Global Integrability for Solutions to Obstacle Problems

Abstract

Denote  $$ {\cal K} _{\psi, \theta} (\Omega) =\left\{ v\in W^{1,p} (\Omega) : v\ge \psi, \mbox { a.e. and }  v-\theta \in W_0^{1,p} (\Omega) \right\}, $$ where $\psi$ is any function in $\Omega \subset \mathbb R^N$, $N\ge 2$, with values in $\mathbb R \cup \{\pm \infty\}$ and $\theta $ is a measurable function. This paper deals with global integrability for $u \in {\cal K}_{\psi, \theta}$ such that

$$ \int_\Omega \langle {\cal A} (x,\nabla u), \nabla (w-u) \rangle {\rm d}x \ge \int_\Omega \langle F, \nabla (w-u) \rangle {\rm d}x, \ \ \forall\ w \in {\cal K}_{\psi,\theta} (\Omega), $$ with $|{\cal A} (x,\xi)| \approx |\xi| ^{p-1}$, $1