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

Abstract

We obtain the maximal regularity for the mixed Dirichlet-conormal problem in cylindrical domains with time-dependent separations, which is the first of its kind. The boundary of the domain is assumed to be Reifenberg-flat and the separation is locally sufficiently close to a Lipschitz function of m variables, where m = 0, . . . , d − 2, with respect to the Hausdorff distance. We consider solutions in both L p -based Sobolev spaces and L q,p -based mixed-norm Sobolev spaces.

The dirichlet-conormal problem with homogeneous and inhomogeneous boundary conditions

“…More precisely, when the base Ω is Lipschitz, under mild conditions on Λ, we obtain the L 1 and L q solvability for q sufficiently close to 1. Furthermore, based on our earlier results under homogeneous boundary conditions in [9], when ∂Ω is also Reifenberg flat and Λ is close to a Lipschitz graph in m variables in the sense of Hausdorff distance, the solvability range can be extended to q ∈ (1, (m + 2)/(m + 1)), which achieves the aforementioned optimal range when m = 0.…”

“…First, we define spaces for weak solutions. As in [9], by u ∈ H −1 p (Q) we mean that there exist g = (g 1 , . .…”

“…Also, such problems were commonly studied in the combustion theory and in modelling exocytosis, which have a form of active transport mechanism. For applications, see [9] and the references therein.…”

“…However, there are very few results regarding parabolic equations with mixed boundary conditions. See [9] and the references therein. Here, we consider such problems on domains with some time dependency -Lipschitz cylinders with timevarying Λ.…”

“…Also, we do not have a good control of u t close to Λ. Such difficulties are overcome by a duality argument combined with the De Giorgi-Nash-Moser estimate, and a dedicated decomposition in [9]. After the current project, we also plan to study more complicated time varying domains.…”