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A priori estimates of stable and finite Morse index solutions to elliptic equations that arise in Physics

J. Silverio Martínez-Baena

TL;DR

This thesis investigates a priori estimates for stable and finite Morse index solutions to semilinear elliptic equations with Dirichlet boundary conditions, emphasizing regularity, stability, and multiplicity and linking these questions to physical field theories. It advances the theory through a counterexample showing that finite radial Morse index does not guarantee regularity in dimensions $3\le N\le 9$, develops sharp regularity results for radial solutions of the nonautonomous Hardy–Hénon equation, and proves existence/multiplicity results for equations with degenerate nonlinearities using variational and topological methods. The work situates these mathematical insights within a broader physical motivation about stability in field theories and potential applications to cosmology and gravity models, outlining several promising directions for extending stability-based techniques to nonautonomous problems and strongly coupled regimes. Overall, the results illuminate fundamental limits of Morse-index-based regularity and provide robust tools for establishing regularity and multiplicity in nonlinear elliptic problems with physically relevant structure. The findings have potential implications for both rigorous PDE analysis and theoretical physics, including models with spatially varying coefficients and nonuniform boundary data.

Abstract

This thesis studies qualitative properties of solutions to nonlinear elliptic equations of Poisson type with Dirichlet boundary conditions that arise from some physical phenomena, with a particular focus on regularity, stability, and multiplicity of solutions. Building on the modern framework of solution stability and Morse index theory, the work investigates how these notions influence regularity in nonlinear elliptic problems. A central contribution is the construction of a counterexample showing that bounded radial Morse index does not prevent singular behavior of solutions in dimensions three through nine, challenging a natural extension of the Brezis-Vázquez regularity conjecture. In addition, optimal regularity results are established for radial solutions of a non-autonomous Hardy-Hénon equation, identifying the precise range of dimensions for which regularity holds. The thesis also addresses existence and multiplicity results for elliptic equations involving nonlinearities with spatially vanishing coefficients. Under suitable assumptions, the existence of multiple distinct solutions is proved using variational and topological methods. Finally, the thesis outlines several directions for future research, including extensions of stability-based regularity techniques to non-autonomous problems and potential applications of these techniques to field theories arising in theoretical physics.

A priori estimates of stable and finite Morse index solutions to elliptic equations that arise in Physics

TL;DR

This thesis investigates a priori estimates for stable and finite Morse index solutions to semilinear elliptic equations with Dirichlet boundary conditions, emphasizing regularity, stability, and multiplicity and linking these questions to physical field theories. It advances the theory through a counterexample showing that finite radial Morse index does not guarantee regularity in dimensions , develops sharp regularity results for radial solutions of the nonautonomous Hardy–Hénon equation, and proves existence/multiplicity results for equations with degenerate nonlinearities using variational and topological methods. The work situates these mathematical insights within a broader physical motivation about stability in field theories and potential applications to cosmology and gravity models, outlining several promising directions for extending stability-based techniques to nonautonomous problems and strongly coupled regimes. Overall, the results illuminate fundamental limits of Morse-index-based regularity and provide robust tools for establishing regularity and multiplicity in nonlinear elliptic problems with physically relevant structure. The findings have potential implications for both rigorous PDE analysis and theoretical physics, including models with spatially varying coefficients and nonuniform boundary data.

Abstract

This thesis studies qualitative properties of solutions to nonlinear elliptic equations of Poisson type with Dirichlet boundary conditions that arise from some physical phenomena, with a particular focus on regularity, stability, and multiplicity of solutions. Building on the modern framework of solution stability and Morse index theory, the work investigates how these notions influence regularity in nonlinear elliptic problems. A central contribution is the construction of a counterexample showing that bounded radial Morse index does not prevent singular behavior of solutions in dimensions three through nine, challenging a natural extension of the Brezis-Vázquez regularity conjecture. In addition, optimal regularity results are established for radial solutions of a non-autonomous Hardy-Hénon equation, identifying the precise range of dimensions for which regularity holds. The thesis also addresses existence and multiplicity results for elliptic equations involving nonlinearities with spatially vanishing coefficients. Under suitable assumptions, the existence of multiple distinct solutions is proved using variational and topological methods. Finally, the thesis outlines several directions for future research, including extensions of stability-based regularity techniques to non-autonomous problems and potential applications of these techniques to field theories arising in theoretical physics.
Paper Structure (43 sections, 41 theorems, 206 equations, 1 figure)

This paper contains 43 sections, 41 theorems, 206 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Organisation of the Thesis
  3. Background
  4. Stability and Morse index in semilinear elliptic equations
  5. Dynamical stability vs elliptic stability
  6. Infinite dimensional Morse theory
  7. Motivation
  8. The Brezis-Vázquez conjecture
  9. The De Giorgi's conjecture
  10. The saddle-shaped solution vanishing on the Simons cone
  11. Physical motivation: Stability of Field Theories in physics
  12. Types of Instabilities in Field Theories
  13. Literature Review
  14. Regularity of stable and finite Morse index solutions
  15. Smoothness of stable solutions up to dimension 9
  16. ...and 28 more sections

Key Result

Proposition 1.6

Let $u$ be a stable weak solution of main_problem and assume $f\in C^1(\mathbb{R})$, $f\geq 0$ and $f$ is convex. Then there exists a sequence $\lbrace u_\epsilon\rbrace$ of classical solutions to the problem such that $u=\lim_{\epsilon\to 0}u_\epsilon$ in $H^1_0(\Omega)$.

Figures (1)

  • Figure 1: $\Gamma^-$-convergence to a characteristic function limiting interface between two substances described by the Cahn-Hilliard model (figure from Cabré2018)

Theorems & Definitions (70)

  • Definition 1.1: Weak solution
  • Definition 1.2: $L^1$-weak solution
  • Remark 1.3
  • Definition 1.4: Stable solution
  • Definition 1.5: Local stability
  • Proposition 1.6: Theorem 3.2.1, dupaigne
  • Definition 1.7: Morse index
  • Remark 1.8
  • Proposition 1.9: Chapter 1, dupaigne
  • Proposition 1.10: Proposition B.1, cabre-figalli-rosoton-serra
  • ...and 60 more