Families of functionals representing Sobolev norms
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2021
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914787383836672 |
|---|---|
| author | Brezis, Haim Seeger, Andreas Van Schaftingen, Jean Yung, Po-Lam |
| author_facet | Brezis, Haim Seeger, Andreas Van Schaftingen, Jean Yung, Po-Lam |
| contents | We obtain new characterizations of the Sobolev spaces $\dot W^{1,p}(\mathbb{R}^N)$ and the bounded variation space $\dot{BV}(\mathbb{R}^N)$. The characterizations are in terms of the functionals $ν_γ (E_{λ,γ/p}[u])$ where \[ E_{λ,γ/p}[u]= \Big\{(x,y )\in \mathbb{R}^N \times \mathbb{R}^N \colon x \neq y, \, \frac{|u(x)-u(y)|}{|x-y|^{1+γ/p}}>λ\Big\} \] and the measure $ν_γ$ is given by $\mathrm{d} ν_γ(x,y)=|x-y|^{γ-N} \mathrm{d} x \mathrm{d} y$. We provide characterizations which involve the $L^{p,\infty}$-quasi-norms $\sup_{λ>0} λ\, ν_γ (E_{λ,γ/p}[u]) ^{1/p}$ and also exact formulas via corresponding limit functionals, with the limit for $λ\to\infty$ when $γ>0$ and the limit for $λ\to 0^+$ when $γ<0$. The results unify and substantially extend previous work by Nguyen and by Brezis, Van Schaftingen and Yung. For $p>1$ the characterizations hold for all $γ\neq 0$. For $p=1$ the upper bounds for the $L^{1,\infty}$ quasi-norms fail in the range $γ\in [-1,0) $; moreover in this case the limit functionals represent the $L^1$ norm of the gradient for $C^\infty_c$-functions but not for generic $\dot W^{1,1}$-functions. For this situation we provide new counterexamples which are built on self-similar sets of dimension $γ+1$. For $γ=0$ the characterizations of Sobolev spaces fail; however we obtain a new formula for the Lipschitz norm via the expressions $ν_0(E_{λ,0}[u])$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_02930 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Families of functionals representing Sobolev norms Brezis, Haim Seeger, Andreas Van Schaftingen, Jean Yung, Po-Lam Functional Analysis Analysis of PDEs Classical Analysis and ODEs 26D10 (26A33, 35A23, 42B25, 42B35, 46E30, 46E3) We obtain new characterizations of the Sobolev spaces $\dot W^{1,p}(\mathbb{R}^N)$ and the bounded variation space $\dot{BV}(\mathbb{R}^N)$. The characterizations are in terms of the functionals $ν_γ (E_{λ,γ/p}[u])$ where \[ E_{λ,γ/p}[u]= \Big\{(x,y )\in \mathbb{R}^N \times \mathbb{R}^N \colon x \neq y, \, \frac{|u(x)-u(y)|}{|x-y|^{1+γ/p}}>λ\Big\} \] and the measure $ν_γ$ is given by $\mathrm{d} ν_γ(x,y)=|x-y|^{γ-N} \mathrm{d} x \mathrm{d} y$. We provide characterizations which involve the $L^{p,\infty}$-quasi-norms $\sup_{λ>0} λ\, ν_γ (E_{λ,γ/p}[u]) ^{1/p}$ and also exact formulas via corresponding limit functionals, with the limit for $λ\to\infty$ when $γ>0$ and the limit for $λ\to 0^+$ when $γ<0$. The results unify and substantially extend previous work by Nguyen and by Brezis, Van Schaftingen and Yung. For $p>1$ the characterizations hold for all $γ\neq 0$. For $p=1$ the upper bounds for the $L^{1,\infty}$ quasi-norms fail in the range $γ\in [-1,0) $; moreover in this case the limit functionals represent the $L^1$ norm of the gradient for $C^\infty_c$-functions but not for generic $\dot W^{1,1}$-functions. For this situation we provide new counterexamples which are built on self-similar sets of dimension $γ+1$. For $γ=0$ the characterizations of Sobolev spaces fail; however we obtain a new formula for the Lipschitz norm via the expressions $ν_0(E_{λ,0}[u])$. |
| title | Families of functionals representing Sobolev norms |
| topic | Functional Analysis Analysis of PDEs Classical Analysis and ODEs 26D10 (26A33, 35A23, 42B25, 42B35, 46E30, 46E3) |
| url | https://arxiv.org/abs/2109.02930 |