Steklov-type eigenvalues associated with best Sobolev trace constants: domain perturbation and overdetermined systems

• Published in: arXiv.org
• Main Author: Lamberti, Pier Domenico
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 31.01.2011
• Subjects: Domains; Eigenvalues; Mathematical analysis; Mathematics - Analysis of PDEs; Mathematics - Spectral Theory; Sobolev space
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1101.5930

We consider a variant of the classic Steklov eigenvalue problem, which arises in the study of the best trace constant for functions in Sobolev space. We prove that the elementary symmetric functions of the eigenvalues depend real-analytically upon variation of the underlying domain and we compute the corresponding Hadamard-type formulas for the shape derivatives. We also consider isovolumetric and isoperimetric domain perturbations and we characterize the corresponding critical domains in terms of appropriate overdetermined systems. Finally, we prove that balls are critical domains for the elementary symmetric functions of the eigenvalues subject to volume or perimeter constraint.