvature

Existence et unicité des graphes Lipschitz continus à courbure de Levi prescrite

Francesca Da Lio

, Annamaria Montanari

aDipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy bDipartimento di Matematica, Università di Bologna, piazza di Porta S. Donato 5, 40127 Bologna, Italy

Received 19 January 2004; accepted 27 October 2004 Available online 7 April 2005

Abstract

In this paper we prove comparison principles between viscosity semicontinuous sub- and supersolutions of the generalized Dirichlet problem (in the sense of viscosity solutions) for the Levi Monge–Ampère equation. As a consequence of this result and of the Perron’s method we get the existence of a continuous solution of the Dirichlet problem related to the prescribed Levi curvature equation under suitable assumptions on the boundary data and on the Levi curvature of the domain. We also show that such a solution is Lipschitz continuous by building Lipschitz continuous barriers and by applying a weak Bernstein method introduced by Barles in [Differential Integral Equations 4 (2) (1991) 241].

2005 Elsevier SAS. All rights reserved.

Résumé

Dans cet article, nous prouvons des principes de comparaison entre sous et sursolutions du problème de Dirichlet généralisé (dans le sens des solutions de viscosité) pour l’équation de Levi Monge–Ampère. Comme conséquence de ces résultats, nous obtenons l’existence d’une solution continue du problème de Dirichlet associé à l’équation de la courbure de Levi sous des hypothèses convenables sur les conditions au bord et sur l’ouvert. Nous prouvons que la solution est lipschitzienne par la méthode de Bernstein faible introduite par Barles dans [Differential Integral Equations 4 (2) (1991) 241].

2005 Elsevier SAS. All rights reserved.

MSC: 35J70; 35B50; 49L25

✩ This work is partially supported by M.I.U.R., project “Viscosity, metric, and control theoretic methods for nonlinear partial differential equations”.

* Corresponding author.

E-mail addresses: francesca.dalio@unito.it (F. Da Lio), montanar@dm.unibo.it (A. Montanari).

1 The second author was partially supported by University of Bologna. Funds for selected research topics.

0294-1449/$ – see front matter 2005 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpc.2004.10.006

1. Introduction

If M is a hypersurface in Rⁿ, and ifΠ is its second fundamental form, then the eigenvalues ofΠ are the principal curvatures of M. The trace ofΠ is called the mean curvature of M and the determinant of Π is the Gauss–Kronecker curvature. The Dirichlet problem for a convex graph with prescribed curvature is classical (see for example [13]). It has been considered by many authors in the past (see [24] for a list of references), starting from the pioneering works by A.D. Aleksandrov and I.Ya. Bakelman. For a real hypersurfaceM⊂Cⁿ⁺¹, letH denote then-dimensional complex subspace of the tangent space toM. The restriction of the second fundamental form ofMonHis a Hermitian formΛ, which is called the Levi form. More precisely, ifMis a real manifold of classC²which is locally defined byρ, then the Levi formΛ(ρ)is the restriction to the complex tangent spaceH of the Hermitian form associated with the complex Hessian matrixHess_Cρ=(_∂z^∂2ρ

∂z¯p)ⁿ_,p⁺¹₌₁ofρ. The Levi form is of great importance in the study of envelopes of holomorphy in the theory of holomorphic functions inCⁿ⁺¹(see [11,14,18,21] for details on this matter). By using the biholomorphic invariant analogue of Euclidean convexity (see for example [18]), it can be shown that the Levi form is the biholomorphic invariant part of the real Hessian of the defining function. Since Λis obtained from part of the second fundamental form of M, one can expect that it will have some properties similar to curvatures. However,Λitself depends on the defining function for the domain. This obstacle can be avoided as follows. IfMis given locally as{ρ=0}with∂ρ=0, then one can define the normalized Levi form asL(ρ)=^Λ(ρ)_|∂ρ|.Easy calculations show thatLis independent of the defining function ρ and depends only on the domain (a proof of this assertion can be found in [19, Proposition A.1]). Bedford and Gaveau [6] were the first to remark this fact and they used the normalized Levi form to bound the domain over whichM can be defined as the graph of a function of classC².The signature ofLis a biholomorphic invariant ofM, althoughLitself is not invariant. We recall that a domain{ρ <0}is pseudoconvex (strongly pseudoconvex) if the Levi formρ (or equivalently the normalized Levi form) is semidefinite positive (positive definite) on the boundary. The eigenvalues ofLcorrespond to mean curvatures in certain complex directions and, more generally, symmetric functions in the eigenvalues ofLare complex curvatures ofM. The product of the eigenvalues ofL, corresponding to the complex version of the Gauss–Kronecker curvature ofM, is the scalar functionk_M(·)defined by

kM(z)= −∂ρ(z)⁻ⁿ⁻²det







0 ∂₁ρ(z) · · · ∂_n+1ρ(z)

∂₁ρ(z) ∂₁₁ρ(z) · · · ∂_1n+1ρ(z) ... ... . .. ...

