A two-phase parabolic free boundary problem with coefficients below the Lipschitz threshold (Q438868)

The following problem is considered: \[ Hu=\Delta u -u_t=\lambda_1(x,t)\chi_{\{ u>0 \}} -\lambda_2(x,t)\chi_{\{ u<0 \}}, \, \mathrm{in } \, Q^-_1=B_1\times (-1,0), \] where \(\lambda_1\), \(\lambda_2\) are strictly positive Hölder continuous functions. Let \(\Gamma^{\pm}(u)=\partial \{ \pm u >0 \} \cap Q^-_1\) and \((0,0)\) is a branching point, i.e. \((0,0)\in \Gamma^+(u)\cap \Gamma^{-}(u)\cap\{ \nabla u=0\} \). The authors prove that \(u\in C^{1,1}_x\cap C^{0,1}_t(Q^-_{r_0})\), where \(Q^-_{r_0}=B_{r_0}\times (-r_o,0]\), under some additional assumptions on \(\lambda_1\), \(\lambda_2\).