A general class of free boundary problems for fully nonlinear elliptic equations

الأصل · EN

In this paper we study the fully nonlinear free boundary problem arrayll F(D²u)=1 & a.e. inB₁ ∩ Ω|D² u| ≤ K & a.e. inB₁Ω, array. 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 Ω⊃ {D u≠ 0 }, and a uniform thickness assumption on the coincidence set {D u = 0 }, we also show local regularity for the free boundary ∂Ω∩ B₁.