On the first eigenvalue and eigenfunction of the Laplacian with mixed boundary conditions

We consider the eigenvalue problem for the Laplacian with mixed Dirichlet and Neumann boundary conditions. For a certain class of bounded, simply connected planar domains we prove monotonicity properties of the first eigenfunction. As a consequence, we establish a variant of the hot spots conjecture...

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Bibliographic Details
Published in:Journal of Differential Equations Vol. 427; pp. 689 - 718
Main Authors: Aldeghi, Nausica, Rohleder, Jonathan
Format: Journal Article
Language:English
Published: Elsevier Inc 15.05.2025
Subjects:
Eigenfunctions
Eigenvalue inequalities
Hot spots
Laplacian
matematik
Mathematics
Mixed boundary conditions
Variational principles
Eigenfunctions
Mixed boundary conditions
Variational principles
Hot spots
Laplacian
Eigenvalue inequalities
ISSN:0022-0396, 1090-2732
Online Access:Get full text
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Summary:We consider the eigenvalue problem for the Laplacian with mixed Dirichlet and Neumann boundary conditions. For a certain class of bounded, simply connected planar domains we prove monotonicity properties of the first eigenfunction. As a consequence, we establish a variant of the hot spots conjecture for mixed boundary conditions. Moreover, we obtain an inequality between the lowest eigenvalue of this mixed problem and the lowest eigenvalue of the corresponding dual problem where the Dirichlet and Neumann boundary conditions are interchanged. The proofs are based on a novel variational principle, which we establish.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2025.02.006