Eigenvalues of the Laplacian with moving mixed boundary conditions: the case of disappearing Neumann region

2021/07/31 by Veronica Felli, Benedetta Noris, Roberto Ognibene · 10 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Boundary (topology) #Boundary value problem #Combinatorics #Complement (music) #Dirichlet boundary condition #Dirichlet distribution #Dirichlet eigenvalue #Dirichlet's principle #Eigenvalues and eigenvectors #Homogeneous #Laplace operator #Mathematical analysis #Mathematics #Neumann boundary condition #Nonlinear Partial Differential Equations #Physics #Pure mathematics #Spectral Theory in Mathematical Physics #math.AP #math.SP #msc:35B25 #msc:35J25 #msc:35P15

paper · pdf · doi:10.1016/j.jde.2022.02.052

published in Journal of Differential Equations 320, 247-315 (Elsevier BV)

openalex created_date 2020/01/30 · openalex publication_date 2022/03/09 · arxiv created 2022/03/10 · arxiv updated 2022/03/11 · openalex updated_date 2026/08/05

Abstract

We deal with eigenvalue problems for the Laplacian with varying mixed boundary conditions, consisting in homogeneous Neumann conditions on a vanishing portion of the boundary and Dirichlet conditions on the complement. By the study of an Almgren type frequency function, we derive upper and lower bounds of the eigenvalue variation and sharp estimates in the case of a strictly star-shaped Neumann region.

Citations

• Sobolev Spaces
• Characterizations of Sobolev Functions that vanish on a part of the boundary

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