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Revista Matemática Iberoamericana

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Published online: 2018-05-28

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

Martin Dindoš^[1] and Sukjung Hwang^[2]

(1) Edinburgh University, UK
(2) Yonsei University, Seoul, Republic of Korea

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.

Keywords: Parabolic boundary value problem, Carleson condition, $L^p$ solvability

Dindoš Martin, Hwang Sukjung: The Dirichlet boundary problem for second order parabolic operators satisfying a Carleson condition. Rev. Mat. Iberoam. 34 (2018), 767-810. doi: 10.4171/RMI/1003