Almost minimizers for semilinear free boundary problems with variable coefficients

We study regularity results for almost minimizers of the functionalF γ ( v ; Ω ) = ∫ Ω⟨ A ( x ) ∇ v , ∇ v ⟩ + q + ( x )( v + ) γ + q − ( x )( v − ) γ d x ,0 ≤ γ ≤ 1 ,where A is a matrix with Hölder continuous coefficients. In the case 0 < γ ≤ 1 we show that an almost minimizer belongs to C 1 , η , where the exponent η is related with the competition between the Hölder continuity of the matrix A , the parameter of almost minimization and γ. In some sense, this regularity is optimal. As far as the case γ = 0 is concerned, our results show that an almost minimizer is C 1 , θlocally in each phase { u > 0 } and { u < 0 } , improving in some sense a recent result of David & Toro.