We study the divergence form second-order elliptic equations with mixed Dirichlet-conormal boundary conditions. The unique W 1,p solvability is obtained with p being in the optimal range (4/3, 4). The leading coefficients are assumed to have small mean oscillations and the boundary of domain is Reifenberg flat. We also assume that the two boundary conditions are separated by some Reifenberg flat set of co-dimension 2 on the boundary. 2010 Mathematics Subject Classification. Primary 35J25, 35B65; Secondary 35J15. Key words and phrases. Mixed boundary value problem, second-order elliptic equations of divergence form, Reifenberg flat domains, W 1,p estimate and solvability. H. Dong and Z. Li were partially supported by the NSF under agreement DMS-1600593.
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.
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