{"title":"四阶Steklov特征值问题的形状优化","authors":"Changwei Xiong, Jinglong Yang, Jinchao Yu","doi":"10.1007/s00245-025-10277-z","DOIUrl":null,"url":null,"abstract":"<div><p>We study three types of fourth-order Steklov eigenvalue problems. For the first two of them, we derive the asymptotic expansion of the eigenvalues on Euclidean annular domains <span>\\(\\mathbb {B}^n_1\\setminus \\overline{\\mathbb {B}^n_\\epsilon }\\)</span> as <span>\\(\\epsilon \\rightarrow 0\\)</span>, in turn yielding some interesting results regarding the shape optimization of the eigenvalues. For these two problems, we also compute the respective spectra on cylinders over closed Riemannian manifolds. For the third problem, we obtain a sharp upper bound for its first non-zero eigenvalue on star-shaped and mean convex Euclidean domains.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"92 1","pages":""},"PeriodicalIF":1.7000,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Shape Optimization for Fourth Order Steklov eigenvalue Problems\",\"authors\":\"Changwei Xiong, Jinglong Yang, Jinchao Yu\",\"doi\":\"10.1007/s00245-025-10277-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We study three types of fourth-order Steklov eigenvalue problems. For the first two of them, we derive the asymptotic expansion of the eigenvalues on Euclidean annular domains <span>\\\\(\\\\mathbb {B}^n_1\\\\setminus \\\\overline{\\\\mathbb {B}^n_\\\\epsilon }\\\\)</span> as <span>\\\\(\\\\epsilon \\\\rightarrow 0\\\\)</span>, in turn yielding some interesting results regarding the shape optimization of the eigenvalues. For these two problems, we also compute the respective spectra on cylinders over closed Riemannian manifolds. For the third problem, we obtain a sharp upper bound for its first non-zero eigenvalue on star-shaped and mean convex Euclidean domains.</p></div>\",\"PeriodicalId\":55566,\"journal\":{\"name\":\"Applied Mathematics and Optimization\",\"volume\":\"92 1\",\"pages\":\"\"},\"PeriodicalIF\":1.7000,\"publicationDate\":\"2025-06-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Applied Mathematics and Optimization\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00245-025-10277-z\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics and Optimization","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00245-025-10277-z","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
On Shape Optimization for Fourth Order Steklov eigenvalue Problems
We study three types of fourth-order Steklov eigenvalue problems. For the first two of them, we derive the asymptotic expansion of the eigenvalues on Euclidean annular domains \(\mathbb {B}^n_1\setminus \overline{\mathbb {B}^n_\epsilon }\) as \(\epsilon \rightarrow 0\), in turn yielding some interesting results regarding the shape optimization of the eigenvalues. For these two problems, we also compute the respective spectra on cylinders over closed Riemannian manifolds. For the third problem, we obtain a sharp upper bound for its first non-zero eigenvalue on star-shaped and mean convex Euclidean domains.
期刊介绍:
The Applied Mathematics and Optimization Journal covers a broad range of mathematical methods in particular those that bridge with optimization and have some connection with applications. Core topics include calculus of variations, partial differential equations, stochastic control, optimization of deterministic or stochastic systems in discrete or continuous time, homogenization, control theory, mean field games, dynamic games and optimal transport. Algorithmic, data analytic, machine learning and numerical methods which support the modeling and analysis of optimization problems are encouraged. Of great interest are papers which show some novel idea in either the theory or model which include some connection with potential applications in science and engineering.