On a Unilateral Problem Associated with Elliptic Operators

Abstract

Let $\mathcal {A}$ be a uniformly elliptic linear differential expression of second order, defined on the bounded domain $\Omega \subset {R^m}$, and let $\beta \subset R \times R$ be a maximal monotone graph. Under some growth assumption on $\beta$ it is shown that for any given $f \in {L^2}(\Omega )$ the problem: $\mathcal {A}u + \beta (u) \backepsilon f$ on $\Omega ,u = 0$ on $\partial \Omega$, admits a strong solution. It is not required that $\mathcal {A}$ is monotone.