Existence of solutions of two-phase free boundary problems for fully nonlinear elliptic equations of second order (Q1429306)

The author establishes the existence of a classical solution for the two-phase free boundary problem in the case of a fully nonlinear elliptic equation of second order. The strategy consists in using Perron's method of the least supersolution to secure the existence of a weak solution for the nonlinear problem. Then the free boundary is proved to have finite \(H^{n-1}\) Hausdorff measure and a normal in a measure-theoretic sense \(H^{n-1}\) almost everywhere. By the regularity theory with respect to the free boundary problem proved by the author previously, it follows that the free boundary is a \(C^{1,\alpha}\) surface near \(H^{n-1}\)-a.e. point hence the weak solution found is actually a classical one.