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 $u\mapsto\mbox{div} \Big(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\Big)$

Bereanu, C.

Jebelean, Petru

Serban, Calin-Constantin

Electronic journal of qualitative theory of differential equations

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Electronic Journal of Qualitative Theory of Differential Equations2015, No. 35, 1–7; doi: 10.14232/ejqtde.2015.1.35 http://www.math.u-szeged.hu/ejqtde/The Dirichlet problem for discontinuous perturbationsof the mean curvature operator in Minkowski spaceCristian Bereanu1,2, Petru Jebelean3 and Ca˘lin S, erbanB 31Faculty of Mathematics and Computer Science, University of Bucharest,14 Academiei Street, Bucharest, RO-70109, Romania2Institute of Mathematics “Simion Stoilow” of the Romanian Academy,21 Calea Grivit,ei, Sector 1, Bucharest, RO-010702, Romania3Department of Mathematics, West University of Timis, oara,4 Blvd. V. Pârvan, Timis, oara, RO-300223, RomaniaReceived 15 April 2015, appeared 6 July 2015Communicated by Gabriele BonannoAbstract. Using the critical point theory for convex, lower semicontinuous perturba-tions of locally Lipschitz functionals, we prove the solvability of the discontinuousDirichlet problem involving the operator u 7→ div(∇u√1−|∇u|2).Keywords: nonsmooth critical point theory, discontinuous Dirichlet problem, meancurvature operator, Palais–Smale condition.2010 Mathematics Subject Classification: 34A60, 49J40, 49J52.1 IntroductionLet Ω be an open bounded set inRN (N ≥ 2) with boundary ∂Ω of class C2 and f : Ω×R→ Rbe a measurable function satisfying the growth condition| f (x, s)| ≤ C(1+ |s|q−1), a.e. x ∈ Ω and all s ∈ R, (1.1)with some q ∈ (1,∞) and C a positive constant. For a.e. x ∈ Ω and all s ∈ R, we denotef (x, s) := limδ↘0ess inf{ f (x, t) : |t− s| < δ}andf (x, s) := limδ↘0ess sup{ f (x, t) : |t− s| < δ}.In this paper we consider the discontinuous Dirichlet problem with mean curvature oper-ator in Minkowski space:M(u) := div(∇u√1− |∇u|2)∈[f (x, u), f (x, u)]in Ω, u|∂Ω = 0. (1.2)BCorresponding author. Email: cserban2005@yahoo.com2 C. Bereanu, P. Jebelean and C. S, erbanWe assume thatf and f are N-measurable (1.3)(recall, a function h : Ω ×R → R is called N-measurable if h(·, v(·)) : Ω → R is measurablewhenever v : Ω→ R is measurable [3]).By a solution of (1.2) we mean a function u ∈W2,p(Ω) for some p > N, such that ‖∇u‖∞ <1, which satisfiesM(u)(x) ∈[f (x, u(x)), f (x, u(x))], a.e. x ∈ Ωand vanishes on ∂Ω. At our best knowledge, this type of solutions, but for differential inclu-sions was firstly considered by A. F. Filippov [7]. Also, for partial differential inclusions werefer the reader to the pioneering works of I. Massabo and C. A. Stuart [12], J. Rauch [14],C. A. Stuart and J. F. Toland [16].This work is motivated by the recent advances in the study of boundary value problemsinvolving the operator M (see [2, 6] and the references therein) and by the seminal paperof K.-C. Chang [4] where the classical critical point theory is extended to locally Lipschitzfunctionals in order to study the problem∆u ∈[f (x, u), f (x, u)]in Ω, u|∂Ω = 0.It is worth to point out that the operators M and ∆ have essentially different structures andthe theory developed in [4] appears as not being applicable to problem (1.2). Thus, we shalluse a more general critical point theory, namely the one concerning convex, lower semicontin-uous perturbations of locally Lipschitz functionals, which was developed by D. Motreanu andP. D. Panagiotopoulos [13] (also, see [10,11]). It should be noticed that, using this theory, vari-ous existence results concerning Filippov type solutions for Dirichlet, periodic and Neumannproblems involving the “p-relativistic” operatoru 