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Optimal domains for elliptic eigenvalue problems with rough coefficients

Stanley Snelson, Eduardo V. Teixeira

Abstract

We prove the existence of an open set minimizing the first Dirichlet eigenvalue of an elliptic operator with bounded, measurable coefficients, over all open sets of a given measure. Our proof is based on a free boundary approach: we characterize the eigenfunction on the optimal set as the minimizer of a penalized functional, and derive openness of the optimal set as a consequence of a Hölder estimate for the eigenfunction. We also prove that the optimal eigenfunction grows at most linearly from the free boundary, i.e. it is Lipschitz continuous at free boundary points.

Optimal domains for elliptic eigenvalue problems with rough coefficients

Abstract

We prove the existence of an open set minimizing the first Dirichlet eigenvalue of an elliptic operator with bounded, measurable coefficients, over all open sets of a given measure. Our proof is based on a free boundary approach: we characterize the eigenfunction on the optimal set as the minimizer of a penalized functional, and derive openness of the optimal set as a consequence of a Hölder estimate for the eigenfunction. We also prove that the optimal eigenfunction grows at most linearly from the free boundary, i.e. it is Lipschitz continuous at free boundary points.
Paper Structure (9 sections, 11 theorems, 94 equations)

This paper contains 9 sections, 11 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Related work
  4. Proof summary
  5. Open questions about the optimal domain
  6. Organization of the paper
  7. The penalized functional
  8. Hölder continuity
  9. Linear growth at boundary points

Key Result

Theorem 1.1

With $A\in M_{\theta,\Theta}(B)$, there exists a set $\Omega_* \in \mathcal{C}$ minimizing $\lambda_1^A(\Omega)$ over $\mathcal{C}$. Furthermore, the eigenfunction $u_* \in H_0^1(\Omega_*)$ corresponding to $\lambda_1^A(\Omega_*)$, extended by zero in $B\setminus \Omega_*$, is locally Hölder continu with $\alpha\in (0,1)$ depending only on $d$, $\theta$, and $\Theta$, and $C>0$ depending on $d$, $

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 10 more