Existence and comparison results for variational-hemivariational inequalities
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We consider a prototype of quasilinear elliptic variational-hemivariational inequalities involving the indicator function of some closed convex set and a locally Lipschitz functional. We provide a generalization of the fundamental notion of sub- and supersolutions on the basis of which we then develop the sub-supersolution method for variationalhemivariational inequalities. Furthermore, we give an example to illustrate the abstract theory developed in this paper. 1. Introduction Let Ω ⊂ RN be a bounded domain with Lipschitz boundary ∂Ω, and let V = W 1,p (Ω) and 1,p V0 = W0 (Ω), 1 < p < ∞, denote the usual Sobolev spaces with their dual spaces V ∗ and ∗ V0 , respectively. In this paper, we deal with the following variational-hemivariational inequality:
u ∈ K : − ∆ p u − f ,v − u +
Ω
j o (u;v − u)dx ≥ 0,
∀v ∈ K,
(1.1)
where j o (s;r) denotes the generalized directional derivative of the locally Lipschitz function j : R → R at s in the direction r given by j o (s;r) = limsup y →s, t ↓0
j(y + tr) − j(y) , t
(1.2)
(cf., e.g., [3, Chapter 2]), and K ⊂ V0 is some closed and convex subset. The operator ∆ p u = div(|∇u| p−2 ∇u) is the p-Laplacian, 1 < p < ∞, and f ∈ V0∗ . The main goal of this paper is to develop the sub-supersolution method for variationalhemivariational inequalities of the form (1.1) which may be considered as a prototype of more general problems of this kind. Problem (1.1) includes various special cases:
Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:1 (2005) 33–40 DOI: 10.1155/JIA.2005.33
34
Variational-hemivariational inequalities (i) for K = V0 and j : R → R smooth, (1.1) is the weak formulation of the Dirichlet problem u ∈ V0 : −∆ p u + j (u) = f in V0∗ ,
(1.3)
for which the sub-supersolution method is well known, (ii) if K = V0 , and j : R → R not necessarily smooth, then (1.1) is a hemivariational inequality of the form
u ∈ V0 : − ∆ p u − f ,v − u +
Ω
j o (u;v − u)dx ≥ 0,
∀v ∈ V 0 ,
(1.4)
for which an extension of the sub-supersolution method has been given recently in [2], (iii) if j = 0, then (1.1) becomes a variational inequality for which a sub-supersolution method has been developed in [4, 5], and an extension of this method to systems of variational inequalities has been proved recently in [1]. This paper provides a unified theory on the sub-supersolution method for variationalhemivariational inequalities that includes all the above cited special cases. 2. Notation and hypotheses For functions w, z : Ω → R and sets W and Z of functions defined on Ω, we use the notations: w ∧ z = min{w,z}, w ∨ z = max{w,z}, W ∧ Z = {w ∧ z | w ∈ W, z ∈ Z }, W ∨ Z = {w ∨ z | w ∈ W, z ∈ Z }, and w ∧ Z = {w} ∧ Z, w ∨ Z = {w} ∨ Z. Next we introduce our basic notion of sub-supersolution. Definition 2.1. A function u ∈ V is called a subsolution of (1.1) if the following holds: (i) u ≤ 0 on ∂Ω, (ii) −∆ p u − f ,v − u + Ω j o (u;v − u)dx ≥ 0, ∀v ∈ u ∧ K. Definition 2.2. u¯ ∈ V is a supersolution of (1.1) if the following holds: (i)
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