Nonsmooth variational problems in the limit case and duality
Given an extended real-valued function \(f\) defined on a reflexive Banach space (typically of the form \(f=\Phi+\alpha,\) where \(\Phi\) is a locally Lipschitz function and \(\alpha\) a lower semicontinuous convex function), and a nonempty subset \(S\) of \(X,\) the authors give conditions guaranteeing that the value \(a=\inf_{S} f\) is critical and attained in \(S\), that is, there exists \(u\in S\) with \(f(u)=a\) and such that for all \(v\in X,\) \(\Phi ^{0}(u,v-u)+\alpha(v)\geq\alpha(u)\), where \(\Phi^{0}(u;.)\) denotes the Clarke directional derivative of \(\Phi\). The paper uses duality (minimax principles) and results from [\textit{S. Marano} and \textit{D. Motreanu}, Topol. Methods Nonlinear Anal. 22, 139--158 (2003; Zbl 1213.58010)]. The last part discusses applications to boundary value problems.
scientific article; zbMATH DE number 2153473