Abstract
We study elliptic and parabolic boundary value problems in spaces of mixed scales with mixed smoothness on the half-space. The aim is to solve boundary value problems with boundary data of negative regularity and to describe the singularities of solutions at the boundary. To this end, we derive mapping properties of Poisson operators in mixed scales with mixed smoothness. We also derive mathcal {R}-sectoriality results for homogeneous boundary data in the case that the smoothness in normal direction is not too large.
Highlights
In recent years, there were some efforts to generalize classical results on the bounded H∞-calculus ([7,8,13,14]) and maximal regularity ([8,9,11,12,21]) of elliptic and parabolic equations to cases in which rougher boundary data can be considered
Including weights which fall outside the A p-range, i.e., weights with r ∈/ (−1, p − 1), provides a huge flexibility concerning the smoothness of the boundary data which can be considered
The elliptic and parabolic equations we are interested in are of the form Mathematics Subject Classification: Primary: 35B65; Secondary: 35K52, 35J58, 46E40, 35S05 Keywords: Boundary value problems, Boundary data of negative regularity, Mixed scales, Mixed smoothness, Poisson operators, Singularities at the boundary, Weights
Summary
There were some efforts to generalize classical results on the bounded H∞-calculus ([7,8,13,14]) and maximal regularity ([8,9,11,12,21]) of elliptic and parabolic equations to cases in which rougher boundary data can be considered. Maximal regularity results for the heat equation with inhomogeneous boundary data have been obtained in [30]. The elliptic and parabolic equations we are interested in are of the form Mathematics Subject Classification: Primary: 35B65; Secondary: 35K52, 35J58, 46E40, 35S05 Keywords: Boundary value problems, Boundary data of negative regularity, Mixed scales, Mixed smoothness, Poisson operators, Singularities at the boundary, Weights
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