The Dirichlet-conormal problem with homogeneous and inhomogeneous boundary conditions

Abstract

We consider the mixed Dirichlet-conormal problem on irregular domains in ${\mathbb{R}}^d$. 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 co-dimension $2$ or is locally sufficiently close to a Lipschitz function of $m$ variables, where $m=1,\dots ,d-2$. For the non-tangential maximal function estimate, we also require the domain to be Lipschitz.

References in corpus (4)

• The L^p Boundary Value Problems on Lipschitz Domains
• Estimates for the $L^q$-mixed problem in $C^{1,1}$-domains
• Optimal regularity for a Dirichlet-conormal problem in Reifenberg flat domain
• The conormal derivative problem for higher order elliptic systems with irregular coefficients