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Hausdorff measure bounds for nodal sets of Steklov eigenfunctions

Stefano Decio

Abstract

We study nodal sets of Steklov eigenfunctions in a bounded domain with $\mathcal{C}^2$ boundary. Our first result is a lower bound for the Hausdorff measure of the nodal set: we show that for $u_λ$ a Steklov eigenfunction, with eigenvalue $λ\neq 0$, $\mathcal{H}^{d-1}(\{u_λ=0\})\geq c_Ω$, where $c_Ω$ is independent of $λ$. We also prove an almost sharp upper bound, namely $\mathcal{H}^{d-1}(\{u_λ=0\})\leq C_Ωλ\log(λ+e)$.

Hausdorff measure bounds for nodal sets of Steklov eigenfunctions

Abstract

We study nodal sets of Steklov eigenfunctions in a bounded domain with boundary. Our first result is a lower bound for the Hausdorff measure of the nodal set: we show that for a Steklov eigenfunction, with eigenvalue , , where is independent of . We also prove an almost sharp upper bound, namely .

Paper Structure

This paper contains 12 sections, 27 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Lower bound on nodal sets
  3. Proof of Theorem \ref{['nadirashvili']}
  4. Elliptic estimates
  5. Frequency function and doubling index
  6. An informal outline of the proof
  7. Local asymmetry
  8. Counting doubling indices
  9. Estimates in a spherical shell
  10. Finding many balls around the zero set
  11. Proof of the lower bound
  12. Upper bound

Key Result

Theorem \oldthetheorem

Let $\Omega$ be a bounded domain in ${\mathbf R}^d$ with $\mathcal{C}^2$-smooth boundary, and let $u_{{\lambda}}$ be a solution of problem in $\Omega$, ${\lambda}\neq 0$. Then there exists a constant $c_{\Omega}>0$ independent of ${\lambda}$ such that

Theorems & Definitions (54)

  • Theorem \oldthetheorem
  • Theorem A
  • Remark
  • Theorem \oldthetheorem
  • Remark
  • Theorem \oldthetheorem
  • Remark
  • Theorem \oldthetheorem: GT, Theorem 9.1
  • Corollary 1
  • proof
  • ...and 44 more