Reverse Faber-Krahn inequalities for the Logarithmic potential operator

2025/01/23 by T. V. Anoop, Anoop, T. V., Johnson, Jiya Rose
Mathematics · #Mathematical Inequalities and Applications #Spectral Theory in Mathematical Physics #Analytic and geometric function theory

paper · pdf · doi:10.48550/arxiv.2501.13569

Abstract

For a bounded open set Ω⊂ ℝ², we consider the largest eigenvalue τ₁(Ω) of the Logarithmic potential operator L. If diam(Ω)≤ 1, we prove reverse Faber-Krahn type inequalities for τ₁(Ω) under polarization and Schwarz symmetrization. Further, we establish the monotonicity of τ₁(Ω\setminusO) with respect to certain translations and rotations of the obstacle O within Ω. The analogous results are also stated for the largest eigenvalue of the Riesz potential operator. Furthermore, we investigate properties of the smallest eigenvalue τ₁(Ω) for a domain whose transfinite diameter is greater than 1. Finally, we characterize the eigenvalues of L on B_R, including the τ₁(B_R) when R>1.

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