The Dirichlet-conormal problem with homogeneous and inhomogeneous boundary conditions

Abstract

We consider the mixed Dirichlet-conormal problem on irregular domains in R d . Two types of regularity results will be discussed: the W 1,p regularity and a non-tangential maximal function estimate. The domain is assumed to be Reifenberg-flat, and the interfacial boundary is either Reifenberg-flat of codimension 2 or is locally sufficiently close to a Lipschitz function of m variables, where m = 1, . . . , d − 2. For the non-tangential maximal function estimate, we also require the domain to be Lipschitz.

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

“…, x d ). This will be achieved by a boundary Caccioppoli type inequality and the corresponding elliptic estimate obtained in [8], which in turn is a consequence of the Besov type estimate established in [24,2]. Since ∂Ω and Γ are not smooth, the usual flattening boundary argument does not work in our case.…”

“…In the literature, elliptic equations with mixed boundary conditions have been studied quite extensively both from the PDE perspective and harmonic analysis point of view. We refer the reader to [26,22,1,24] and [28,2,5,8] and the references therein. In these papers, regularity of solutions in H ölder, Sobolev, and Besov spaces as well as the non-tangential maximal function estimates were obtained.…”

“…To approximate the domain by domains with flat boundaries and separations, the third step in the proof is to apply the cutoff and reflection argument first used in [6,7] for equations with the pure Dirichlet or conormal boundary condition. Such argument was also used recently in [4,8] for elliptic equations with mixed boundary conditions. Here an additional difficulty is the extra u t term which appears on the right-hand side after the reflection.…”