Besov Regularity for the Neumann Problem

Abstract

This paper is concerned with the regularity of the solutions to the Neumann problem in Lipschitz domains SZ contained in Rd. Especially, we consider the specific scaleB r s(L (Q)) 117 = s/d + 1/p, of Besov spaces. The regularity of the variational solution in these Besov spaces determines the order of approximation that can be achieved by adaptive and nonlinear numerical schemes. We show that the solution to the Neumann problem is much smoother in the specific Besov scale than in the usual LP-Sobolev scale which justifies the use of adaptive schemes. The proofs are performed by combining some recent regularity results derived by Zanger [23] with some specific properties of harmonic Besov spaces.