2018/09/04 by Steinerberger, Stefan
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.1809.01044
A central result of Sturm-Liouville theory (also called the Sturm-Hurwitz Theorem) states that if ϕk is a sequence of eigenfunctions of a second order differential operator on the interval I ⊂ ℝ, then any linear combination satisfies a uniform bound on the roots # \x ∈ I:∑k ≥ n ak ϕk(x) = 0 \ ≥ n-1. We provide a sharp (up to logarithmic factors) generalization to two dimensions: let (M,g) be a compact two-dimensional manifold (with or without boundary), let (ϕk) denote the sequence of eigenfunctions of a uniformly elliptic operator -div(a(⋅) ∇) (with Dirichlet or Neumann boundary conditions). Then, for any linear combination of eigenfunctions above a certain index n, f = ∑k ≥ nak ϕk ~ we have H1 \ x: f(x) = 0\ \gtrsim \frac√(n)√logn log (n \frac‖f‖L2(M)‖f‖L1(M) )-1/2 \frac‖f‖L1(M)‖ f ‖L∞(M) . Examples on M=\mathbbT2 and M=\mathbbS2 shows that this is optimal up to the logarithmic factors. The proof is using optimal transport and a new inequality for the Wasserstein metric Wp: 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 ≤ p