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

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, . . ., d−2, with respect to the Hausdorff distance, we prove the unique solvability for p ∈ (2(m+2/(m+3), 2(m+ 2)/(m + 1))). In the case when m = 0, the range p ∈ (4/3, 4) is optimal in view of the known results for Laplace equations. © 2022 Society for Industrial and Applied Mathematics.