On the Dirichlet boundary value problem on Cartan-Hadamard manifolds

TL;DR

This paper studies the Dirichlet boundary value problem on Cartan-Hadamard manifolds, demonstrating non-existence of bounded solutions for certain semi-linear elliptic equations by developing new comparison techniques that account for the manifold's curvature and boundary geometry.

Contribution

Introduces a novel comparison method for semi-linear elliptic equations on Cartan-Hadamard manifolds, extending non-existence results beyond hyperbolic spaces.

Findings

Non-existence of bounded solutions under certain conditions

New comparison technique based on convex hypersurfaces

Highlights the role of curvature and boundary geometry

Abstract

In this paper, we investigate the Dirichlet boundary value problem on Cartan-Hadamard manifolds, focusing on the non-existence of bounded (viscosity) solutions to semi-linear elliptic equations of the form $\Delta u + f(u) = 0$ in domains with prescribed asymptotic boundary, extending previous results by Bonorino and Klaser originally established for hyperbolic spaces. Using a novel comparison technique based on convex hypersurfaces inspired by Choi, G\'alvez, and Lozano, we overcome the absence of totally geodesic foliations, which are instrumental in the hyperbolic space. Our results highlight the interplay between curvature, the spectrum of the Laplacian, and the geometry of the asymptotic boundary.