∂_n+1ρ(z) ∂_n+11ρ(z) · · · ∂_n+1n+1ρ(z)





. (1.1)

We will call k_M(z)the total Levi curvature ofM at a pointz∈M. In (1.1)∂_j, ∂_j¯, ∂_j¯ denote respectively the derivatives _∂z^∂

j,_∂^∂_z¯

j,_∂z^∂2

∂¯zj and∂ρ =(∂₁ρ, . . . , ∂n+1ρ). To convince the reader that the total Levi curvature is the analogous of the Gauss curvature for the complex structure, we propose the following example.

Example 1.1 (Total Levi curvature of a ball). IfMis the ball of radiusrwith center at zero, then by choosing as defining functionρ= |z₁|²+ · · · + |z_n+1|²−r², an easy calculation showsk_M≡r⁻ⁿ.

However, a cylinder inCⁿ⁺¹may not have zero total Levi curvature, as the following example shows.

Example 1.2 (Total Levi curvature of a cylinder). LetB(0, r)⊂Cⁿ×Rbe a ball of radius r. We consider the following cylinder

B(0, r)×iR=

(z, w)∈Cⁿ×C: |z|²+ w+ w 2

2

−r²<0

.

It is easy to check that 1

2rⁿ k_{∂B(0,r)×iR}(z, w)=r²+(Rew)² 2rⁿ⁺² 1

rⁿ for every(z, w)∈∂B(0, r)×iR.

IfMis the graph of a functionu:Ω→R,Ω⊂Cⁿ×R, we say thatuis Levi convex inΩ if epi(u)= {(z, w)∈ Cⁿ⁺¹: Imw > u(z,Rew)}is pseudoconvex at every point ofM. In this situation for every(z, w)∈epi(u)we have

k_M(z, w)= −2ⁿ⁺²

1+ |Du|²₋(n+2)/2

det

0 u_z¯ u_Rew−i/2 u_z u_zz¯ u_zw u_Rew+i/2 u_wz¯ u_ww

where all the partial derivatives ofu are computed at(z,Rew). The determinant on the right-hand side is often called the Levi Monge–Ampère operator LMA(u), to emphasize the comparison with the Euclidean Monge–

Ampère operator. Even if the Levi curvature has some geometric properties similar to the Euclidean Gauss curvature we must stress that the Levi Monge–Ampère operator is never strictly elliptic, not even on the class of strictly convex functions. In this paper we consider the Dirichlet problem of finding a non parametric hyper- surface with prescribed total Levi curvaturek on a domainΩ ⊂Cⁿ×R⊂Cⁿ×CwhereΩ×iRis strongly pseudoconvex. The problem can be formulated as follows. Givenϕ∈C(∂Ω)andk0 continuous, findu∈C(Ω) Levi convex such that

u|∂Ω=ϕ and LMA(u)=k(·, u)

1+ |Du|²(n+2)/2

onΩ. (1.2)

In the sequel we denote byDu andD²u the gradient and the Hessian matrix of u respectively. The Dirichlet problem for LMA forn=1 was considered first by A. Debiard and Gaveau [12], who gave an estimate for the modulus of continuity of the solution and by Z. Slodkowski and G. Tomassini in [22].

Z. Slodkowski and G. Tomassini defined in [23] viscosity solutions of the Dirichlet problem (1.2) with(1+

|Du|²)raised to the 3n/2 power. The technique developed in [23] is to reduce (1.2) to a Bellman problem for a family of quasilinear degenerate elliptic operatorL_ν and to provide a priori estimates of the solutions of the uniformly elliptic equation

Lν(u)+εu=k^1/n

1+ |Du|²1/2

independent of ε and of ν. The main result is the existence of a Lipschitz continuous viscosity solution u to (1.2). However, the method in [23] requires very strong conditions onkand on the growth of its first and second derivatives (see [23, Theorem 2.4 and condition (2.5), p. 488]). In addition such a solution is shown to be unique only in the particular casek≡0. We reacll that ifuis the solution of (1.2) withk≡0, then forλmax_Ωu, the setΓ₊^λ(u)= {(x,is)∈Ω×iR:u(x)sλ}is both the holomorphic hull and the envelope of holomorphy of C_ϕ,λ=(Ω× {iλ})∪ {(x,is)∈∂Ω×iR: ϕ(x)sλ}. In this paper we consider also the casek≡0 which is seems to be significative above all from the point of view of the regularity theory of pde’s. In [20] it is shown that LMA is degenerate elliptic in the set of Levi convex functions, namely, ifu, vare Levi convex andD²uD²vthen 0L(u)L(v)and LMA(u)LMA(v).Therefore one cannot expect in generalC^∞regularity result for this equation. We notice that ifk≡0 then every real function of the last variableu(Rew)is a solution LMA(u)=0.

