Besov Regularity for the Poisson Equation in Smooth and Polyhedral Cones

Abstract

The regularity of solutions to the Dirichlet and Neumann problems in smooth and polyhedral cones contained in R 3 is studied with particular attention to the specific scale B s T(L T), 1/t = s/3 + 1/2, of Besov spaces. The regularity of the solution in these Besov spaces determines the order of approximation that can be achieved by adaptive and nonlinear numerical schemes. We show that the solutions are much smoother in the specific Besov scale than in the usual L 2-Sobolev scale, which justifies the use of adaptive schemes. The proofs are performed by combining weighted Sobolev estimates with characterizations of Besov spaces by wavelet expansions.