A boxing Inequality for the fractional perimeter
| Autor: | Daniel Spector, Augusto C. Ponce |
|---|---|
| Přispěvatelé: | UCL - SST/IRMP - Institut de recherche en mathématique et physique |
| Rok vydání: | 2020 |
| Předmět: |
Trace (linear algebra)
010102 general mathematics Dimension (graph theory) Hausdorff space 010103 numerical & computational mathematics First order 01 natural sciences Functional Analysis (math.FA) Theoretical Computer Science Mathematics - Functional Analysis Sobolev space Combinatorics Perimeter Mathematics - Analysis of PDEs Mathematics (miscellaneous) Bounded function Content (measure theory) FOS: Mathematics 0101 mathematics Primary: 46E30 46E35 Secondary: 31B15 42B35 Analysis of PDEs (math.AP) Mathematics |
| Zdroj: | Scuola Normale Superiore di Pisa. Annali. Classe di Scienze, Vol. 20, p. 107–141 (2020) |
| ISSN: | 2036-2145 0391-173X |
| DOI: | 10.2422/2036-2145.201711_012 |
| Popis: | We prove the Boxing inequality: $$\mathcal{H}^{d-\alpha}_\infty(U) \leq C\alpha(1-\alpha)\int_U \int_{\mathbb{R}^{d} \setminus U} \frac{\mathrm{d}y \, \mathrm{d}z}{|y-z|^{\alpha+d}},$$ for every $\alpha \in (0,1)$ and every bounded open subset $U \subset \mathbb{R}^d$, where $\mathcal{H}^{d-\alpha}_\infty(U)$ is the Hausdorff content of $U$ of dimension $d -\alpha$ and the constant $C > 0$ depends only on $d$. We then show how this estimate implies a trace inequality in the fractional Sobolev space $W^{\alpha, 1}(\mathbb{R}^d)$ that includes Sobolev's $L^{\frac{d}{d - \alpha}}$ embedding, its Lorentz-space improvement, and Hardy's inequality. All these estimates are thus obtained with the appropriate asymptotics as $\alpha$ tends to $0$ and $1$, recovering in particular the classical inequalities of first order. Their counterparts in the full range $\alpha \in (0, d)$ are also investigated. |
| Databáze: | OpenAIRE |
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