Optimal Regularity for a Dirichlet-Conormal Problem in Reifenberg Flat Domain

Abstract

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.

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

“…This step is standard, by multiplying a cut-off function in the t variable with sufficiently small support, using H ölder's inequality, and then choosing λ large enough. Such argument can be found in the proof of [5,Corollary 5.2]. Combining this with the H 1 p solvability in Theorem 2.4, we immediately obtain the H 1 q,p -solvability and estimate for equations with a i = b i = c = 0.…”

“…Proof. The estimate (3.7) is a simple consequence of H ölder's inequality and (3.6), the proof of which is the same as that of [5,Corollary 3.2 (a)]. Here the chain of inclusions…”

“…See Section 2.1 for the definitions of function spaces. In the special case when Γ is Reifenberg-flat of co-dimension 2, i.e., m = 0, we get the solvability when p ∈ (4/3, 4), which is an optimal range in view of the known results for the Laplace equation in [26] and for elliptic equations in [5]. Furthermore, under the same conditions, in Theorem 2.5 we also obtain the solvability in mixed-norm space L t q (L x p ) for any q ∈ [p, ∞) when p ∈ 2(m+2) m+3 , 2 .…”

“…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.…”

The Dirichlet-conormal problem with homogeneous and inhomogeneous boundary conditions

“…In this paper, we continue our discussion in [5] on the mixed Dirichlet-conormal boundary value problems. On a domain Ω ⊂ R d , we consider the following secondorder symmetric divergence form elliptic equation, with two types of boundary conditions prescribed on two different parts of the boundary:…”