The dirichlet-conormal problem with homogeneous and 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 ⊂ R d 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 L q is solvable. In other words, for any given Dirichlet data in the parabolic Riesz potential space L 1 q 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 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 L q solvability in the optimal range q ∈ (1, 2). For the Laplace equation, such results were established in [6,5,7] and [14].