Boundary value problems with rough boundary data

Abstract

We consider linear boundary value problems for higher-order parameter-elliptic equations, where the boundary data do not belong to the classical trace spaces. We employ a class of Sobolev spaces of mixed smoothness that admits a generalized boundary trace with values in Besov spaces of negative order. We prove unique solvability for rough boundary data in the half-space and in sufficiently smooth domains. As an application, we show that the operator related to the linearized Cahn--Hilliard equation with dynamic boundary conditions generates a holomorphic semigroup in $L^p({\mathbb{R}}_{+}^n)\times L^p({\mathbb{R}}^{n-1})$.

41 pages

References in corpus (2)

• On the trace problem for Triebel--Lizorkin spaces with mixed norms
• The Bi-Laplacian with Wentzell boundary conditions on Lipschitz domains