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An overdetermined problem in 2D linearised hydrostatics

Giovanni Franzina

Abstract

In two spatial dimensions, we discuss the relation between the solvability of Schiffer's overdetermined problem and the optimality, among sets of prescribed area, of the first eigenvalue in the buckling problem for a clamped plate and that of the first eigenvalue of the Stokes operator. For the latter, we deduce that the minimisers under area constraint that are smooth and simply connected must be discs from the fact that a pressureless velocity is a necessary condition of optimality.

An overdetermined problem in 2D linearised hydrostatics

Abstract

In two spatial dimensions, we discuss the relation between the solvability of Schiffer's overdetermined problem and the optimality, among sets of prescribed area, of the first eigenvalue in the buckling problem for a clamped plate and that of the first eigenvalue of the Stokes operator. For the latter, we deduce that the minimisers under area constraint that are smooth and simply connected must be discs from the fact that a pressureless velocity is a necessary condition of optimality.
Paper Structure (10 sections, 10 theorems, 25 equations)

This paper contains 10 sections, 10 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Overdetermined problems
  3. Proof of Proposition \ref{['prop0']}
  4. $\mathfrak{S}^B(\Omega)\subseteq \mathfrak{S}_0^H(\Omega)$.
  5. $\mathfrak{S}_g^H(\Omega)\subseteq\mathfrak{S}^B(\Omega)$ for all harmonic $g$
  6. $\mathfrak{S}^B(\Omega)\subseteq \mathfrak{S}_p^S(\Omega)$ for an appropriate $p$
  7. The last statement
  8. Proof of Theorem \ref{['thm1']}
  9. Proof of Proposition \ref{['prop1']}
  10. Relations between the least eigenvalues

Key Result

Proposition 1

We have If $\Omega$ is simply connected, then the second inclusion holds as an equality.

Theorems & Definitions (18)

  • Proposition 1
  • Theorem 1
  • Remark 1
  • Remark 2
  • Proposition 2
  • Proposition 3
  • Remark 3: Optimality conditions for $\lambda_1^S(\Omega)$
  • Corollary 1
  • Corollary 2
  • Lemma 1
  • ...and 8 more