Hence, in this case the regularity of a solution comes from the regularity of the boundary data. However, ifk=0 the missing ellipticity direction can be recovered by taking into account the CR structure of the hypersurface. This fact has been used by F. Lascialfari and the second author in [20] to prove that the Levi Monge Ampère operator

has some hypoelliptic properties analogous to the Monge–Ampère operator. Precisely, ifC^m,α denote the usual Hölder space with respect to the Euclidean metric, the following regularity result was proved in [20].

Theorem. LetΩ⊂R²ⁿ⁺¹be an open set andq∈C^∞(Ω×R×R²ⁿ⁺¹),q >0. Ifu∈C^2,α(Ω)is a strictly Levi convex solution to the Levi Monge–Ampère equation

LMA(u)=q(·, u, Du), (1.3)

thenu∈C^∞(Ω).

The existence of classical solution of Eq. (1.3) for n >1 is an interesting open problem while forn=1 it has been solved by Citti, Lanconelli and the second author in [7]. The main aim of this paper is to show the existence and the uniqueness of a Lipschitz continuous viscosity solution of (1.2) under far less restrictive regularity assumptions on the prescribed functionk. To this purpose we use the main tools of the theory of viscosity solutions.

We recall that the theory of viscosity solutions, which was initiated in the early 80’s by the paper of M.G. Crandall and P.L. Lions [8] not only provides a convenient partial differential equations framework for dealing with the lack of the existence of classical solutions, but also leads to the correct formulation of the “generalized” boundary conditions of fully nonlinear elliptic and parabolic pde’s. For a complete survey of the results obtained within the theory of viscosity solutions for the first-order case we refer to the books of Bardi and Capuzzo Dolcetta [1] and Barles [2], while for the second-order case we refer to the “User’s guide” of Crandall, Ishii and Lions [9].

In framework of viscosity solutions the standard Dirichlet boundary conditions have to be relaxed and read in the viscosity sense as

min

−LMA(u)+2ⁿk(·, u)

1+ |Du|²(n+2)/2

, u−ϕ

0 on∂Ω (1.4)

and max

−LMA(u)+2ⁿk(·, u)

1+ |Du|²(n+2)/2

, u−ϕ

0 on∂Ω. (1.5)

Roughly speaking, these relaxed conditions mean that the equations has to hold up to the boundary, when the boundary condition is not assumed in the classical sense.

One of the main tools to prove the existence and the uniqueness of a continuous solution to (1.2) is to provide a comparison principle between semicontinuous sub and supersolutions to (1.2). Indeed the existence follows easily through the Perron’s method by Ishii [15] with the version up to the boundary obtained by the first author in [10].

Hereafter we suppose thatΩ ⊂R²ⁿ⁺¹is a bounded domain with boundary of classC². We list below some basic assumptions we use throughout the paper.

We assume thatk:Ω×R→ [0,+∞)is a continuous bounded function satisfying (H1) for allR >0,there exists_R>0, such that, for everyx∈Ω, and−RvuR

R(u−v)k^1/n(·, u)−k^1/n(·, v), (1.6)

(H2) for allR >0,for all(x, y)∈Ω and|u|R, there exists a modulus of continuityω_Rsuch thatω_R(s)→0 ass→0⁺and

k^1/n(x, u)−k^1/n(y, u)ω_R(|x−y|).

Conditions (H1) and (H2) will be used in Section 3 to prove a comparison principle between viscosity semicontin- uous sub- and supersolution to the problem (1.2). In Section 4 to solve the Dirichlet problem by using the Perron’s method we will use the following additional assumptions onkandΩ:

(H3) Ω×iRis strongly pseudoconvex and, for allξ₀∈∂Ω, sup_Ω×Rk < k_∂Ω×iR(ξ₀).

(H4) sup_Ω×Rk <1/rⁿ, whereris the radius of the minimum sphere containingΩ.

Condition (H3) will guarantee that there is no loss of boundary condition and will also allow to build local barriers to the problem (1.2). Condition (H4) will be used to build a particular global subsolution to (1.2) and thus it will permit to get the existence of a continuous solution by the Perron’s method.

We prove the following theorems.

Theorem 1.1 (The strictly monotone case). Assume (H1)–(H4) hold. Then for anyϕ∈C(∂Ω)there exists a unique continuous viscosity solutionu of (1.2). Moreover, ifk∈C^0,1(Ω×W ) for everyW⊂⊂R, andϕ∈C^1,1(∂Ω), thenu∈C^0,1(Ω).

In order to include the case when the prescribed functionkis constant, (H1) may be modified and relaxed to (H5) for allR >0, for everyx∈Ω, and−RvuR

0k(·, u)−k(·, v). (1.7)

Indeed, whenkis constant inx, i.e.

