Extremal solutions of hemivariational inequalities with D. C. -superpotentials (Q2754428)

Let \(X\) denote the Sobolev space \(W^{1,p}_0(\Omega)\), where \(\Omega\subset\mathbb{R}^n\) is a bounded domain with Lipschitz boundary and \(1<p<\infty\). The author studies hemivariational inequalities of the form NEWLINE\[NEWLINE u\in X: \;\langle Au,\varphi-u\rangle +\int_\Omega\Phi^0(u,\varphi-u) dx \geq \langle h,\varphi-u\rangle \quad \forall \varphi\in X,NEWLINE\]NEWLINE where \(A\) is a nonpotential elliptic operator of Leray-Lions type, \(h\in X^*\), \(\Phi^0\) denotes the generalized directional derivative of \(\Phi\) in the sense of Clarke and \(\Phi:\mathbb{R}\to\mathbb{R}\) can be represented in the form \(\Phi_1-\Phi_2\) with \(\Phi_1,\Phi_2\) being convex and having some additional properties. Assuming existence of an upper and a lower solution \(\overline{u}\) and \(\underline{u}\), respectively, satisfying \(\underline{u}\leq\overline{u}\) and some integrability conditions, the author proves the existence of an extremal solution in the interval \([\underline{u},\overline{u}]\). An example with \(p=2\), \(Au=-\Delta u\) and \(\Phi\) satisfying certain growth conditions is discussed.NEWLINENEWLINEFor the entire collection see [Zbl 0961.00018].