7→( |u′|p−2u′(1− |u′|p)1−1/p)′were obtained in the recent paper [9].A first existence result for the Dirichlet problem involving the operator M was obtainedby F. Flaherty in [8], where it is shown that problemM(u) = 0 in Ω, u|∂Ω = ϕ,has at least one solution, provided that ∂Ω has non-negative mean curvature and ϕ ∈ C2(Ω)with ‖∇ϕ‖∞ < 1. The result was generalized in [1] by R. Bartnik and L. Simon, proving thatproblemM(u) = g(x, u) in Ω, u|∂Ω = 0 (1.4)is solvable, provided that the Carathéodory function g : Ω×R→ R is bounded. More general,if g satisfies the L∞-growth condition:for each ρ > 0 there is some αρ ∈ L∞(Ω) such that|g(x, s)| ≤ αρ(x) for a.e. x ∈ Ω, ∀ s ∈ R with |s| ≤ ρ,it is shown in [2, Theorem 2.1] that (1.4) is still solvable. The approach in [2] relies on Szulkin’scritical point theory [17]. The aim of the present paper is to obtain a similar result for theDiscontinuous perturbations of the mean curvature operator 3discontinuous problem (1.2). Precisely, we show in the main result (Theorem 4.1) that underassumptions (1.1) and (1.3) problem (1.2) always has at least one solution.The rest of the paper is organized as follows. In Section 2 we recall some notions fromnonsmooth analysis which will be needed in the sequel. The variational formulation of prob-lem (1.2) is a key step in our approach and it is given in Section 3. Section 4 is devoted to theproof of the main result.2 PreliminariesLet (X, ‖ · ‖) be a real Banach space and X∗ its topological dual. A functional G : X → R iscalled locally Lipschitz if for each u ∈ X, there is a neighborhood Nu of u and a constant k > 0depending on Nu such that|G(w)− G(z)| ≤ k‖w− z‖, ∀ w, z ∈ Nu.For such a function G, the generalized directional derivative at u ∈ X in the direction of v ∈ X isdefined byG0(u; v) = lim supw→u, t↘0G(w + tv)− G(w)tand the generalized gradient (in the sense of Clarke [5]) of G at u ∈ X is defined as being thesubset of X∗∂G(u) = {η ∈ X∗ : G0(u; v) ≥ 〈η, v〉, ∀ v ∈ X} ,where 〈·, ·〉 stands for the duality pairing between X∗ and X. For more details concerning theproperties of the generalized directional derivative and of the generalized gradient we referto [5].If I : X → (−∞,+∞] is a functional having the structureI = Φ+ G, (2.1)with G : X → R locally Lipschitz and Φ : X → (−∞,+∞] proper, convex and lower semicon-tinuous, then an element u ∈ X is said to be a critical point of I provided thatG0(u; v− u) +Φ(v)−Φ(u) ≥ 0, ∀ v ∈ X.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 iff0 ∈ ∂G(u) + ∂Φ(u),where ∂Φ(u) stands for the subdifferential of Φ at u ∈ X in the sense of convex analysis [15],i.e.,∂Φ(u) = {η ∈ X∗ : Φ(v)−Φ(u) ≥ 〈η, v− u〉, ∀ v ∈ X} .Also, I in (2.1) is said to satisfy the Palais–Smale condition (in short, (PS) condition) if everysequence (un) ⊂ X for which (I(un)) is bounded andG0(un; v− un) +Φ(v)−Φ(un) ≥ −εn‖v− un‖, ∀ v ∈ X,for a sequence (εn) ⊂ R+ with εn → 0, possesses a convergent subsequence.Theorem 2.1. ([11, Theorem 1]) If I is bounded from below and satisfies the (PS) condition thenc = infX I is a critical value of I .4 C. Bereanu, P. Jebelean and C. S, erban3 The variational settingIn the sequel we shall give the variational formulation of problem (1.2). With this aim, weintroduce the setK0 ={v ∈W1,∞(Ω) : ‖∇v‖∞ ≤ 1, v = 0 on ∂Ω}.Notice that since W1,∞(Ω) is continuously (in fact, compactly) embedded into C(Ω), the eval-uation at ∂Ω is understood in the usual sense. According to [2], K0 is compact in C(Ω) andone has‖v‖∞ ≤ c(Ω) for all v ∈ K0, (3.1)with c(Ω) a positive constant. Also, the functional Ψ : C(Ω)→ (−∞,+∞] given byΨ(v) =∫Ω[1−√1− |∇v|2], for v ∈ K0,+∞, for v ∈ C(Ω) \ K0(3.2)is proper, convex and lower semicontinuous [2, Lemma 2.4].Having in view