The Dirichlet problem for discontinuous perturbations of the mean curvature operator in Minkowski space

Using the critical point theory for convex, lower semicontinuous perturbations of locally Lipschitz functionals, we prove the solvability of the discontinuous Dirichlet problem involving the operator $u\mapsto{div} (\frac{\nabla u}{\sqrt{1-|\nabla u|^2}})$.

By a solution of (1.2) we mean a function u ∈ W 2,p (Ω) for some p > N , such that ∇u ∞ < 1, which satisfies a.e. x ∈ Ω and vanishes on ∂Ω. At our best knowledge, this type of solutions, but for differential inclusions was firstly considered by A.F. Filippov [8]. Also, for partial differential inclusions we refer the reader to the pioneering works of I. Massabo and C.A. Stuart [12], J. Rauch [14], C.A. Stuart and J.F. Toland [17].
This work is motivated by the recent advances in the study of boundary value problems involving the operator M (see [2], [6] and the references therein) and by the seminal paper of K.-C. Chang [4] where the classical critical point theory is extended to locally Lipschitz functionals in order to study the problem in Ω, u| ∂Ω = 0.
It is worth to point out that the operators M and ∆ have essentially different structures and the theory developed in [4] appears as not being applicable to problem (1.2). Thus, we shall use a more general critical point theory, namely the one concerning convex, lower semicontinuous perturbations of locally Lipschitz functionals, which was developed by D. Motreanu and P.D. Panagiotopoulos [13] (also, see [10], [11]). It should be noticed that, using this theory, various existence results concerning Filippov type solutions for Dirichlet, periodic and Neumann problems involving the "p-relativistic" operator were obtained in the recent paper [9].
A first existence result for the Dirichlet problem involving the operator M was obtained by F. Flaherty in [7], where it is shown that problem has at least one solution, provided that ∂Ω has non-negative mean curvature and ϕ ∈ C 2 (Ω) with ∇ϕ ∞ < 1. The result was generalized in [1] by R. Bartnik and L. Simon, proving that problem is solvable, provided that the Carathéodory function g : Ω × R → R is bounded. More general, if g satisfies the L ∞ -growth condition: it is shown in [2, Theorem 2.1] that (1.4) is still solvable. The approach in [2] relies on Szulkin's critical point theory [16]. The aim of the present paper is to obtain a similar result for the discontinuous problem (1. The rest of the paper is organized as follows. In Section 2 we recall some notions from nonsmooth analysis which will be needed in the sequel. The variational formulation of problem (1.2) is a key step in our approach and it is given in Section 3. Section 4 is devoted to the proof of the main result.

Preliminaries
Let (X, · ) be a real Banach space and X * its topological dual. A functional G : X → R is called locally Lipschitz if for each u ∈ X, there is a neighborhood N u of u and a constant k > 0 depending on N u such that For such a function G, the generalized directional derivative at u ∈ X in the direction of v ∈ X is defined by and the generalized gradient (in the sense of Clarke [5]) of G at u ∈ X is defined as being the subset of X * where ·, · stands for the duality pairing between X * and X. For more details concerning the properties of the generalized directional derivative and of the generalized gradient we refer to [5].
If I : X → (−∞, +∞] is a functional having the structure with G : X → R locally Lipschitz and Φ : X → (−∞, +∞] proper, convex and lower semicontinuous, then an element u ∈ X is said to be a critical point of I provided that The number c = I(u) is called a critical value of I corresponding to the critical point u. According to Kourogenis et al. [10], u ∈ X is a critical point of I iff where ∂Φ(u) stands for the subdifferential of Φ at u ∈ X in the sense of convex analysis [15], i.e., Also, I in (2.1) is said to satisfy the Palais-Smale condition (in short, (P S) condition) if every sequence (u n ) ⊂ X for which (I(u n )) is bounded and for a sequence (ε n ) ⊂ R + with ε n → 0, possesses a convergent subsequence.

The variational setting
In the sequel we shall give the variational formulation of problem (1.2). With this aim, we introduce the set Notice that since W 1,∞ (Ω) is continuously (in fact, compactly) embedded into C(Ω), the evaluation at ∂Ω is understood in the usual sense. According to [2], K 0 is compact in C(Ω) and one has v ∞ ≤ c(Ω) for all v ∈ K 0 , (3.1) with c(Ω) a positive constant. Also, the functional Ψ : C(Ω) → (−∞, +∞] given by is proper, convex and lower semicontinuous [2,Lemma 2.4]. Having in view the growth condition (1.1), we define F : L q (Ω) → R by and, on account of the embedding C(Ω) ⊂ L q (Ω), we introduce the functional From [4, Theorem 2.1], one has that F is locally Lipschitz in L q (Ω) and for all v ∈ L q (Ω). Then, by the continuity of the embedding C(Ω) ⊂ L q (Ω) it is clear that F is locally Lipschitz on C(Ω). Also, since C(Ω) is dense in L q (Ω), it holds (see [5], p. 47): ) for a.e. x ∈ Ω and for all w ∈ C(Ω).
The functional framework of Section 2 fits the following choices: X = C(Ω), Φ = Ψ in (3.2), G = F in (3.3) and Notice that, the compactness of K 0 ⊂ C(Ω) implies that I satisfies the (P S) condition.
Next, for arbitrary u ∈ K 0 , by (1.1) and (3.1), the primitive F satisfies |F (x, u(x))| ≤ C (c(Ω) + c(Ω) q /q) =: C 2 , for a.e. x ∈ Ω. Hence, We deduce that the functional I is bounded from below on C(Ω). Then, using that I verifies the (P S) condition and Theorem 2.1, we have that I is a critical value of I and the proof is complete.