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A hot spots theorem for the mixed eigenvalue problem with small Dirichlet region

Lawford Hatcher

TL;DR

This work addresses hot spots for the mixed Laplacian eigenproblem on convex domains with a small Dirichlet region $D$. It combines variational eigenvalue bounds, Neumann convexity, and nodal-set analysis to show that if $D$ is connected and sufficiently small, the first mixed eigenfunction $u_1^D$ has no interior critical points in $\Omega\setminus D$, with explicit diameter-based bounds for $D$ involving $|\,\Omega\,|$, $d$, and $j_0$ (the first zero of $J_0$). The authors also derive upper bounds for $\lambda_1^D$ via annulus comparisons and long, narrow geometries, proving $\lambda_1^D(\Omega)\to 0$ as the diameter of $D$ shrinks and providing constructive examples. They apply these results to produce new hot-spots instances for mixed problems and present a suite of examples illustrating the sharpness and scope of the connectivity requirement, including domains built from polygons and boundary arcs and results on second Neumann eigenfunctions for certain domains.

Abstract

We prove that on convex domains, first mixed Laplace eigenfunctions have no interior critical points if the Dirichlet region is connected and sufficiently small. We also find two seemingly new estimates on the first mixed eigenvalue to give explicit examples of when the Dirichlet region is sufficiently small.

A hot spots theorem for the mixed eigenvalue problem with small Dirichlet region

TL;DR

This work addresses hot spots for the mixed Laplacian eigenproblem on convex domains with a small Dirichlet region . It combines variational eigenvalue bounds, Neumann convexity, and nodal-set analysis to show that if is connected and sufficiently small, the first mixed eigenfunction has no interior critical points in , with explicit diameter-based bounds for involving , , and (the first zero of ). The authors also derive upper bounds for via annulus comparisons and long, narrow geometries, proving as the diameter of shrinks and providing constructive examples. They apply these results to produce new hot-spots instances for mixed problems and present a suite of examples illustrating the sharpness and scope of the connectivity requirement, including domains built from polygons and boundary arcs and results on second Neumann eigenfunctions for certain domains.

Abstract

We prove that on convex domains, first mixed Laplace eigenfunctions have no interior critical points if the Dirichlet region is connected and sufficiently small. We also find two seemingly new estimates on the first mixed eigenvalue to give explicit examples of when the Dirichlet region is sufficiently small.
Paper Structure (4 sections, 10 theorems, 45 equations, 2 figures)

This paper contains 4 sections, 10 theorems, 45 equations, 2 figures.

Table of Contents

  1. Introduction
  2. First mixed eigenvalue estimates
  3. Proofs of main results
  4. Examples

Key Result

Theorem 1.1

Suppose that $\Omega$ is convex. There exists $\epsilon>0$ such that if $D$ is connected with diameter at most $\epsilon$, then $u_1^D$ has no critical points in $\Omega\setminus D$.Recall that a domain is, by definition, open, so the theorem does not prohibit critical points in $\partial\Omega$. Mo

Figures (2)

  • Figure 1: The pairs $(\Omega,D)$ constructed in Examples \ref{['connectedisnec']} and \ref{['connectedisnec2']}---each with $\epsilon=0.1$. The dotted lines show the lines of symmetry for each figure. These graphics were created with Desmos (2024).
  • Figure 2: An example of a Neumann convex pair $(\Omega,D)$ discussed in Example \ref{['wild']}. The dashed line segments show the square $R_-$, and the solid black curve denotes $D=\gamma$. Created with Adobe Illustrator (2024).

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 2.3
  • proof
  • Proposition 2.4
  • ...and 18 more