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Polygonal Faber-Krahn inequality: Local minimality via validated computing

Beniamin Bogosel, Dorin Bucur

Abstract

The main result of the paper shows that the regular $n$-gon is a local minimizer for the first Dirichlet-Laplace eigenvalue among $n$-gons having fixed area for $n \in \{5,6\}$. The eigenvalue is seen as a function of the coordinates of the vertices in $\Bbb R^{2n}$. Relying on fine regularity results of the first eigenfunction in a convex polygon, an explicit a priori estimate is given for the eigenvalues of the Hessian matrix associated to the discrete problem, whose coefficients involve the solutions of some Poisson equations with singular right hand sides. The a priori estimates, in conjunction with certified finite element approximations of these singular PDEs imply the local minimality for $n \in \{5,6\}$. All computations, including the finite element computations, are realized using interval arithmetic.

Polygonal Faber-Krahn inequality: Local minimality via validated computing

Abstract

The main result of the paper shows that the regular -gon is a local minimizer for the first Dirichlet-Laplace eigenvalue among -gons having fixed area for . The eigenvalue is seen as a function of the coordinates of the vertices in . Relying on fine regularity results of the first eigenfunction in a convex polygon, an explicit a priori estimate is given for the eigenvalues of the Hessian matrix associated to the discrete problem, whose coefficients involve the solutions of some Poisson equations with singular right hand sides. The a priori estimates, in conjunction with certified finite element approximations of these singular PDEs imply the local minimality for . All computations, including the finite element computations, are realized using interval arithmetic.
Paper Structure (15 sections, 13 theorems, 179 equations, 11 figures, 2 tables)

This paper contains 15 sections, 13 theorems, 179 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dirichlet-Laplacian eigenvalues on polygons: first and second derivatives
  4. General approximation framework
  5. Material derivative decomposition: piecewise $H^2$ regularity and estimates
  6. Estimations for Hessian eigenvalues
  7. Numerical validation
  8. Validation strategy, implementation and results
  9. Detailed description of the validation procedure
  10. Simulations and Results
  11. Implementation
  12. Conclusion, Perspectives
  13. Morley element and interpolation constants
  14. Description of the code
  15. Fast validation of eigenvalues for large sparse matrices

Key Result

Theorem 2.1

For $0\leq k \leq n-1$, $\theta=2\pi/n$ denote and The eigenvalues of the Hessian matrix of $\lambda_1({\bf x})\mathcal{A}({\bf x})$ given in are given by $\mu_j$, $j=0,...,2n-1$.

Figures (11)

  • Figure 1: Examples of admissible triangulations used for defining perturbations on a polygon and graphical view of the function $\varphi_1$.
  • Figure 2: Graphical representation of finite element solutions for problems \ref{['eq:U0']}. Discontinuities of normal derivatives are visible across certains rays connecting the center of the regular $n$-gon to the vertices.
  • Figure 3: Reflection procedure: for the pentagon at least four reflections are needed (left), for $n \geq 6$ three reflections suffice (right). The thresholds for the application of the cutoff function are also illustrated with dashed lines. The cutoff function equals $1$ inside the small disk, zero outside the big disk and is affine in the radial direction in-between.
  • Figure 4: Symmetric mesh for a regular pentagon for $m=5$: the ray $[{\bf o} {\bf a}_0]$ is divided into $m$ equal segments.
  • Figure 5: Mesh of a triangular slice $\mathcal{T}_h^0$ of the regular $n$-gon (left). Mapping from the slice $\mathcal{T}_h^0$ to the whole mesh used to extend a function by dihedral symmetry (right).
  • ...and 6 more figures

Theorems & Definitions (18)

  • Theorem 2.1
  • Lemma 3.1
  • Lemma 3.2: Extension operator with computed norm
  • Remark 3.3
  • Theorem 3.4
  • Lemma 3.5
  • proof
  • Theorem 3.6
  • proof
  • Remark 3.7
  • ...and 8 more