Nonlocal Problems with Local Boundary Conditions I: Function Spaces and Variational Principles

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Mathematical Analysis Pub Date : 2024-06-10 DOI:10.1137/23m1588111

James M. Scott, Qiang Du

引用次数: 0

Abstract

SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 4185-4222, June 2024.
Abstract. We present a systematic study on a class of nonlocal integral functionals for functions defined on a bounded domain and the naturally induced function spaces. The function spaces are equipped with a seminorm depending on finite differences weighted by a position-dependent function, which leads to heterogeneous localization on the domain boundary. We show the existence of minimizers for nonlocal variational problems with classically defined, local boundary constraints, together with the variational convergence of these functionals to classical counterparts in the localization limit. This program necessitates a thorough study of the nonlocal space; we demonstrate properties such as a Meyers–Serrin theorem, trace inequalities, and compact embeddings, which are facilitated by new studies of boundary-localized convolution operators.

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具有局部边界条件的非局部问题 I：函数空间与变分原理

SIAM 数学分析期刊》，第 56 卷，第 3 期，第 4185-4222 页，2024 年 6 月。摘要。我们系统地研究了一类定义在有界域上的函数的非局部积分函数以及自然诱导的函数空间。这些函数空间配备了由位置相关函数加权的取决于有限差分的半规范，从而导致域边界上的异质局部化。我们展示了具有经典定义的局部边界约束的非局部变分问题的最小化存在，以及这些函数在局部化极限中对经典对应函数的变分收敛。这一方案需要对非局部空间进行深入研究；我们展示了梅耶斯-塞林定理、迹不等式和紧凑嵌入等性质，而对边界局部卷积算子的新研究则促进了这些性质的研究。

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

SIAM Journal on Mathematical Analysis 数学-应用数学

CiteScore

3.30

自引率

5.00%

发文量

175

审稿时长

12 months

期刊介绍： SIAM Journal on Mathematical Analysis (SIMA) features research articles of the highest quality employing innovative analytical techniques to treat problems in the natural sciences. Every paper has content that is primarily analytical and that employs mathematical methods in such areas as partial differential equations, the calculus of variations, functional analysis, approximation theory, harmonic or wavelet analysis, or dynamical systems. Additionally, every paper relates to a model for natural phenomena in such areas as fluid mechanics, materials science, quantum mechanics, biology, mathematical physics, or to the computational analysis of such phenomena. Submission of a manuscript to a SIAM journal is representation by the author that the manuscript has not been published or submitted simultaneously for publication elsewhere. Typical papers for SIMA do not exceed 35 journal pages. Substantial deviations from this page limit require that the referees, editor, and editor-in-chief be convinced that the increased length is both required by the subject matter and justified by the quality of the paper.