Abstract
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 variational-hemivariational inequalities. Furthermore, we give an example to illustrate the abstract theory developed in this paper.
Highlights
Let Ω ⊂ RN be a bounded domain with Lipschitz boundary ∂Ω, and let V = W1,p(Ω) and V0 = W01,p(Ω), 1 < p < ∞, denote the usual Sobolev spaces with their dual spaces V ∗ and V0∗, respectively
We deal with the following variational-hemivariational inequality: u ∈ K : − ∆pu − f, v − u + jo(u; v − u)dx ≥ 0, ∀v ∈ K, Ω
The main result of this paper is given by the following theorem which provides an existence and comparison result for the variational-hemivariational inequality (1.1)
Summary
Let Ω ⊂ RN be a bounded domain with Lipschitz boundary ∂Ω, and let V = W1,p(Ω) and V0 = W01,p(Ω), 1 < p < ∞, denote the usual Sobolev spaces with their dual spaces V ∗ and V0∗, respectively. We deal with the following variational-hemivariational inequality:. Where jo(s;r) denotes the generalized directional derivative of the locally Lipschitz function j : R → R at s in the direction r given by jo(s; r). For which the sub-supersolution method is well known, (ii) if K = V0, and j : R → R not necessarily smooth, (1.1) is a hemivariational inequality of the form u ∈ V0 : − ∆pu − f , v − u + jo(u; v − u)dx ≥ 0, ∀v ∈ V0,. This paper provides a unified theory on the sub-supersolution method for variationalhemivariational inequalities that includes all the above cited special cases
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