Boundary value problems with rough boundary data
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866908336839983104 |
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| author | Denk, Robert Ploß, David Rau, Sophia Seiler, Jörg |
| author_facet | Denk, Robert Ploß, David Rau, Sophia Seiler, Jörg |
| contents | 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 $L^p(\mathbb R^n_+)\times L^p(\mathbb R^{n-1})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_13540 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Boundary value problems with rough boundary data Denk, Robert Ploß, David Rau, Sophia Seiler, Jörg Analysis of PDEs 35J40 (primary), 46E35, 47D06, 35K35 (secondary) 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 $L^p(\mathbb R^n_+)\times L^p(\mathbb R^{n-1})$. |
| title | Boundary value problems with rough boundary data |
| topic | Analysis of PDEs 35J40 (primary), 46E35, 47D06, 35K35 (secondary) |
| url | https://arxiv.org/abs/2211.13540 |