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.
We consider second-order elliptic equations in non-divergence form with oblique derivative boundary conditions. We show that any strong solutions to such problems are twice continuously differentiable up to the boundary provided that the mean oscillations of coefficients satisfy the Dini condition and the boundary is locally represented by a C 1 function whose first derivatives are Dini continuous. This improves a recent result in [6]. An extension to fully nonlinear elliptic equations is also presented.2010 Mathematics Subject Classification. Primary 35J25, 35B65; Secondary 35J15.
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].
The aim of this paper is to establish $W^2_p$ estimate for non-divergence form 2nd-order elliptic equations with the oblique derivative boundary condition in domains with small Lipschitz constants. Our result generalizes those in [ 16] and [ 17], which work for $C^{1,\alpha }$ domains with $\alpha> 1-1/p$. As an application, we also obtain a solvability result. An extension to fully nonlinear elliptic equations with the oblique derivative boundary condition is also discussed.
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