A Remark on the Global Lipschitz Regularity of Solutions to Inner Obstacle Problems Involving Degenerate Functionals of p-Growth

Abstract

We extend some recent results of Jagodzinski, Olek and Szczepaniak (Irish Math. Soc. Bull. 61 (2008), 15- 27) on the Lipschitz character of solutions to inner obstacle problems associated to a uniformly elliptic operator to the case of nonlinear, degenerate operators. In a recent paper Jagodzinski, Olek and Szczepaniak (6) investi- gated the Lipschitz regularity of solutions to so-called inner obsta- cle problems extending earlier work of Jordanov (7). By definition we are confronted with an inner obstacle problem if one or several side conditions imposed on the comparison functions are required to hold only on certain specified subregions of the domain of definition, where in case of several obstacles from above and/or below some natural conditions for compatibility have to be satisfied. The basic ideas for these kind of obstacle problems with obstacles defined only on a portion of are explained in the textbook of Kinderlehrer and Stampacchia (8) (see pp. 137-139). The purpose of our short note now is the analysis of the inner obstacle problem for the p-energy functional I(u,) :=