H-log spaces of continuous functions, potentials, and elliptic boundary value problems
In these notes we study a family of Banach spaces, denoted $\, D^{0,\,\al}(\Ov)\,,$ $\,\al \in\,\R^+\,,$ and called H-log spaces. For $\,0<\,\la\leq\,1\,,$ one has $ C^{0,\,\la}(\Ov)\subset D^{0,\,\al}(\Ov) \subset\,C(\Ov)\,,$ with compact embedding. These spaces present the following "inter...
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| Main Author | |
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| Format | Journal Article |
| Language | English |
| Published |
13.03.2015
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| Subjects | |
| Online Access | Get full text |
| DOI | 10.48550/arxiv.1503.04173 |
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| Summary: | In these notes we study a family of Banach spaces, denoted $\,
D^{0,\,\al}(\Ov)\,,$ $\,\al \in\,\R^+\,,$ and called H-log spaces. For
$\,0<\,\la\leq\,1\,,$ one has $ C^{0,\,\la}(\Ov)\subset D^{0,\,\al}(\Ov)
\subset\,C(\Ov)\,,$ with compact embedding. These spaces present the following
"intermediate" regularity behavior. Solutions $\,u\,$ of second order linear
elliptic boundary value problems, under "external forces" $\,f\in\,
D^{0,\,\al}(\Ov)\,$ for some $\,\al>\,1\,,$ satisfy $\,\na^2\,u\in\,
D^{0,\,\al-\,1}(\Ov)\,$. This result is optimal, since $\,\na^2\,u\in\,
D^{0,\,\beta}(\Ov)\,,$ for some $\,\beta >\,\al-1\,,$ is false in general. We
present a preliminary study on this subject. |
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| DOI: | 10.48550/arxiv.1503.04173 |