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Global bifurcation of solutions to elliptic systems with system and domain symmetries

Piotr Stefaniak

TL;DR

We address global bifurcation of weak solutions for parameterized elliptic systems on symmetric domains, exploiting the degree for equivariant gradient maps to detect bifurcation from a constant branch without requiring nondegeneracy. The method reduces to spectral data through the set $\Lambda=\{\beta_k/\alpha_j\}$ and uses the Euler ring of a torus to compute a bifurcation index $\mathcal{BIF}_{\mathbb{T}}(\lambda_0)$, guaranteeing global bifurcation when it is nontrivial. Symmetry breaking occurs at every nonzero bifurcation level on compact symmetric spaces, and under additional structural assumptions the bifurcating continua are unbounded, extending previous results to multi-eigenspace settings. The results provide a unified, symmetry-aware framework for global bifurcation in elliptic systems with group actions, with explicit implications for symmetric-space geometries such as spheres.

Abstract

We study parameterized elliptic systems on symmetric domains with additional system symmetries. We prove the existence of continua of nontrivial solutions bifurcating from the constant branch determined by a critical point of the potential, without assuming nondegeneracy, via the degree for equivariant gradient maps. Our assumptions are formulated in terms of the right-hand side. When the domain is a compact symmetric space, the bifurcating solutions break symmetry at every nonzero level. Under additional assumptions on the right-hand side, the continua are unbounded.

Global bifurcation of solutions to elliptic systems with system and domain symmetries

TL;DR

We address global bifurcation of weak solutions for parameterized elliptic systems on symmetric domains, exploiting the degree for equivariant gradient maps to detect bifurcation from a constant branch without requiring nondegeneracy. The method reduces to spectral data through the set and uses the Euler ring of a torus to compute a bifurcation index , guaranteeing global bifurcation when it is nontrivial. Symmetry breaking occurs at every nonzero bifurcation level on compact symmetric spaces, and under additional structural assumptions the bifurcating continua are unbounded, extending previous results to multi-eigenspace settings. The results provide a unified, symmetry-aware framework for global bifurcation in elliptic systems with group actions, with explicit implications for symmetric-space geometries such as spheres.

Abstract

We study parameterized elliptic systems on symmetric domains with additional system symmetries. We prove the existence of continua of nontrivial solutions bifurcating from the constant branch determined by a critical point of the potential, without assuming nondegeneracy, via the degree for equivariant gradient maps. Our assumptions are formulated in terms of the right-hand side. When the domain is a compact symmetric space, the bifurcating solutions break symmetry at every nonzero level. Under additional assumptions on the right-hand side, the continua are unbounded.
Paper Structure (11 sections, 21 theorems, 60 equations)

This paper contains 11 sections, 21 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Elements of representation theory
  4. Torus representations
  5. Euler ring of a torus
  6. The degree
  7. Main result
  8. Setting and assumptions
  9. Global bifurcations
  10. Symmetry breaking
  11. Elliptic systems on a symmetric space

Key Result

Lemma 2.1

With the above notation, the orthogonal $\Gamma\times G$–representations $\bigoplus_{j=1}^{N}\mathbb{V}$ and $\mathbb{W}\otimes\mathbb{V}$ are equivalent.

Theorems & Definitions (40)

  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 30 more