A General Class of Free Boundary Problems for Fully Nonlinear Elliptic Equations

By Alessio Figalli, H. Shahgholian. 2012

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AbstractIn this paper we study the fully nonlinear free boundary problem $$\left\{\begin{array}{ll}F(D^{2}u) = 1 & {\rm almost \, everywhere \, in}\, B_{1} \cap \Omega\\ |D^{2} u| \leqq K & {\rm almost \, everywhere \, in} \, B_{1} \setminus \Omega,\end{array}\right.$$F(D2u)=1almosteverywhereinB1∩Ω|D2u|≦KalmosteverywhereinB1\Ω,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 W2,n solutions are locally C1,1 inside B1. Under the extra condition that $${\Omega \supset \{D{u} \neq 0 \}}$$Ω⊃{Du≠0} and a uniform thickness assumption on the coincidence set {Du = 0}, we also show local regularity for the free boundary $${\partial \Omega \cap B_1}$$∂Ω∩B1.

Subfields: Applied Mathematics

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