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

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