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Flat free boundaries regularity in two-phase problems for a class of fully nonlinear elliptic operators with variable coefficients. (English) Zbl 1179.35349

The author consider the following free boundary problem: \[ \begin{cases} F(D^2u(x),x)=0 &\text{in }\Omega^+(u)=\{x\in \Omega\subset \mathbb R^n:\;u>0\},\\ F(D^2u(x),x)=0 &\text{in }\Omega^-(u)=\{x\in \Omega\subset \mathbb R^n:\;u\leq 0\},\\ u=0&\text{on }\partial\Omega^+\cap\Omega,\\ u_\nu^+=G(u_\nu^-)=0&\text{on }\partial\Omega^+\cap\Omega,\end{cases} \] where \(u\) is a viscosity solution of the equation \(F(D^2u(x),x)=0\). It is proved that flat free boundaries are \(C^{1,\gamma}\).

MSC:

35R35 Free boundary problems for PDEs
35B65 Smoothness and regularity of solutions to PDEs
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