## Abstract

Let $\Omega$ be a smooth bounded open set of ${\bf R}^n,A$ a uniformly elliptic second-order operator and $f$ a function belonging to $W^{1,p}(\Omega)\ (p>n)$. For $\varphi,\psi\in H_0^1(\Omega)\cap W^{2,\infty}(\Omega)$, $\varphi\leq\psi$, we prove that any solution $u$ of the problem $(-Au,v-u)\geq(f,v-u)$ for all $v\in K$, $u\in K=${$v\in H_0^1(\Omega)\colon\varphi\leq v\leq\psi$ a.e. in $\Omega$}, belongs to $W^{2,\infty}(\Omega)$. This generalizes the results of H. Brézis and D. Kinderlehrer [Indiana Univ. Math. J. 23 (1973/74), 831--844] in the case of two barriers.