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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2406.04431 |
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Sommario:
- 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 $Ω$.