Mappings of generalized finite distortion and continuity
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2022
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913237724823552 |
|---|---|
| author | Doležalová, Anna Kangasniemi, Ilmari Onninen, Jani |
| author_facet | Doležalová, Anna Kangasniemi, Ilmari Onninen, Jani |
| contents | We study continuity properties of Sobolev mappings $f \in W_{\mathrm{loc}}^{1,n} (Ω, \mathbb{R}^n)$, $n \ge 2$, that satisfy the following generalized finite distortion inequality \[\lvert Df(x)\rvert^n \leq K(x) J_f(x) + Σ(x)\] for almost every $x \in \mathbb{R}^n$. Here $K \colon Ω\to [1, \infty)$ and $Σ\colon Ω\to [0, \infty)$ are measurable functions. Note that when $Σ\equiv 0$, we recover the class of mappings of finite distortion, which are always continuous. The continuity of arbitrary solutions, however, turns out to be an intricate question. We fully solve the continuity problem in the case of bounded distortion $K \in L^\infty (Ω)$, where a sharp condition for continuity is that $Σ$ is in the Zygmund space $Σ\log^μ(e + Σ) \in L^1_{\mathrm{loc}}(Ω)$ for some $μ> n-1$. We also show that one can slightly relax the boundedness assumption on $K$ to an exponential class $\exp(λK) \in L^1_{\mathrm{loc}}(Ω)$ with $λ> n+1$, and still obtain continuous solutions when $Σ\log^μ(e + Σ) \in L^1_{\mathrm{loc}}(Ω)$ with $μ> λ$. On the other hand, for all $p, q \in [1, \infty]$ with $p^{-1} + q^{-1} = 1$, we construct a discontinuous solution with $K \in L^p_{\mathrm{loc}}(Ω)$ and $Σ/K \in L^q_{\mathrm{loc}}(Ω)$, including an example with $Σ\in L^\infty_{\mathrm{loc}}(Ω)$ and $K \in L^1_{\mathrm{loc}}(Ω)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_14141 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Mappings of generalized finite distortion and continuity Doležalová, Anna Kangasniemi, Ilmari Onninen, Jani Analysis of PDEs Complex Variables Functional Analysis 30C65 (primary) 35R45 (secondary) We study continuity properties of Sobolev mappings $f \in W_{\mathrm{loc}}^{1,n} (Ω, \mathbb{R}^n)$, $n \ge 2$, that satisfy the following generalized finite distortion inequality \[\lvert Df(x)\rvert^n \leq K(x) J_f(x) + Σ(x)\] for almost every $x \in \mathbb{R}^n$. Here $K \colon Ω\to [1, \infty)$ and $Σ\colon Ω\to [0, \infty)$ are measurable functions. Note that when $Σ\equiv 0$, we recover the class of mappings of finite distortion, which are always continuous. The continuity of arbitrary solutions, however, turns out to be an intricate question. We fully solve the continuity problem in the case of bounded distortion $K \in L^\infty (Ω)$, where a sharp condition for continuity is that $Σ$ is in the Zygmund space $Σ\log^μ(e + Σ) \in L^1_{\mathrm{loc}}(Ω)$ for some $μ> n-1$. We also show that one can slightly relax the boundedness assumption on $K$ to an exponential class $\exp(λK) \in L^1_{\mathrm{loc}}(Ω)$ with $λ> n+1$, and still obtain continuous solutions when $Σ\log^μ(e + Σ) \in L^1_{\mathrm{loc}}(Ω)$ with $μ> λ$. On the other hand, for all $p, q \in [1, \infty]$ with $p^{-1} + q^{-1} = 1$, we construct a discontinuous solution with $K \in L^p_{\mathrm{loc}}(Ω)$ and $Σ/K \in L^q_{\mathrm{loc}}(Ω)$, including an example with $Σ\in L^\infty_{\mathrm{loc}}(Ω)$ and $K \in L^1_{\mathrm{loc}}(Ω)$. |
| title | Mappings of generalized finite distortion and continuity |
| topic | Analysis of PDEs Complex Variables Functional Analysis 30C65 (primary) 35R45 (secondary) |
| url | https://arxiv.org/abs/2210.14141 |