On the solutions of nonlocal 1-Laplacian equation with $ L^1 $-data

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-11-01 DOI:10.3934/dcds.2023148

Dingding Li, Chao Zhang

引用次数: 0

Abstract

We study the solutions to a nonlocal 1-Laplacian equation given by $$ 2\text{P.V.}\int_{\mathbb{R}^N}\frac{u(x)-u(y)}{|u(x)-u(y)|} \frac{dy}{|x-y|^{N+s}}=f(x) \quad \textmd{for } x\in \Omega, $$ with Dirichlet boundary condition $u(x)=0$ in $\mathbb R^N\backslash \Omega$ and nonnegative $L^1$-data. By investigating the asymptotic behaviour of renormalized solutions $u_p$ to the nonlocal $p$-Laplacian equations as $p$ goes to $1^+$, we introduce a suitable definition of solutions and prove that the limit function $u$ of $\{u_p\}$ is a solution of the nonlocal $1$-Laplacian equation above.

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关于有 $ L^1 $ 数据的非局部 1-拉普拉斯方程的解

我们研究了由 $$ 2\text{P.V.}\int_{mathbb{R}^N}\frac{u(x)-u(y)}{|u(x)-u(y)|} 所给出的非局部 1 拉普拉斯方程的解。\frac{dy}{|x-y|^{N+s}}=f(x) \quad \textmd{for } x\in \Omega, $$ 在 $\mathbb R^N\backslash \Omega$ 中具有德里赫特边界条件 $u(x)=0$ 和非负 $L^1$ 数据。通过研究非局部 $p$ 拉普拉斯方程的重正化解 $u_p$ 在 $p$ 达到 1^+$ 时的渐近行为，我们引入了一个合适的解的定义，并证明了 $\{u_p\}$ 的极限函数 $u$ 是上述非局部 1$ 拉普拉斯方程的解。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.