Regularity of solutions to a parabolic free boundary problem with variable coefficients (Q329798)

In this paper the author proves optimal Lipschitz regularity of viscosity solutions for a type of parabolic free boundary problem with variable coefficients, under the assumption that the free boundary is Lipschitz and assuming a non-degeneracy condition.NEWLINENEWLINEThe family of parabolic free boundary problems studied in this paper is NEWLINE\[NEWLINE \mathcal{L}u-u_t=0 \text{ in } (\{u>0\}\cup \{u\leq 0\}^{\circ})\subset \Omega, NEWLINE\]NEWLINENEWLINENEWLINENEWLINE\[NEWLINE G(u_{\nu}^+,u_{\nu}^-)=1 \text{ along } \partial\{u>0\}\subset\Omega, NEWLINE\]NEWLINE where \(u_{\nu}^{\pm}\) denote the inner normal derivative relative to \(\{u>0\}\) and \(\{u\leq 0\}^{\circ}\), the operator \(\mathcal{L}= \sum_{i,j} a_{ij}(x,t)D_{ij}\) is assumed to have Hölder continuous coefficients, and \(G(a,b)\) is Lipschitz in both variables, strictly increasing in the first variable, and strictly decreasing in the second variable.NEWLINENEWLINEUnder the assumptions that the free boundary of a viscosity solution \(u\) is Lipschitz and that there exists \(m>0\) such that if there exists a space-time ball \(B_R\subset \{u>0\}\) with \(B_R\cap \partial \{u>0\}=\{(x_0,t_0)\}\), then NEWLINE\[NEWLINE \frac{1}{|B_r'(x_0)|}\int_{B_r'(x_0)}u^+ dx\geq mr, NEWLINE\]NEWLINE the author proves that \(u\) is Lipschitz continuous.NEWLINENEWLINEThe key ingredient of this paper is the proof that there exists a cone of full monotonicity up to the free boundary of \(u\). This allows the author to control the time derivative by the spatial gradient, and proceed to show boundedness of \(|\nabla u|\), establishing the Lipschitz continuity of the solution.