(H6) k(x, u)=k(u)for all(x, u)∈Ω×R, the result can be strengthened as follows.

Theorem 1.2 (Thex-independent case). Assume that (H2)–(H6) hold. Then, for everyϕ∈C(∂Ω), there exists a unique continuous viscosity solutionuof (1.2). Moreover, ifϕ∈C^1,1(∂Ω), thenu∈C^0,1(Ω).

The proofs of Theorems 1.1 and 1.2 follow classical arguments from the theory of viscosity solutions (see e.g. [9]). The Lipschitz continuity of the solution is obtained by building local barriers on the boundary and by adapting to our setting the weak Bernstein method, which was introduced by Barles in [3] to get gradient bound for viscosity solutions to fully nonlinear degenerate elliptic pde’s.

If in addition the prescribed functionksatisfies (H7) k∈C^0,1(Ω×W )for everyW⊂⊂R, and

(H8) there areα0, L >0 such that D_xk·p+D_uk|p|²

(1+ |p|²)^1/2 +gnk^1+1/nα (1.8)

for almost every(x, u)∈Ω×Rand for all|p|L, for somegg₀,whereg₀ is the universal constant

√2(2−√ 2)(√

2+1)⁻¹,

we prove the existence of a solution of (1.2) by an approximation argument. More precisely, by using local barriers and the weak Bernstein method, we get a priori estimates of the Lipschitz constant and of theL_∞-norm of the solution of the approximating problem. The result is

Theorem 1.3 (The Lipschitz continuous case). Assume (H3)–(H5) and (H7)–(H8) hold. For everyϕ∈C^1,1(∂Ω) there exists a Lipschitz continuous viscosity solutionuof (1.2). Moreover, ifk >0, then the solution is unique.

We should stress that for general fully nonlinear pde’s the weak Bernstein method requires the inequality in (1.8) to hold for some α >0. Because of the particular structure of the Levi Monge–Ampère operator here the constantαcan be zero.

The uniqueness statement in Theorem 1.3 is obtained via a comparison principle between continuous sub- and supersolutions. In the case k >0 a strong comparison principle betweenC² sub- and supersolutions has been proved in [19], by using the fact that the nonellipticity direction can be recovered by commutations.

Our paper is organized as follows. In Section 2 we give a precise viscosity formulation of the Dirichlet problem (1.2) and we show the equivalence with the one given in [23]. In Section 3 we analyze the loss of boundary conditions for the Dirichlet problem (1.2). The question of loss of boundary conditions have been addressed by the first author in [10] for general fully nonlinear second order degenerate elliptic and parabolic equations. As it is well known, this fact may depend on various aspects such as the geometry of the domains, the structural properties of the operator appearing in the equation and the value of the boundary data. The main result of this section is that under the hypothesis (H3) there is no loss of boundary condition for the Dirichlet problem (1.2). In Section 4 we prove comparison principles between viscosity semicontinuous sub- and supersolutions to the problem (1.2) assuming either conditions (H1) and (H2), or (H2) and (H5)–(H6). Using a geometric property of the Levi curvature in this section we also prove a comparison principle between continuous sub- and supersolution for Lipschitz continuous k >0 satisfying (H7). This result yields the uniqueness of a solution of problem (1.2) in Theorem 1.3. As a by- product of the comparison results and the Perron’s method, under the hypothesis (H4) we get the existence of a continuous solution to (1.2) for all continuous boundary data. In Section 5, we show the existence of a Lipschitz continuous solution to (1.2). Moreover by using an approximation argument, together with some a priori estimate for the Lipschitz constant of a solution, we also prove the existence part of Theorem 1.3. In Section 6 we give an estimate of the maximum ball contained inΩ for which the problem (1.2) is solvable in the class of Lipschitz continuous viscosity solutions. Our argument is inspired from [6, Theorem 1] and to [19, Corollary 1.1]. Moreover, when the domainΩ is a ball, we shall prove a nonexistence result which shows that conditions (H3) and (H4) cannot be significantly relaxed.

2. Graphs with prescribed Levi curvature in a viscosity sense

In this section we give the definition of pseudoconvex domains and Levi convex functions in a suitable weak sense. We also give a precise formulation of the Dirichlet problem (1.2) in a viscosity sense.

We start with the following

Definition 2.1. An open setD⊂Cⁿ⁺¹is pseudoconvex in a generalized sense if for everyz₀∈∂Dand for every φ∈C²(Cⁿ⁺¹)such that∂_zφ(z₀)=0 and{φ(z) < φ(z₀)} ⊆Dnearz₀,we haveL(φ)(z₀)0.

One can see that Definition 2.1 is equivalent to the definition of Hartogs pseudoconvexity given in the literature (see e.g. [18]). More precisely we have the following equivalences.

