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arXiv · 2211.13540

Boundary value problems with rough boundary data

Robert Denk·David Ploß·Sophia Rau·Jörg Seiler
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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 $L^p(\mathbb R^n_+)\times L^p(\mathbb R^{n-1})$.

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Robert Denk, David Ploß, Sophia Rau, Jörg Seiler. 2022-11-24. Boundary value problems with rough boundary data. https://doi.org/10.1016/j.jde.2023.04.001

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