Title: 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 $u_t-\mbox{div}(A\nabla u) - {B}\cdot\nabla u =0$ on
 time-varying domains with coefficient matrices $A=[a_{ij}]$ and ${B} =
 [b_{i}]$ that satisfy a small Carleson condition. The result is
 motivated by similar results for the elliptic equation
 $\mbox{div}(A\nabla u) + {B}\cdot \nabla u =0$ that were established in
 the papers Kenig, Pipher, Petermichl and myself. The result complements
 the papers of Hofmann 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.
 
 The main result says that given $p\in (2,\infty)$ there exists $C(p)>0$
 such that if the Carleson norm of coefficients is small then the
 Dirichlet boundary value problem for a parabolic equation is solvable in
 $L^p$. This is a join work with S. Hwang.
 
 I shall also discuss a second result (with J. Pipher and S. Petermich)
 that shows that $C(p)\to \infty$ as $p\to\infty$ which is a quantitative
 result complementing the previous results on $L^p$ solvability under
 large Carleson condition (where only existence of $p<\infty$ is
 established, but not how it might depend on the Carleson norm).