Optimal regularity for the variable coefficients parabolic Signorini problem

Abstract

In this paper we discuss the optimal regularity of the variable coefficient parabolic Signorini problem with $W^{1,1}_p$ coefficients and $L^p$ inhomogeneity, where $p>n+2$ with $n$ being the space dimension. Relying on an parabolic Carleman estimate and an epiperimetric inequality, we show the optimal regularity of the solutions as well as the regularity of the regular free boundary.