The Finiteness Principle for the boundary values of $C^2$-functions
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913380458037248 |
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| author | Shvartsman, Pavel |
| author_facet | Shvartsman, Pavel |
| contents | Let $Ω$ be a domain in $R^n$, and let $N=3\cdot 2^{n-1}$. We prove that the trace of the space $C^2(Ω)$ to the boundary of $Ω$ has the following finiteness property: A function $f:\partialΩ\to R$ is the trace to the boundary of a function $F\in C^2(Ω)$ provided there exists a constant $λ>0$ such that for every set $E\subset\partialΩ$ consisting of at most $N$ points there exists a function $F_E\in C^2(Ω)$ with $\|F_E\|_{C^2(Ω)}\leλ$ whose trace to $\partialΩ$ coincides with $f$ on $E$. We also prove a refinement of this finiteness principle, which shows that in this criterion we can use only $N$-point subsets $E\subset\partialΩ$ which have some additional geometric ``visibility'' properties with respect to the domain $Ω$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_04431 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Finiteness Principle for the boundary values of $C^2$-functions Shvartsman, Pavel Functional Analysis 46E35 Let $Ω$ be a domain in $R^n$, and let $N=3\cdot 2^{n-1}$. We prove that the trace of the space $C^2(Ω)$ to the boundary of $Ω$ has the following finiteness property: A function $f:\partialΩ\to R$ is the trace to the boundary of a function $F\in C^2(Ω)$ provided there exists a constant $λ>0$ such that for every set $E\subset\partialΩ$ consisting of at most $N$ points there exists a function $F_E\in C^2(Ω)$ with $\|F_E\|_{C^2(Ω)}\leλ$ whose trace to $\partialΩ$ coincides with $f$ on $E$. We also prove a refinement of this finiteness principle, which shows that in this criterion we can use only $N$-point subsets $E\subset\partialΩ$ which have some additional geometric ``visibility'' properties with respect to the domain $Ω$. |
| title | The Finiteness Principle for the boundary values of $C^2$-functions |
| topic | Functional Analysis 46E35 |
| url | https://arxiv.org/abs/2406.04431 |