A spectral isoperimetric inequality on the n-sphere for the Robin-Laplacian with negative boundary parameter

2024/07/08 by Paolo Acampora, Antonio Celentano, Acampora, Paolo +7
Computer Science · Mathematics · #35P15 #52A55 #58J50 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2407.05987

openalex publication_date 2024/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For every given β<0, we study the problem of maximizing the first Robin eigenvalue of the Laplacian λ_β(Ω) among convex (not necessarily smooth) sets Ω⊂\mathbbSⁿ with fixed perimeter. In particular, denoting by σ_n the perimeter of the n-dimensional hemisphere, we show that for fixed perimeters P<σ_n, geodesic balls maximize the eigenvalue. Moreover, we prove a quantitative stability result for this isoperimetric inequality in terms of volume difference between Ω and the ball D of the same perimeter.

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