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

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arxiv 2403.17717 v2 pith:IJ426K6E submitted 2024-03-26 math.SP math.AP

Nausica Aldeghi , Jonathan Rohleder This is my paper · ORCID
classification math.SPmath.AP
keywords boundaryconditionseigenvaluemixedproblemdirichleteigenfunctionestablish
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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.

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  1. Hot spots in domains of constant curvature

    math.SP 2025-08 unverdicted novelty 6.0 of 10

    The hot spots conjecture holds for all non-acute geodesic triangles of constant negative curvature, with additional critical point and monotonicity results for other constant curvature triangles and polygons.

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