Spectral and stochastic properties of the $f$-Laplacian, solutions of PDE's at infinity, and geometric applications

TL;DR

This paper explores the spectral and stochastic characteristics of the $f$-Laplacian to understand solutions of semilinear elliptic PDEs outside compact sets, linking spectral theory with stochastic analysis for geometric insights.

Contribution

It introduces a novel approach combining spectral and stochastic methods to study PDE solutions at infinity, providing new geometric applications.

Findings

Spectral properties influence the behavior of PDE solutions at infinity.

Stochastic analysis reveals links between operator properties and solution qualitative features.

New geometric applications are derived from spectral-stochastic connections.

Abstract

The aim of this paper is to suggest a new viewpoint to study qualitative properties of solutions of semilinear elliptic PDE's defined outside a compact set. The relevant tools come from spectral theory and from a combination of stochastic properties of the relevant differential operators. Possible links between spectral and stochastic properties are analyzed in detail.