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Local version of Courant's nodal domain theorem

S. Chanillo, A. Logunov, E. Malinnikova, D. Mangoubi

Abstract

Let $(M, g)$ be a closed Riemannian manifold, where g is $C^1$-smooth metric. Consider the sequence of eigenfunctions $u_k$ of the Laplace operator on M. Let $B$ be a ball on $M$. We prove a sharp estimate of the number of nodal domains of $u_k$ that intersect $B$. The problem of local bounds for the volume and for the number of nodal domains was raised by Donnelly and Fefferman, who also proposed an idea how one can prove such bounds. We combine their idea with two ingredients: the recent sharp Remez type inequality for eigenfunctions and the Landis type growth lemma in narrow domains.

Local version of Courant's nodal domain theorem

Abstract

Let be a closed Riemannian manifold, where g is -smooth metric. Consider the sequence of eigenfunctions of the Laplace operator on M. Let be a ball on . We prove a sharp estimate of the number of nodal domains of that intersect . The problem of local bounds for the volume and for the number of nodal domains was raised by Donnelly and Fefferman, who also proposed an idea how one can prove such bounds. We combine their idea with two ingredients: the recent sharp Remez type inequality for eigenfunctions and the Landis type growth lemma in narrow domains.

Paper Structure

This paper contains 8 sections, 6 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. The main tools.
  3. Remez type inequality for eigenfunctions.
  4. Weak maximum principle and a version of Landis' growth lemma.
  5. Proof of Theorem \ref{['main']}.
  6. Local bound for the volume of nodal domain.
  7. Local bound for the number of nodal domains.
  8. Appendix.

Key Result

Theorem \oldthetheorem

Consider a ball $B=\{ x \in M: d_g(x,x_0) < r \}$ with center at $x_0\in M$ and radius $r<r_0(M)$. For any eigenfunction $u_k$, the number of connected components of $B \setminus Z_{u_k}$ that intersect $\frac{1}{2}B$(the local number of nodal domains in $B$) is not greater than where $C_1,C_2$ depend only on $(M,g)$ and are independent of $k$ and $B$.

Theorems & Definitions (12)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem: L63, p.24
  • Corollary \oldthetheorem
  • proof
  • Lemma \oldthetheorem: a version of weak maximum principle
  • proof
  • ...and 2 more