Proposition 2.1. LetD⊂Cⁿ⁺¹be an open set. The following conditions are equivalent:

(1) Dis pseudoconvex in a generalized viscosity sense;

(2) For everyz₀∈∂Dand for every quadratic polynomialq withq(z₀)=0, ∂_zq(z₀)=0,such that{z: q(z) <0} is contained inDnearz₀, thenL(q)(z₀)0;

(3) Dis Hartogs pseudoconvex.

Proof. The proof of Proposition 2.1 is implicitly contained in [17, Theorem 4.1.27]) and we leave details to the reader. 2

LetMbe a nonparametric hypersurface, which is the graph of aC²functionu:Ω→R, whereΩ⊂R²ⁿ⁺¹is an open bounded set, namely,M= {s=u(x₁, y₁, . . . , x_n, y_n, t )}. In this case the coefficientsA_p¯(Du, D²u)of the Levi formL(u)are quasilinear partial differential operators whose real and imaginary parts are given by

Re

Ap¯(Du, D²u)

=

∂xx_pu+∂yy_pu+a∂x_ptu+ap∂xtu +b∂y_ptu+bp∂ytu+(aap+bbp)∂_t²u

,

(2.1) Im

A_p¯(Du, D²u)

=

∂_xypu−∂_xpyu−a_p∂_ytu+a∂_yptu +b_p∂_xtu−b∂_xptu+(b_pa−ba_p)∂_t²u where

a=a(Du)=∂_yu−∂_xu ∂_tu

1+(∂_tu)² , b=b(Du)=−∂_xu−∂_yu ∂_tu

1+(∂_tu)² . (2.2)

In particular for every=1, . . . , n, the diagonal coefficientA(Du, D²u)is a degenerate elliptic second order operator, whose characteristic form

ξ=(ξ₁, . . . , ξ2n+1)−→(ξ2l−1+aξ2n+1)²+(ξ2l+bξ2n+1)²,

is nonnegative definite for everyξ ∈R²ⁿ⁺¹, but has 2n−1 eigenvalues identically zero. In accordance with the notations of the Introduction, we define the Levi Monge–Ampère operator as

LMA(u)=(1+u²_t)det

A,p(Du, D²u)

. (2.3)

Definition 2.2. We say that a functionu∈C²(Ω)is Levi convex (strictly Levi convex) atξ₀∈Ω ifL(u)(ξ₀)0 (>0) and Levi convex (strictly Levi convex) inΩ ifL(u)(ξ )0 (>0) for everyξ∈Ω.

Remark 2.1. The following conditions are equivalent (see [20]):

(1) uis Levi convex inΩ,

(2) the matrixA(Du, D²u)=(A_,p¯(Du, D²u))_,p=1,...,nis nonnegative definite inΩ, (3) the epigraph ofuis pseudoconvex.

In [20] it has been proved that ifu∈C²(Ω)is convex in the classical sense, thenuis Levi convex. In particular, ifD²u0, thenA(Du, D²u)0.The converse obviously is not true.

For anyO⊆R^m, we denote by USC(O)the set of upper semicontinuous functions inO and by LSC(O)the set of lower semicontinuous functions inO.

Definition 2.2 can be generalized to upper semicontinuous functions as follows (see also [23]).

Definition 2.3. We say that a functionu∈USC(Ω)is Levi convex (strictly Levi convex) in a viscosity sense at ξ₀∈Ωif for allφ∈C²(Ω)and for all local maximumξ₀ofu−φwe haveL(φ)(ξ₀)0 (>0) and Levi convex (strictly Levi convex) inΩ ifL(φ)(ξ )0 (>0).

Now we give the definition of viscosity subsolution and supersolution to the following equation

detA(Du, D²u)=k(ξ, u)f (Du) inΩ, (2.4)

wherek:Ω×R→ [0,+∞)is a given continuous function and f (Du)=2ⁿ(1+ |Du|²)^(n+2)/2

(1+(∂_tu)²) .

Our definition extends the one given by Ishii and Lions in [16] in the case of the classical Monge–Ampère equation, and it is analogous to that introduced by Slodkowski and Tomassini in [23] for the Dirichlet problem (1.2).

Definition 2.4. We say thatu∈USC(Ω)(resp.v∈LSC(Ω)) is a viscosity subsolution (resp. supersolution) of (2.4) if for allφ∈C²(Ω)the following holds: at each local maximumξ₀(resp. local minimum ) point ofu−φ (v−φ) then

detA(Dφ, D²φ)(ξ0)k

ξ0, u(ξ0) f

Dφ(ξ0) and

L(φ)(ξ₀)0

(resp. eitherL(φ)(ξ₀)is not semidefinite positive or detA(Dφ, D²φ)(ξ₀)k

ξ₀, u(ξ₀) f

Dφ(ξ₀) and

L(φ)(ξ₀)0).

