Coercive and semicoercive semivariational inequalities on convex sets (Q2719449)

The authors discuss the following problem. To find \(u\in K \) such that the inequality NEWLINE\[NEWLINE a(u,\nu-u)+\int_\Omega(D_cj)(u)\cdot(\nu-u) d\Omega\geq (l,\nu-u),\quad \forall \nu\in K\tag{1} NEWLINE\]NEWLINE holds true. Here \(V\) is a real Hilbert space such that \(V\subset L^2(\Omega)\subset V^*\), where \(V^*\) denotes the dual space \(V\), \(\Omega\) is an open bounded domain in \(\mathbb{R}^n\), and the injections are continuous \((\|\nu\|_{L^2(\Omega)}\leq c_1\|\nu\|) \) and dense. Moreover, \(V\subset L^2(\Omega) \) is compact; \(K\) is a convex closed subset of \(V\), such that \(0\in K\); \(a\: V\times V\to \mathbb{R} \) is a bilinear, continuous, positive form and \(l\in V^*\); NEWLINE\[NEWLINE(D_cf)(x)\cdot h = \limsup\limits_{_{\substack{ y\to x\\ \lambda\to 0_+}}} \frac{f(y+\lambda h)-f(y)}{\lambda}.NEWLINE\]NEWLINE The authors establish the existence and solvability conditions for the inequalities of type (1).