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Reverse isoperimetric inequality for the lowest Robin eigenvalue of a triangle

David Krejcirik, Vladimir Lotoreichik, Tuyen Vu

TL;DR

This work investigates reverse spectral isoperimetric questions for the Robin Laplacian on triangles, showing that the equilateral triangle locally maximises the ground state for negative Robin parameter when the area is fixed. The authors reduce the problem to an equivalent one on the equilateral triangle through a geometry-preserving transform, compute first- and second-order derivatives, and derive Hessian bounds to prove local optimality for a range of negative couplings. They further explore global validity for small and large couplings by constructing explicit trial functions, including the equilateral ground state for small couplings and sector/Neumann-type states for large couplings, with complementary numerical evidence. Taken together, the results provide partial confirmations of the reverse isoperimetric inequality in the triangle class, with explicit bounds and a clear methodology for extending the regime of validity, illuminating the interplay between geometry and boundary interactions in spectral optimization.

Abstract

We consider the Laplace operator on a triangle, subject to attractive Robin boundary conditions. We prove that the equilateral triangle is a local maximiser of the lowest eigenvalue among all triangles of a given area provided that the negative boundary parameter is sufficiently small in absolute value, with the smallness depending on the area only. Moreover, using various trial functions, we obtain sufficient conditions for the global optimality of the equilateral triangle under fixed area constraint in the regimes of small and large couplings. We also discuss the constraint of fixed perimeter.

Reverse isoperimetric inequality for the lowest Robin eigenvalue of a triangle

TL;DR

This work investigates reverse spectral isoperimetric questions for the Robin Laplacian on triangles, showing that the equilateral triangle locally maximises the ground state for negative Robin parameter when the area is fixed. The authors reduce the problem to an equivalent one on the equilateral triangle through a geometry-preserving transform, compute first- and second-order derivatives, and derive Hessian bounds to prove local optimality for a range of negative couplings. They further explore global validity for small and large couplings by constructing explicit trial functions, including the equilateral ground state for small couplings and sector/Neumann-type states for large couplings, with complementary numerical evidence. Taken together, the results provide partial confirmations of the reverse isoperimetric inequality in the triangle class, with explicit bounds and a clear methodology for extending the regime of validity, illuminating the interplay between geometry and boundary interactions in spectral optimization.

Abstract

We consider the Laplace operator on a triangle, subject to attractive Robin boundary conditions. We prove that the equilateral triangle is a local maximiser of the lowest eigenvalue among all triangles of a given area provided that the negative boundary parameter is sufficiently small in absolute value, with the smallness depending on the area only. Moreover, using various trial functions, we obtain sufficient conditions for the global optimality of the equilateral triangle under fixed area constraint in the regimes of small and large couplings. We also discuss the constraint of fixed perimeter.
Paper Structure (19 sections, 9 theorems, 108 equations, 5 figures)

This paper contains 19 sections, 9 theorems, 108 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Robin Laplacian on a triangle
  4. Geometric setting
  5. The equilateral triangle
  6. Local optimality of the equilateral triangle
  7. The first derivative with respect to a
  8. The second derivative with respect to a
  9. The mixed second derivative
  10. The first derivative with respect to c
  11. The second derivative with respect to c
  12. Conclusions from the Hessian estimates
  13. Small couplings
  14. The ground state of the equilateral triangle as a trial function
  15. Limitations of the trial function
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\Omega^*$ be an equilateral triangle and let $\Omega$ denote any triangle of the same area. There exists a negative number $\alpha_0$ depending solely on the fixed area such that, for all $\alpha \in [\alpha_0,0)$, provided that $|\Omega\setminus\Omega^*|$ is sufficiently small (with the smallness depending also on the area $|\Omega|=|\Omega^*|$ fixed).

Figures (5)

  • Figure 1: The graph of the function $g$ defined in \ref{['g']}.
  • Figure 2: The blue colour indicates the region of validity of Conjecture \ref{['Conj.triangle']}: the equilateral triangle eigenfunction as a trial function.
  • Figure 3: The blue colour indicates the region of validity of Conjecture \ref{['Conj.triangle']}: the constant function as a trial function.
  • Figure 4: The blue colour indicates the region where the sufficient condition \ref{['eq:condition']} for validity of Conjecture \ref{['Conj.triangle']} is satisfied.
  • Figure 5: The blue colour indicates the region of validity of Conjecture \ref{['Conj.triangle']}: the function $u_\star$ in \ref{['eq:ustar']} as a trial function.

Theorems & Definitions (20)

  • Conjecture 1
  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Corollary 3.3
  • ...and 10 more