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

Abstract

We consider the mixed Dirichlet-conormal problem for the heat equation on cylindrical domains with a bounded and Lipschitz base Ω⊂Rd and a time-dependent separation Λ. Under certain mild regularity assumptions on Λ, we show that for any q>1 sufficiently close to 1, the mixed problem in Lq is solvable. In other words, for any given Dirichlet data in the parabolic Riesz potential space Lq1 and the Neumann data in Lq, there is a unique solution and the non-tangential maximal function of its gradient is in Lq 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 Ω is 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 also prove the unique solvability result for any q∈(1,(m+2)/(m+1)). In particular, when m=0, i.e., Λ is Reifenberg-flat of co-dimension 2, we derive the Lq solvability in the optimal range q∈(1,2). For the Laplace equation, such results were established in [25,24,5] and [13].