Optimal Regularity for the No‐Sign Obstacle Problem |
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Authors: | John Andersson Erik Lindgren Henrik Shahgholian |
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Affiliation: | 1. Mathematics Institute, University of Warwick, Coventry CV4 7AL, UNITED KINGDOM;2. Department of Mathematical Sciences, Norwegian University of Sciences and Technology, Sentralbygg 2, Alfred Getz vei 1, 7491 Trondheim, NORWAY;3. Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, SWEDEN |
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Abstract: | In this paper we prove the optimal $C^{1,1}(B_{1/2})$ ‐regularity for a general obstacle‐type problem under the assumption that $f*N$ is $C^{1,1}(B_1)$ , where N is the Newtonian potential. This is the weakest assumption for which one can hope to get $C^{1,1}$ ‐regularity. As a by‐product of the $C^{1,1}$ ‐regularity we are able to prove that, under a standard thickness assumption on the zero set close to a free boundary point $x^0$ , the free boundary is locally a $C^1$ ‐graph close to $x^0$ 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. |
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