the growth condition (1.1), we define F̂ : Lq(Ω)→ R byF̂ (v) =∫ΩF(x, v), ∀ v ∈ Lq(Ω),whereF(x, s) =∫ s0f (x, ξ) dξ (x ∈ Ω, s ∈ R)and, on account of the embedding C(Ω) ⊂ Lq(Ω), we introduce the functionalF = F̂ |C(Ω). (3.3)From [4, Theorem 2.1], one has that F̂ is locally Lipschitz in Lq(Ω) and∂F̂ (v) ⊂[f (·, v(·)), f (·, v(·))], (3.4)for all v ∈ Lq(Ω). Then, by the continuity of the embedding C(Ω) ⊂ Lq(Ω) it is clear thatF is locally Lipschitz on C(Ω). Also, since C(Ω) is dense in Lq(Ω), the following holds (see[5, p. 47]):∂F̂ (v) = ∂F (v), ∀ v ∈ C(Ω). (3.5)Lemma 3.1. Let v ∈ K0. If ` ∈ ∂F (v), then there is some ζ` ∈ L∞(Ω) such that ζ`(x) ∈[f (x, v(x)), f (x, v(x))]for a.e. x ∈ Ω and〈`, w〉 =∫Ωζ`w (3.6)for all w ∈ C(Ω).Proof. From (3.5) and (3.4) we infer that there is a function ζ` ∈ Lq′(Ω) with 1/q + 1/q′ = 1,such that ζ`(x) ∈[f (x, v(x)), f (x, v(x))]for a.e. x ∈ Ω and (3.6) holds true for all w ∈ Lq(Ω).To see that ζ` ∈ L∞(Ω), from (1.1) and (3.1), one gets−C1 ≤ f (x, v(x)) ≤ f (x, v(x)) ≤ C1, for a.e. x ∈ Ω,with C1 = C(1 + c(Ω)q−1). This shows that |ζ`(x)| ≤ C1 for a.e. x ∈ Ω and the proof iscomplete.Discontinuous perturbations of the mean curvature operator 5The functional framework of Section 2 fits the following choices: X = C(Ω), Φ = Ψ in(3.2), G = F in (3.3) andI := Ψ+F .Notice that, the compactness of K0 ⊂ C(Ω) implies that I satisfies the (PS) condition.4 Main resultWe have the following theorem.Theorem 4.1. Assume that (1.1) and (1.3) hold true. If u is a critical point of I , then u is a solutionof problem (1.2). Moreover, I is bounded from below and attains its infimum at some u0 ∈ K0, whichsolves problem (1.2).Proof. Let u be a critical point of I . Then u ∈ K0 and there exist hu ∈ ∂Ψ(u) and `u ∈ ∂F (u)such that〈hu, w〉+ 〈`u, w〉 = 0, ∀ w ∈ C(Ω).This and the fact that hu ∈ ∂Ψ(u) yieldΨ(w)−Ψ(u) + 〈`u, w− u〉 ≥ 0, ∀ w ∈ C(Ω). (4.1)Using Lemma 3.1 we deduce that there is some ζu = ζ(`u) ∈ L∞(Ω) such thatζu(x) ∈[f (x, u(x)), f (x, u(x))], a.e. x ∈ Ω (4.2)and〈`u, w〉 =∫Ωζuw, ∀ w ∈ C(Ω). (4.3)By virtue of (4.3), inequality (4.1) becomesΨ(w)−Ψ(u) +∫Ωζu(w− u) ≥ 0, ∀ w ∈ C(Ω). (4.4)On account of Lemma 2.2 in [6], for each function e ∈ L∞(Ω), the Dirichlet problemM(v) = e(x) in Ω, v|∂Ω = 0has a unique solution ve ∈ W2,p(Ω) for all 1 ≤ p < ∞. Then, from Lemma 2.3 in [2], one hasthat ve is the unique solution in K0 of the variational inequality∫Ω[√1− |∇v|2 −√1− |∇w|2 + e(w− v)]≥ 0, ∀ w ∈ K0and hence,Ψ(w)−Ψ(ve) +∫Ωe(w− ve) ≥ 0, ∀ w ∈ C(Ω).From this and (4.4), we infer that u = ve, with e = ζu. But, on account of (4.2), this means thatu solves problem (1.2).Next, for arbitrary u ∈ K0, by (1.1) and (3.1), the primitive F satisfies|F(x, u(x))| ≤ C (c(Ω) + c(Ω)q/q) =: C2, for a.e. x ∈ Ω.6 C. Bereanu, P. Jebelean and C. S, erbanHence,|F (u)| ≤∫Ω|F(x, u)| ≤ C2 vol(Ω), ∀ u ∈ K0.We deduce that the functional I is bounded from below on C(Ω). Then, using that I verifiesthe (PS) condition and Theorem 2.1, we have thatc = infC(Ω)I = infK0Iis a critical value of I and the proof is complete.AcknowledgementsThe authors are grateful to the anonymous referee for providing a useful reference for furtherdevelopments of the subject. The work of Ca˘lin S, erban was supported by the strategic grantPOSDRU/159/1.5/S/137750, “Project Doctoral and Postdoctoral programs support for in-creased competitiveness in Exact Sciences research” co-financed by the European Social Fundwithin the Sectoral Operational Programme Human Resources Development 2007-2013.References[1] R. Bartnik, L. 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The Dirichlet problem for discontinuous perturbations of the mean curvature operator in Minkowski space

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