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We establish , , solvability of the Dirichlet~boundary value problem for a parabolic equation on time-varying domains with coefficient matrices and that satisfy a small Carleson condition. The results are sharp in the following sense. For a given value of there exists operators that satisfy Carleson condition but fail to have 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 (for some large ) Dirichlet boundary value problem for coefficients that satisfy large Carleson condition was established. We also give a new (substantially shorter) proof of these results.
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Martin Dindoš, Sukjung Hwang, The Dirichlet boundary problem for second order parabolic operators satisfying a Carleson condition. Rev. Mat. Iberoam. 34 (2018), no. 2, pp. 767–810