Lipschitz Character of Solutions to the Double Inner Obstacle Problems

Abstract

In our paper we consider the inner problem with l 2 N impediments from below, the inner problem with m 2 N impediments from above and the double inner problem with l + m impediments. Assuming the Lipschitz character of the obstacles we show that the corresponding solutions are also Lipschitz. We extend here the result given in (SV), where the author considered the inner obstacle problem with a single impediment from below. Our work is based on the ideas introduced by J. Jordanov from 1982 who investigated H 1;p () regularity of solutions to inner obstacle problems. In the 1970's there was considerable interest in the analysis of obstacle problems. This was connected with the development of research on variational inequalities and has been studied by many authors (see (BC), (BS), (T) and references therein). The majority of results concentrated on, natural from a mathematical point of view, problems of existence and uniqueness of the solutions. However, in case of variational inequalities corresponding to obstacle problems additional questions regarding, e.g., the coincidence set (cf. (DS1), (DS2)) or regularity of the solutions (cf. (BS)) can be posed. These problems seem to be interesting due to possible applications.