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 $u_t-\mathrm{div}(A\nabla u)-\mathbf{B}\cdot \nabla u=0$ on time-varying domains with coefficient matrices $A=[a_{ij}]$ and $\mathbf{B}=[b_i]$ 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.