REGULARITY OF THE FREE BOUNDARY FOR SOME ELLIPTIC AND PARABOLIC PROBLEMS. II

In this paper we extend the results of the first one to solutions of some obstacle problem in the semilinear elliptic case that are used as a model for a gas problem. More precisely, we prove that the points of the free boundary, where the zero set has no density, lie in a Lipschitz surface. Furthermore, we get the C ¹ regularity for singular points with some (n ? 1)-density.

We also investigate the free boundary at points with density. We show that the set of these points is locally a C ¹ surface. This result is an extension of those achieved by Alt and Phillips [3] Alt, H. W. and Phillips, D. 1986. A free boundary problem for semilinearelliptic equations. J. Reine Angew. Math., 368: 63–107.  [Google Scholar], where it is used a concept stronger than the “density” applied here.