Optimal Regularity for the No-Sign Obstacle Problem



In this paper we prove the optimal equation image-regularity for a general obstacle-type problem

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under the assumption that equation image is equation image, where N is the Newtonian potential. This is the weakest assumption for which one can hope to get equation image-regularity. As a by-product of the equation image-regularity we are able to prove that, under a standard thickness assumption on the zero set close to a free boundary point equation image, the free boundary is locally a equation image-graph close to equation image provided f is Dini. This completely settles the question of the optimal regularity of this problem, which has been the focus of much attention during the last two decades. © 2012 Wiley Periodicals, Inc.