Reverse Isoperimetric Inequality for the Lowest Robin Eigenvalue of a Triangle

0575046 - ÚJF 2024 RIV US eng J - Journal Article
Krejčiřík, D. - Lotoreichik, Vladimir - Vu, T.
Reverse Isoperimetric Inequality for the Lowest Robin Eigenvalue of a Triangle.
Applied Mathematics and Optimization. Roč. 88, č. 2 (2023), č. článku 63. ISSN 0095-4616. E-ISSN 1432-0606
R&D Projects: GA ČR(CZ) GA21-07129S
Institutional support: RVO:61389005
Keywords : Robin Laplacian * Lowest eigenvalue * Spectral optimisation * Triangles
OECD category: Applied mathematics
Impact factor: 1.8, year: 2022
Method of publishing: Open access
https://doi.org/10.1007/s00245-023-10033-1

We consider the Laplace operator on a triangle, subject to attractive Robin boundary conditions. We prove that the equilateral triangle is a local maximiser of the lowest eigenvalue among all triangles of a given area provided that the negative boundary parameter is sufficiently small in absolute value, with the smallness depending on the area only. Moreover, using various trial functions, we obtain sufficient conditions for the global optimality of the equilateral triangle under fixed area constraint in the regimes of small and large couplings. We also discuss the constraint of fixed perimeter.
Permanent Link: https://hdl.handle.net/11104/0344892