The Dirichlet boundary problem for second order parabolic operators satisfying a 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- \mathrm {div}(A\nabla u) - \boldsymbol{B}\cdot\nabla u =0 on time-varying domains with coefficient matrices A=[a_{ij}] and \boldsymbol{B} = [b_{i}] that satisfy a small Carleson condition. The results are sharp in the following sense. For a given value of 1 < p < \infty there exists operators that satisfy Carleson condition but fail to have L^p solvability of the Dirichlet problem. Thus the assumption of smallness is sharp . Our results complements results of Hofmann, Lewis and Rivera-Noriega, where solvability of parabolic L^p (for some large p ) Dirichlet boundary value problem for coefficients that satisfy large Carleson condition was established. We also give a new (substantially shorter) proof of these results.