We recall that for allA∈Sⁿ(the set ofn×nsymmetric matrices) the following matrix identity holds:

(detA)^1/n= inf

Tr(AB): B∈Sⁿ, B0,detB=n⁻ⁿ

, ifA0,

−∞, otherwise. (2.5)

In view of the identity (2.5), we give another viscosity formulation of the Dirichlet problem (1.2). To this purpose we consider the operatorF:Ω×R×R^N×S^N→RwithN=2n+1,defined by

F (ξ, u, p, X):=

k^1/n(ξ, u)f^1/n(p)−

detA(p, X)1/n

, ifA(p, X)0,

+∞, otherwise (2.6)

and the Dirichlet problem

F (ξ, u, Du, D²u)=0 inΩ,

u(ξ )=ϕ(ξ ), in∂Ω, (DP)

whereϕ∈C(∂Ω)the solutionuis a scalar function andDuandD²udenote respectively its gradient and Hessian matrix.

We setF^∗andF_∗the usc and lsc envelope ofF respectively.

Definition 2.5. A functionu∈USC(Ω)(resp.v∈LSC(Ω)) is said to be a viscosity subsolution (resp. supersolu- tion) of (DP) in a generalized sense iff the following property holds:

for allφ∈C²(Ω), at each maximum pointξ₀∈Ω ofu−φwe have F_∗

ξ0, u(ξ0), Dφ(ξ0), D²φ(ξ0)

0 ifξ0∈Ω, min

F_∗

ξ₀, u(ξ₀), Dφ(ξ₀), D²φ(ξ₀)

, u(ξ₀)−ϕ(ξ₀)

0 ifξ₀∈∂Ω, (resp. for allφ∈C²(Ω), at each minimum pointξ0∈Ω ofu−φwe have

F^∗

ξ₀, u(ξ₀), Dφ(ξ₀), D²φ(ξ₀)

0 ifξ₀∈Ω, max

F^∗

ξ₀, u(ξ₀), Dφ(ξ₀), D²φ(ξ₀)

, u(ξ₀)−ϕ(ξ₀)

0 ifξ₀∈∂Ω.

Proposition 2.2. Every solutionuof (2.4) at ξ₀∈Ω in the sense of Definition 2.4 is a solution ofF =0 in the sense of Definition 2.5.

Proof. We show that every solution ofF =0 in the sense of Definition 2.5 is a solution of (2.4) in the sense of Definition 2.4, the other implication being evident. Letube a generalized solution of (DP) and letφ∈C²(Ω)be such thatu−φhas a maximum atξ₀∈Ω, then the inequalityF_∗(ξ₀, u(ξ₀), Dφ(ξ₀), D²φ(ξ₀))0 implies that L(φ)(ξ₀)0. Thus we have

detA(Dφ, D²φ)(ξ₀)k

ξ₀, u(ξ₀) f

Dφ(ξ₀) and

L(φ)(ξ₀)0.

Now suppose thatu−φhas a minimum atξ₀∈Ω and thatF^∗(ξ₀, u(ξ₀), Dφ(ξ₀), D²φ(ξ₀))0. We distinguish the following two cases:

(1)L(φ)(ξ₀)0 andL(φ)(ξ₀)has at least one null eigenvalue. In this case we have 0=detA(Dφ, D²φ)(ξ₀)k

ξ₀, u(ξ₀) f

Dφ(ξ₀) .

(2)L(φ)(ξ₀) >0, then there is a ballB(ξ₀, r),r >0, such thatL(φ)(y) >0 for ally∈B(ξ₀, r). It follows that F^∗

ξ₀, u(ξ₀), Dφ(ξ₀), D²φ(ξ₀)

=F

ξ₀, u(ξ₀), Dφ(ξ₀), D²φ(ξ₀) and the inequalityF^∗(ξ₀, u(ξ₀), Dφ(ξ₀), D²φ(ξ₀))0 implies

detA(Dφ, D²φ)(ξ₀)k

ξ₀, u(ξ₀) f

Dφ(ξ₀) and

L(φ)(ξ₀)0.

Hence we can conclude. 2

In the sequel when we talk about sub- and supersolutions of (DP), we will always mean in a viscosity sense.

We explicitly remark that subsolutions of (DP) are Levi convex in a viscosity sense. Moreover, standard calcu- lations show that ifu∈C²(Ω)∩C(Ω) is Levi convex, thenuis a classical solution of (DP) iffuis a viscosity solution of (DP) (see [23]).

3. Loss of boundary conditions

LetΩ be a bounded open set ofR^N, N=2n+1, withC²boundary. Denote byd a smooth function agreeing in a neighborhoodWof∂Ωwith the signed distance function to∂Ωwhich is positive inΩ and negative inR\Ω and we denote byn(ξ ):= −Dd(ξ )for allξ∈W.Ifξ∈∂Ω, n(ξ )is just the unit outward normal to∂Ωatξ.

