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On the Dirichlet boundary value problem on Cartan-Hadamard manifolds

Marcos P. Cavalcante, José M. Espinar, Diego A. Marín

TL;DR

The article addresses the Dirichlet problem at infinity for semilinear elliptic equations on Cartan--Hadamard manifolds, establishing nonexistence of bounded solutions in domains whose asymptotic boundary has Hausdorff dimension below a computed threshold. It introduces a convex-barrier framework, replacing the unavailable totally geodesic foliations in general manifolds with distance-based barriers $v=higl(d_{ound ext{-} extSigma}igr)$ and an ODE linked to curvature bounds, together with smoothing via Greene--Wu. The main result extends Bonorino--Klaser-type nonexistence from hyperbolic space to pinched Cartan--Hadamard manifolds and applies to nonlinearities such as the Allen--Cahn type $f(u)=u-u^3$, $f(u)= anh u$, and others, including corollaries on bounded Allen--Cahn solutions. The approach illuminates the interplay between curvature, the spectrum of the Laplacian, and the geometry of the asymptotic boundary, with broad applicability to rank-one symmetric spaces and related geometric contexts.

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 $Δu + f(u) = 0$ 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, Gálvez, 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.

On the Dirichlet boundary value problem on Cartan-Hadamard manifolds

TL;DR

The article addresses the Dirichlet problem at infinity for semilinear elliptic equations on Cartan--Hadamard manifolds, establishing nonexistence of bounded solutions in domains whose asymptotic boundary has Hausdorff dimension below a computed threshold. It introduces a convex-barrier framework, replacing the unavailable totally geodesic foliations in general manifolds with distance-based barriers and an ODE linked to curvature bounds, together with smoothing via Greene--Wu. The main result extends Bonorino--Klaser-type nonexistence from hyperbolic space to pinched Cartan--Hadamard manifolds and applies to nonlinearities such as the Allen--Cahn type , , and others, including corollaries on bounded Allen--Cahn solutions. The approach illuminates the interplay between curvature, the spectrum of the Laplacian, and the geometry of the asymptotic boundary, with broad applicability to rank-one symmetric spaces and related geometric contexts.

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 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, Gálvez, 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.
Paper Structure (14 sections, 18 theorems, 180 equations, 4 figures)

This paper contains 14 sections, 18 theorems, 180 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Outline of the paper and statement of the main results.
  3. Preliminary Results
  4. Asymptotic geometry of domains in Cartan--Hadamard manifolds
  5. Hausdorff dimension of a set in the asymptotic boundary
  6. Viscosity solutions on Riemannian manifolds
  7. Construction of super-solutions
  8. Convex barriers in pinched Cartan--Hadamard manifolds
  9. The model case
  10. Cones and half-spaces
  11. Anderson's convex sets and barriers
  12. Non-existence results via convex barriers
  13. Proof of Theorem \ref{['thm:Anderson']}
  14. An Linfty bound for bounded Allen--Cahn solutions

Key Result

Theorem 1.1

Let $(M^n,g)$ be a Cartan--Hadamard manifold with pinched sectional curvatures $-\kappa_1 \le K \le -\kappa_2$, for some $\kappa_1\ge \kappa_2>0$. If $\Omega\subset M^n$ is a domain such that then there exists no nontrivial bounded solution $u\in C^2(\Omega)\cap C^0(\overline{\Omega})$ to the Dirichlet problem where $|f(u)|\le \lambda |u|$, $f(-u)=-f(u)$, and $\lambda \le \frac{(n-1)^2\kappa_2}{

Figures (4)

  • Figure 1: A picture in the Poincaré ball model of $\mathbb{H}^n(-\kappa)$. For $\theta$ small enough, the convex set $\mathcal{C}$ bounded by an equidistant hypersurface $\Sigma$ provides a barrier for the cone $C_o(z,\theta)$.
  • Figure 2: The triangle $\triangle(o,p,c(s))$ used in the proof of Lemma \ref{['lemmaAngle']}.
  • Figure 3: A schematic picture of the barrier construction in the pinched-curvature setting.
  • Figure 4: A schematic picture of the case $\partial_\infty\Omega=\{\xi\}$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 24 more