The problem of studying the regularity of the free boundary that arises when considering the energy minimizing function over the set of those functions bigger than a given has been the subject of intensive research in the last decade. Let me mention H. Lewy and G. Stampacchia [14], D. Kinderlehrer [11], J. C. Nitsche [15] and N. M. Riviere and the author [5] among others. In two dimensions, by the use of analytic reflection techniques due mainly to H. Lewy [13], much was achieved. Recently, the author was able to prove, in a three dimensional filtration problem [4], that the resulting free surface is of class C 1 and all the second derivatives of the variational solution are continuous up to the free boundary, on the non-coincidence set. This fact has not only the virtue of proving that the variational solution is a classical one, but also verifies the hypothesis necessary to apply a recent result due to D. Kinderlehrer and L. Nirenberg, [12] to conclude that the free boundary is as smooth as the obstacle. Nevertheless, in that paper ([4]), strong use was made of the geometry of the problem: this implied that the free boundary was Lipschitz. Also it was apparently essential that the Laplacian of the obstacle was constant. In the first part of this paper we plan to treat the general non-linear free boundary problem as presented in H. Brezis-D. Kinderlehrer [2]. Our main purpose is to prove that if X 0 is a point of density for the coincidence set, in a neighborhood of X 0 the free boundary is a C 1 surface and all the second derivatives of the solution are continuous up to it. In the second part we will s tudy the parabolic case (one phase Stefan problem) as presented by G. Duvaut [7] or A. Friedman and D. Kinderlehrer [9]. There we prove that if for a fixed time, to, the point X 0 is a density point for the coincidence set (the ice) then in a
Search from 250 M+ research papersSearch from 250 M+ research papers
Jan 1, 1986
Luis A Caffarelli 
Open Access