In this section we analyze the loss of boundary conditions for the Dirichlet problem (DP) whereF is given by (2.6). The question of loss of boundary conditions have been addressed by the first author in [10] for general fully nonlinear second order degenerate elliptic and parabolic equations. As it is well known this fact may depend on various aspects, such as the geometry of the domains, the structural properties of the operator appearing in the equation and the value of the boundary data (see e.g. the example in [5]). Here we are going to test on the operator (2.6) the conditions which have been found in [10] implying that the Dirichlet boundary conditions are assumed continuously by the solutions of (DP). To this end we introduce the following subsets of the boundary∂Ω: we denote byΣ₋the set of the pointsξ∈∂Ω such that, for allR >0 either

lim inf

w→ξ α↓0

F w,−R,−n(w)+o_α(1)

α ,−1

α²n(w)⊗n(w)+o_α(1) α²

>0 (3.1)

or

lim inf

w→ξ α↓0

F w,−R,−n(w)+o_α(1)

α ,1

αD²d(w)+o_α(1) α

>0 (3.2)

and we denote byΣ₊the set of the pointsξ ∈∂Ω such that, for allR >0 lim sup

w→ξ α↓0

F w, R,n(w)+o_α(1)

α , 1

α²n(w)⊗n(w)+o_α(1) α²

<0 (3.3)

or

lim sup

w→ξ α↓0

F w, R,n(w)+o_α(1) α ,−1

αD²d(w)+o_α(1) α

<0, (3.4)

whereo_α(1)→0 asα↓0 andp⊗pis the matrix(p_ip_j)^N_i,j=1, for allp=(p₁, . . . , p_N). Finally we set Σ:=∂Ω\(Σ₋∪Σ₊).

We premise some comments on the setsΣ_±. In Section 4 of [10], it is proved that there cannot be loss of boundary conditions respectively for the sub- and supersolutions of (DP), namely for every ξ ∈Σ₋ (resp.Σ₊) and any subsolutionsu(resp. supersolutionsv) we haveu(ξ )ϕ(ξ )(v(x)ϕ(ξ )).

We will show that condition (H3) on∂Ω is enough to guarantee thatΣ= ∅.

Now we are going to test the conditions (3.1)–(3.3) and (3.4) in the case of the operator (2.6). We first note that ifuis a defining function forΩ then the Levi curvature (1.1) ofM= {(ξ, s)∈R²ⁿ⁺²: u(x, y, t )−s=0}can be represented as follows

k_M(ξ, u)= −h(Du)detB(Du, D²u) (3.5)

withh(Du):=2ⁿ⁺²(1+ |Du|²)^−((n+2)/2)and

B(Du, D²u)=











0 ∂₁u · · · ∂_tu−i 2

∂₁u ∂₁₁u · · · ∂1tu .. 2

. ... . .. ...

∂tu+i 2

∂_t1u

2 · · · ∂t tu 4











. (3.6)

Define

k_M^∞(ξ ):= lim

η→∞k_M(ξ, ηs).

Then

k_M^∞(ξ )= −2ⁿ|Du|⁻⁽ⁿ⁺²⁾detB_∞(Du, D²u), (3.7) with

B_∞(Du, D²u):=







0 ∂₁u · · · ∂_tu

∂₁u ∂₁₁u · · · ∂1tu ... ... . .. ...

∂_tu ∂_t1u · · · ∂_{t t}u







andk^∞_M(ξ )is exactly the Levi curvaturek_∂Ω×iRof the cylinder∂Ω×iR= {(ξ, s): u(ξ )=0}. By algebraic com- putations one can rewritek_M also as follows (see e.g [20])

k_M(ξ, u)= 1 2ⁿ

(1+u²_t)

(1+ |Du|²)^(n+2)/2detA(Du, D²u), (3.8)

whereA(p, X)is then×nHermitian matrix defined in (2.1).

We list below some facts on the matrixAthat will be useful to check (3.1)–(3.4) on a boundary point. First we have

A(ηp, ηX)= η

(η⁻²+p²_N)²A(p, X, η) (3.9)

where the coefficients ofA(p, X, η)are given by Re

A_p¯(Du, D²u, η)

=(η⁻²+u²_t)²(∂_xxpu+∂_yypu)+(η⁻²+u²_t)(a∂_xptu+a_p∂_xtu +b∂_yptu+b_p∂_ytu)+(aa_p+bb_p)∂_t²u,

Im

A_p¯(Du, D²u, η)

=(η⁻²+u²_t)²(∂xy_pu−∂x_pyu)+(η⁻²+u²_t)(−a_p∂ytu+a∂y_ptu +b_p∂_xtu−b∂_xptu)+(b_pa−ba_p)∂_t²u

and

a=a(Du, η)=η⁻¹∂_yu−∂_xu ∂_tu, b=b(Du, η)= −η⁻¹∂_xu−∂_yu ∂_tu.

