On the p-Laplace operator on Riemannian manifolds

TL;DR

This thesis explores various properties of the p-Laplace operator on Riemannian manifolds, including potential theory, eigenvalue estimates, and harmonic function critical sets, advancing understanding of nonlinear PDEs in geometric contexts.

Contribution

It provides new insights into potential theory, eigenvalue bounds, and critical set analysis for p-Laplace operators on Riemannian manifolds, integrating geometric and analytic methods.

Findings

Khasmkinskii condition characterization

Sharp eigenvalue bounds under Ricci curvature constraints

Analysis of critical sets of harmonic functions

Abstract

This thesis covers different aspects of the p-Laplace operators on Riemannian manifolds. Chapter 2. Potential theoretic aspects: the Khasmkinskii condition. Chapter 3: sharp eigenvalue estimates with Ricci curvature lower bounds. Chapter 4: Critical sets of (2-)harmonic functions.