The Dirichlet boundary problem for second order parabolic operators satisfying Carleson condition

Abstract

We establish $L^p$, $2\le p\le\infty$ solvability of the Dirichlet boundary value problem for a parabolic equation $u_t-\mbox{div}(A\nabla u)=0$ on time-varying domains with coefficient matrix $A=(a_{ij})$ that satisfy a small Carleson condition. The result is motivated by similar results for the elliptic equation $\mbox{div}(A\nabla u)=0$ that were established previously.