A metric Sturm–Liouville theory in two dimensions

  • Stefan SteinerbergerEmail author


A central result of Sturm–Liouville theory (also called the Sturm–Hurwitz theorem) states that if \(\phi _k\) is a sequence of eigenfunctions of a second order differential operator on the interval \(I \subset \mathbb {R}\), then any linear combination satisfies a uniform bound on the roots
$$\begin{aligned} \# \left\{ x \in I:\sum _{k \ge n}{ a_k \phi _k(x)} = 0 \right\} \ge n-1. \end{aligned}$$
We provide a sharp (up to logarithmic factors) generalization to two dimensions: let (Mg) be a compact two-dimensional manifold (with or without boundary), let \((\phi _k)\) denote the sequence of eigenfunctions of a uniformly elliptic operator \(-\text{ div }(a(\cdot ) \nabla )\) (with Dirichlet or Neumann boundary conditions). Then, for any linear combination of eigenfunctions above a certain index n,
$$\begin{aligned} f = \sum _{k \ge n}{a_k \phi _k} ~ \text{ we } \text{ have } \quad \mathcal {H}^1 \left\{ x: f(x) = 0\right\} \gtrsim _{} \frac{\sqrt{n}}{\sqrt{\log {n}}} \log \left( n \frac{\Vert f\Vert _{L^2(M)}}{\Vert f\Vert _{L^1(M)}} \right) ^{-1/2} \frac{\Vert f\Vert _{L^1(M)}}{\Vert f \Vert _{L^{\infty }(M)}} . \end{aligned}$$
Examples on \(M=\mathbb {T}^2\) and \(M=\mathbb {S}^2\) shows that this is optimal up to the logarithmic factors. The proof is using optimal transport and a new inequality for the Wasserstein metric \(W_p\): if f(x)dx and g(x)dx are two absolutely continuous measures on a two-dimensional domain M with continuous densities and the same total mass, then, for all \(1 \le p <\infty \),
$$\begin{aligned} W_p(f(x)dx, g(x) dx) \cdot \mathcal {H}^1 \left\{ x \in M: f(x) = g(x) \right\} \gtrsim _{M,p} \frac{\Vert f-g\Vert _{L^1(M)}^{1+1/p}}{\Vert f-g\Vert _{L^{\infty }(M)}}. \end{aligned}$$

Mathematics Subject Classification

28A75 34B24 35B05 35P20 49Q20 



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Authors and Affiliations

  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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