Given a function and , in this paper is considered the following obstacle problem: 1. in ; 2. in ; 3. for those such that ; 4. .
The author proves that if is in then the solution is in for all . In the case where the contact set is convex, the optimal regularity is obtained. Moreover, when is in with the solution is in for all . Finally some applications of the results are presented. In particular, interesting considerations on a well known Signorini problem are given.