MoreoverA(p, X, η)converges toA_∞(p, X)asη→ ∞locally uniformly in(p, X),where the real part and the imaginary part of(A_∞)_p(Du, D²u)are given by

Re

(A_∞)_p(Du, D²u)

=(u_t)⁴(∂_xxpu+∂_yypu)+u²_t(a^∞∂_xptu+a_p^∞∂_xtu +b^∞ ∂_yptu+b_p^∞∂_ytu)+(a^∞ a_p^∞+b^∞ b_p^∞)∂_t²u,

(3.10) Im

(A_∞)_p(Du, D²u)

=(u_t)⁴(∂_xypu−∂_xpyu)+u²_t(−a_p^∞∂_ytu+a^∞ ∂_yptu +b^∞_p∂_xtu−b^∞ ∂_xptu)+(b^∞_pa^∞−b^∞ a_p^∞)∂_t²u with

a^∞=a^∞ (Du)= −∂_xu ∂_tu, b^∞ =b^∞ (Du)= −∂_yu ∂_tu. (3.11) Next we start analyzing the two conditions (3.1), (3.3). Standard computations show that detB_∞(n, n⊗n)= detB_∞(−n,−n⊗n)=0.

Now we takeξ0∈∂Ω and we distinguish two cases.

Case 1: for allα >0 small and for allwclose toξ₀the matrixA(^−n(w)_α^+oα(1),−_α¹2n(w)⊗n(w)+^oα_α⁽¹⁾2 )is not semidefinite positive. In this case we trivially have

lim inf

w→ξ α↓0

F w, R,−n(w)+o_α(1)

α ,− 1

α²n(w)⊗n(w)+o_α(1) α²

0.

Case 2: there are subsequencesα_n→0 andw_n→ξ₀(that we continue to denote byαandw) such that the matrixA(^−n(w)_α^+oα(1),−_α¹2n(w)⊗n(w)+^oα_α⁽¹⁾2 )is semidefinite positive. In this case the following estimate holds.

F w,−R,−n(w)+o_α(1)

α ,−1

α²n(w)⊗n(w)+o_α(1) α²

=f^1/n −n(w)+o_α(1) α

k^1/n(w,−R)

−

detA −n(w)+o_α(1)

α ,− 1

α²n(w)⊗n(w)+o_α(1) α²

f⁻¹ −n(w)+o_α(1) α

1/n .

By using the identities (3.7), (3.8) and the fact that detB_∞(−n,−n⊗n)=0,we get lim inf

w→ξ0

α↓0

−

detA −n(w)+oα(1)

α ,− 1

α²n(w)⊗n(w)+oα(1) α²

f⁻¹ −n(w)+oα(1) α

1/n

+k^1/n(w,−R) lim inf

w→ξ₀ α↓0

−2

−det B_∞

−n(w),−n(w)⊗n(w)1/n

+k^1/n(w,−R)

=k^1/n(ξ₀,−R)0.

Thus sincef^1/n0 we finally obtain lim inf

w→ξ₀ α↓0

F w,−R,−n(w)+o_α(1)

α ,− 1

α²n(w)⊗n(w)+o_α(1) α²

lim inf

w→ξ₀ α↓0

f^1/n −n(w)+o_α(1) α

k^1/n(ξ₀,−R)0.

In a similar way one sees that lim sup

w→ξ0

α↓0

F w, R,n(w)+o_α(1)

α , 1

α²n(w)⊗n(w)+o_α(1) α²

0.

Thus the conditions (3.1) and (3.3) are not satisfied.

This implies that we have to impose some suitable conditions on the Levi curvature of the domain in order that both conditions (3.2) and (3.4) hold.

To this end we assume that Ω satisfies (H3). Then since−d is a defining function of Ω the following two conditions holds for everys∈Randξ₀∈∂Ω:

A_∞

n(ξ₀),−D²d(ξ₀)

>0 (3.12)

and 2

−det B_∞

n(ξ₀),−D²d(ξ₀)1/n

> k^1/n(ξ₀, s). (3.13)

Proposition 3.1. Assume (H3) then both the conditions (3.2) and (3.4) are satisfied.

Proof. We first notice thatA_∞(−n(ξ₀), D²d(ξ₀))= −A_∞(n(ξ₀),−D²d(ξ₀))is not semidefinite positive.

We claim that the matrix A −n(w)+o_α(1)

α ,1

αD²d(w)+o_α(1) α

is not semidefinite positive forα→0 andw→ξ₀as well.

Indeed sinceA_∞(−n(ξ₀), D²d(ξ₀))is not semidefinite positive, there is at least one eigenvalue which is strictly negative. Now from (3.9) it follows that

A −n(w)+o_α(1)

α ,1

αD²d(w)+o_α(1) α

= α⁻¹

(α²+(dt+oα(1)²)²A

Dd+o_α(1), D²d+o_α(1), α⁻¹

. (3.14)