Work

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

Year: 2025

Type: article

Abstract: 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.

Source: Journal of Differential Equations

DOI: 10.1016/j.jde.2025.02.006

Authors: Nausica Aldeghi, Jonathan Rohleder

Institution: Stockholm University

Language: English

Topic: Spectral Theory in Mathematical Physics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,550

Funder: Vetenskapsrådet