The Dirichlet-conormal problem for the heat equation with inhomogeneous boundary conditions

TL;DR

This paper proves the solvability of the mixed Dirichlet-conormal problem for the heat equation on cylindrical domains with inhomogeneous boundary conditions in various $L_q$ spaces, including Hardy spaces, under mild regularity assumptions.

Contribution

It extends solvability results for the heat equation's mixed boundary value problem to inhomogeneous boundary conditions with minimal regularity assumptions on the domain and boundary separation.

Findings

Solvability in $L_q$ for $q>1$ close to 1.

Unique solutions with boundary data in parabolic Riesz potential and $L_q$ spaces.

Extension to Hardy space data when $q=1$.

Abstract

We consider the mixed Dirichlet-conormal problem for the heat equation on cylindrical domains with a bounded and Lipschitz base $\Omega\subset \mathbb{R}^d$ and a time-dependent separation $\Lambda$. Under certain mild regularity assumptions on $\Lambda$, we show that for any $q>1$ sufficiently close to 1, the mixed problem in $L_q$ is solvable. In other words, for any given Dirichlet data in the parabolic Riesz potential space $\mathcal{L}_q^1$ and the Neumann data in $L_q$, there is a unique solution and the non-tangential maximal function of its gradient is in $L_q$ on the lateral boundary of the domain. When $q=1$, a similar result is shown when the data is in the Hardy space. Under the additional condition that the boundary of the domain $\Omega$ is Reifenberg-flat and the separation is locally sufficiently close to a Lipschitz function of $m$ variables, where $m=0,\ldots,d-2$, with respect to the Hausdorff distance, we also prove the unique solvability result for any $q\in(1,(m+2)/(m+1))$. In particular, when $m=0$, i.e., $\Lambda$ is Reifenberg-flat of co-dimension $2$, we derive the $L_q$ solvability in the optimal range $q\in (1,2)$. For the Laplace equation, such results were established in [6, 5, 7] and [14].