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Concavity Properties of Solutions of Elliptic Equations under Conformal Deformations

Gabriel Khan, Soumyajit Saha, Malik Tuerkoen

Abstract

We study the Dirichlet problem for the weighted Schrödinger operator \[-Δu +Vu = λρu,\] where $ρ$ is a positive weighting function and $V$ is a potential. Such equations appear naturally in conformal geometry and in the composite membrane problem. Our primary goal is to establish concavity estimates for the principle eigenfunction with respect to conformal connections. Doing so, we obtain new bounds on the fundamental gap problem, which is the difference between the first and second eigenvalues. In particular, we partially resolve a conjecture of Nguyen, Stancu and Wei [IMRN 2022] on the fundamental gap of horoconvex domains. In addition, we obtain a power convexity estimate for solutions to the torsion problem in spherical geometry on convex domains which are not too large.

Concavity Properties of Solutions of Elliptic Equations under Conformal Deformations

Abstract

We study the Dirichlet problem for the weighted Schrödinger operator where is a positive weighting function and is a potential. Such equations appear naturally in conformal geometry and in the composite membrane problem. Our primary goal is to establish concavity estimates for the principle eigenfunction with respect to conformal connections. Doing so, we obtain new bounds on the fundamental gap problem, which is the difference between the first and second eigenvalues. In particular, we partially resolve a conjecture of Nguyen, Stancu and Wei [IMRN 2022] on the fundamental gap of horoconvex domains. In addition, we obtain a power convexity estimate for solutions to the torsion problem in spherical geometry on convex domains which are not too large.
Paper Structure (14 sections, 10 theorems, 73 equations, 1 figure)

This paper contains 14 sections, 10 theorems, 73 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview of the Paper
  3. Structure of the paper
  4. Background and Preliminaries
  5. Conformal deformations of the Laplace operator
  6. The fundamental gap in hyperbolic space
  7. Spectral Gap Estimates
  8. A convexity result for the torsion problem
  9. Log-concavity of the First Eigenfunction
  10. Log-concavity via the barrier PDE approach
  11. Applications of Conformal Deformations
  12. Fundamental Gap Estimates of Horoconvex Domains
  13. Remarks on Theorem \ref{['Fundamental gap of Horoconvex-domains']}
  14. Finding conformal factors

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{H}^2$ be a horoconvex domain whose diameter satisfies Then the fundamental gap of $\Omega$ satisfies

Figures (1)

  • Figure 1: In this figure, the blue spherical geodesic between the two points is contained within the two red horocycles. The previous argument shows that this is always the case when the points are contained within the green circle

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Remark 1
  • Theorem 1.5
  • Remark 2
  • Definition 2.1
  • Theorem 2.2: Theorem 1.1 nguyen2022fundamental
  • Lemma 2.3: singer1985estimate
  • ...and 13 more