Optimal regularity of mixed Dirichlet-conormal boundary value problems for parabolic operators

Jongkeun Choi; Hongjie Dong; Zongyuan Li

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arXiv · 2101.01654

Optimal regularity of mixed Dirichlet-conormal boundary value problems for parabolic operators

Jongkeun Choi·Hongjie Dong·Zongyuan Li

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Abstract

We obtain the regularity of solutions in Sobolev spaces for the mixed Dirichlet-conormal problem for parabolic operators in cylindrical domains with time-dependent separations, which is the first of its kind. Assuming the boundary of the domain to be Reifenberg-flat and the separation to be locally sufficiently close to a Lipschitz function of $m$ variables, where $m=0,\ldots,d-2$, with respect to the Hausdorff distance, we prove the unique solvability for $p\in (2(m+2/(m+3),2(m+2)/(m+1)))$. In the case when $m=0$, the range $p\in(4/3,4)$ is optimal in view of the known results for Laplace equations.

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BibTeX RIS

Jongkeun Choi, Hongjie Dong, Zongyuan Li. 2021-01-05. Optimal regularity of mixed Dirichlet-conormal boundary value problems for parabolic operators. https://arxiv.org/abs/2101.01654

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