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

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 $m=0,\dots ,d-2$, 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.

arXiv comment34 pages, to appear in SIAM J. Math. Anal. The original submission was divided into two parts as suggested by the referee. This is the first part