The free boundary for a fourth order variational inequality

Abstract

Consider the variational inequality $$\min_{v \in k}\left\{\int_{\Omega}{|\Delta v|^{2}}-2\int_{{|\Omega}fv}\right\}=\int_{\Omega}{|\Delta u|^{2}}-2\int_{\Omega}{fu}, \quad u \in K,$$ where $\Omega$ is a bounded domain in $R^{2}$ and $$K=\left\{v \in H_{0}^{2}(\Omega),\,\alpha \leq \beta\right\} \quad (\alpha < 0 < \beta).$$ This problem was studied by Brezis and Stampacchia [3] who proved that the solution $u$ belongs to $W_{\mathrm{loc}^{3,p}}(\Omega)$ if $f \in L^{p}(p > 2)$. In this paper we study the free boundary for this problem. Particular attention will be given to the case $-\alpha=\beta\rightarrow 0$. It will be shown, for a special choice of $f$ and $\Omega$, that $u/\beta\rightarrow w$ where $w$ is the solution of a variational inequality for the Laplace operator with obstacle $\frac{1}{2} d^{2}$ and $d$ is the distance function to $\partial\Omega$.