On the Bossel-Daners inequality for the p-Laplacian on complete Riemannian manifolds

TL;DR

The article advances Bossel-Daners-type eigenvalue comparisons for the Robin $p$-Laplacian on complete Riemannian manifolds with Ricci lower bounds and extends the results to compact minimal submanifolds under AVR and non-negative intermediate Ricci curvature. The authors introduce the $H_\Omega$ functional to link eigenvalues with level-set geometry and develop radial analysis on model spaces to perform sharp comparisons with geodesic balls. They establish that $\lambda_p(\Omega,\beta) \ge \lambda_p(\Omega^\sharp,\beta)$ with equality implying strong rigidity to model spaces, thereby unifying several Euclidean and curved-space cases, including Dirichlet limits when $p=2$ and $\beta=\infty$. The methods hinge on isoperimetric inequalities in curved spaces and a careful transfer of radial eigenfunction properties to general domains and submanifolds. This work broadens the scope of Robin-boundary eigenvalue optimization beyond Euclidean settings and connects geometric and analytic inequalities via a unified framework.

Abstract

In this paper, we obtain the Bossel-Daners inequality for the first eigenvalue of the p-Laplacian with Robin boundary conditions on complete Riemannian manifolds with lower Ricci curvature bounds. Furthermore, we demonstrate that the Bossel-Daners inequality extends to compact submanifolds within complete Riemannian manifolds characterized by positive asymptotic volume ratio and non-negative intermediate Ricci curvature.

Theorems & Definitions (14)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Lemma 3.1
• Lemma 3.2
• proof
• Lemma 4.1
• Remark 4.2
• proof