Abstract
In this paper we study the fully nonlinear free boundary problem
where K > 0, and Ω is an unknown open set. Our main result is the optimal regularity for solutions to this problem: namely, we prove that W 2,n solutions are locally C 1,1 inside B 1. Under the extra condition that \({\Omega \supset \{D{u} \neq 0 \}}\) and a uniform thickness assumption on the coincidence set {D u = 0}, we also show local regularity for the free boundary \({\partial \Omega \cap B_1}\).
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Communicated by F. Lin
A. Figalli was partially supported by NSF Grant DMS-0969962. H. Shahgholian was partially supported by Swedish Research Council. A. Figalli acknowledges the Göran Gustafsson Foundation for his visiting appointment to KTH.
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Figalli, A., Shahgholian, H. A General Class of Free Boundary Problems for Fully Nonlinear Elliptic Equations. Arch Rational Mech Anal 213, 269–286 (2014). https://doi.org/10.1007/s00205-014-0734-0
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DOI: https://doi.org/10.1007/s00205-014-0734-0