Optimal Domains for Elliptic Eigenvalue Problems with Rough Coefficients

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Mathematical Analysis Pub Date : 2024-05-08 DOI:10.1137/22m1523820

Stanley Snelson, Eduardo V. Teixeira

引用次数: 0

Abstract

SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3412-3429, June 2024.
Abstract. We prove the existence of an open set minimizing the first Dirichlet eigenvalue of an elliptic operator with bounded, measurable coefficients, over all open sets of a given measure. Our proof is based on a free boundary approach: we characterize the eigenfunction on the optimal set as the minimizer of a penalized functional, and derive openness of the optimal set as a consequence of a Hölder estimate for the eigenfunction. We also prove that the optimal eigenfunction grows at most linearly from the free boundary, i.e., it is Lipschitz continuous at free boundary points.

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具有粗糙系数的椭圆特征值问题的最佳域

SIAM 数学分析期刊》，第 56 卷，第 3 期，第 3412-3429 页，2024 年 6 月。摘要。我们证明了在给定度量的所有开集上，存在一个开集最小化椭圆算子的第一个 Dirichlet 特征值，该算子的系数是有界的、可测的。我们的证明基于自由边界方法：我们将最优集上的特征函数描述为受惩罚函数的最小化，并推导出最优集的开放性是特征函数霍尔德估计的结果。我们还证明了最优特征函数在自由边界上最多呈线性增长，即在自由边界点上是 Lipschitz 连续的。

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.