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.

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 35J25
• 35B65
• Secondary 35J15

Keywords:

• Inhomogeneous boundary conditions
• mixed Dirichlet-conormal boundary value problem
• nontangential maximal function estimates
• Reifenberg flat domains and separations
• VMO coefficients
• W^{1,p} estimates and solvability

Acknowledgement

The authors would like to thank Jongkeun Choi for helpful discussions on the subject. They are also grateful to the referees for their careful reading and many useful comments.

Additional information

Funding

H. Dong and Z. Li were partially supported by the NSF under agreement DMS-1600593.