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On the first eigenvalue and eigenfunction of the Laplacian with mixed boundary conditions

Nausica Aldeghi, Jonathan Rohleder

TL;DR

This work analyzes the ground state of the Laplacian with mixed Dirichlet–Neumann boundary conditions on planar domains, solving $-\Delta \psi = \lambda \psi$ with $\nu\cdot\nabla\psi = 0$ on $\Gamma$ and $\psi = 0$ on $\Gamma^c$. The authors introduce two novel variational principles—div-curl and curvature—whose minimizers are gradients of eigenfunctions, enabling precise spectral and monotonicity results. They prove a strict inequality $\lambda_1^{\Gamma^c} < \lambda_1^{\Gamma}$ and establish monotonicity of the first eigenfunction in the coordinate directions, yielding a hot-spots-type property for the mixed problem. A Helmholtz-type decomposition clarifies the spectral relation between the two mixed problems, linking the full spectrum to the eigenvalues of $-\Delta_{\Gamma}$ and $-\Delta_{\Gamma^c}$ and providing a robust framework for nodal and boundary behavior in mixed boundary settings.

Abstract

We consider the eigenvalue problem for the Laplacian with mixed Dirichlet and Neumann boundary conditions. For a certain class of bounded, simply connected planar domains we prove monotonicity properties of the first eigenfunction. As a consequence, we establish a variant of the hot spots conjecture for mixed boundary conditions. Moreover, we obtain an inequality between the lowest eigenvalue of this mixed problem and the lowest eigenvalue of the corresponding dual problem where the Dirichlet and Neumann boundary conditions are interchanged. The proofs are based on a novel variational principle, which we establish.

On the first eigenvalue and eigenfunction of the Laplacian with mixed boundary conditions

TL;DR

This work analyzes the ground state of the Laplacian with mixed Dirichlet–Neumann boundary conditions on planar domains, solving with on and on . The authors introduce two novel variational principles—div-curl and curvature—whose minimizers are gradients of eigenfunctions, enabling precise spectral and monotonicity results. They prove a strict inequality and establish monotonicity of the first eigenfunction in the coordinate directions, yielding a hot-spots-type property for the mixed problem. A Helmholtz-type decomposition clarifies the spectral relation between the two mixed problems, linking the full spectrum to the eigenvalues of and and providing a robust framework for nodal and boundary behavior in mixed boundary settings.

Abstract

We consider the eigenvalue problem for the Laplacian with mixed Dirichlet and Neumann boundary conditions. For a certain class of bounded, simply connected planar domains we prove monotonicity properties of the first eigenfunction. As a consequence, we establish a variant of the hot spots conjecture for mixed boundary conditions. Moreover, we obtain an inequality between the lowest eigenvalue of this mixed problem and the lowest eigenvalue of the corresponding dual problem where the Dirichlet and Neumann boundary conditions are interchanged. The proofs are based on a novel variational principle, which we establish.
Paper Structure (10 sections, 19 theorems, 128 equations, 4 figures)

This paper contains 10 sections, 19 theorems, 128 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Function spaces
  4. The Laplacian with mixed boundary conditions
  5. Sesquilinear forms and self-adjoint operators
  6. Variational principles for mixed Dirichlet--Neumann Laplacian eigenvalues
  7. A div-curl variational principle for mixed boundary conditions
  8. A curvature variational principle for mixed boundary conditions
  9. Eigenvalue inequalities and monotonicity of eigenfunctions
  10. A Helmholtz type decomposition

Key Result

Theorem 1.1

Assume that $\Omega$ is a bounded, simply connected Lipschitz domain with piecewise smooth boundary, that $\Gamma$ is connected, and that the unit normal field $\nu$ satisfies $\nu (x) \in \overline{Q_3}$ for almost all $x \in \Gamma^c$ and $\nu (x) \in \overline{Q_2} \cup \overline{Q_4}$ for almost

Figures (4)

  • Figure 1.1: Two domains for which Theorem \ref{['thm:intro1']} holds, rotated as required. The quadrants into which the exterior normal points are indicated. On the domain on the left, $\psi_1$ takes its unique maximum at $P$.
  • Figure 4.1: The eigenfunction $\varphi$ would simultaneously be vanishing on $\Gamma$ and strictly increasing along both pictured paths.
  • Figure 4.2: The first eigenfunction of $-\Delta_{{\Gamma^c}}$ takes its maximum at $P$.
  • Figure A.1: Each $\Gamma_i$ is contained in a level set of the potentials of elements of $H_c$.

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Proposition 3.2
  • ...and 26 more