Estimates of the second-order derivatives for solutions to the two-phase parabolic problem

Abstract

The L_{\infty}-estimates of the second derivatives for solutions of the parabolic free boundary problem with two phases \Delta u-\partial_{t}u=\lambda^{+}\chi_{\left\{ u>0\right\} }-\lambda^{-}\chi_{\left\{ u<0\right\} }\textrm{in }B_{1}^{+}\times]-1,0],\textrm{ }\lambda^{\pm}\geq0,\lambda^{+}+\lambda^{-}>0, satisfying the non-zero Dirichlet condition on \Pi_{1}:=\left\{ (x,t):\left|x\right|\leq1,x_{1}=0,-1<t\leq0\right\}, are proved.