On the Strong Solvability of a Unilateral Boundary Value Problem for Nonlinear Discontinuous Operators in the Plane

Abstract

A uniqueness and existence theorem in the Sobolev space is proved for a unilateral boundary value problem for a class of nonlinear discontinuous operators in the plane. The operator is assumed to satisfy a suitable ellipticity condition, which allows us to apply nearness theory of mappings. Estimate $$ \begin{gathered} \int_\Omega {\sum\limits_{i,j = 1}^2 {\left( {\frac{{\partial ^2 u}} {{\partial x_i \partial x_j }}} \right)} ^2 dx \leqslant \int_\Omega {|\Delta u|^2 dx} } \hfill \\ \forall u \in W^{2,2} (\Omega ):u \geqslant 0,\frac{{\partial u}} {{\partial n}} \geqslant 0, u \cdot \frac{{\partial u}} {{\partial n}} = 0 on \partial \Omega \hfill \\ \end{gathered} $$ having interest in itself, plays a fundamental role.KeywordsSignorini problemunilateral problemstheory of nearness