First Eigenvalue and Torsional Rigidity: Isoperimetric Inequalities for the Fractional Laplacian

TL;DR

The paper extends classical isoperimetric-type inequalities to the fractional Laplacian by introducing a generalized fractional torsional rigidity $Q(\alpha,\Omega)$ and proving a mass-concentration comparison that yields a nonlocal Kohler-Jobin inequality: among bounded Lipschitz sets with fixed fractional torsional rigidity, the ball minimizes the first Dirichlet eigenvalue $\lambda_1(\Omega)$. A key step is showing that for each $\alpha<\lambda_1(\Omega)$ there exists a ball radius $R(\alpha)$ with $Q(\alpha,\Omega)=Q(\alpha,B_{R(\alpha)})$, and that $\alpha \mapsto R(\alpha)$ is decreasing, which leads to $\lambda_1(\Omega) \ge \lambda_1(B_{R(\alpha)})$. The authors also derive a sharp reverse Hölder inequality for the eigenfunction corresponding to $\lambda_1(\Omega)$, tying $L^q$ norms to the $L^1$ norm with explicit dependence on $\lambda_1(\Omega)$ and the parameters $N,s,q$. Overall, the work provides nonlocal analogues of Kohler-Jobin and Saint-Venant-type results, with explicit, analyzable constants and a framework for further shape-optimization studies in fractional settings.

Abstract

We present a fractional counterpart of a generalized Kohler-Jobin inequality, showing that, among all bounded, open sets $Ω\subset \mathbb{R}^N$ with Lipschitz boundary, having the same fractional torsional rigidity, the first Dirichlet eigenvalue $λ_1(Ω)$ of the fractional Laplacian attains its minimum on balls. With the same arguments we also establish a reverse Hölder inequality for an eigenfunction corresponding to $λ_1(Ω)$.

Figures (1)

• Figure 1: On the left, a distribution function which presents a discontinuity and a flat zone; on the right, the corresponding decreasing rearrangement.

Theorems & Definitions (40)

• Definition 2.1
• Proposition 2.1
• Proposition 2.2
• Theorem 3.1
• Proposition 3.1
• Remark 3.1
• Proposition 3.2
• Remark 3.2
• Proposition 3.3
• Remark 3.3