二维度量Sturm-Liouville理论
A Metric Sturm-Liouville theory in Two Dimensions
摘要：
Sturm-Liouville理论的一个中心结果（也称为Sturm-Hurwitz定理）指出，如果$ϕ_k$是区间$I \subset \mathbb{R}$上的二阶微分算子的特征函数序列，则任何线性组合满足根$$ \# \left\{x \in I:\sum_{k \geq n}{ a_k ϕ_k(x)} = 0 \right\} \geq n-1.$$上的一致界。（直到对数因子）推广到两个维度：设$(M,g)$是一个紧致的二维流形（有边界或无边界），用$(ϕ_k)$表示一致椭圆算子$-\mbox{div}(a(\cdot) \nabla)$（具有Dirichlet或Neumann边界条件）的特征函数序列.然后，对于高于某个指数$n$，$$ f = \sum_{k \geq n}{a_k ϕ_k} ~ \mbox{we have} \quad \mathcal{H}^1 \left\{ x: f(x) = 0\right\} \gtrsim_{} \frac{\sqrt{n}}{\sqrt{\log{n}}} \log \left(n \frac{\|f\|_{L^2(M)}}{\|f\|_{L^1(M)}} \right)^{-1/2} \frac{\|f\|_{L^1(M)}}{\| f \|_{L^{\infty}(M)}} .$$的本征函数的任何线性组合，关于$M=\mathbb{T}^2$和$M=\mathbb{S}^2$的例子表明，这是最佳的对数因子。利用最优迁移和Wasserstein度量$W_p$的一个新的不等式证明：如果$f(x)dx$和$g(x)dx$是二维域$M$上的两个绝对连续测度，且密度连续，总质量相同，则对所有$1 \leq p <\infty$，$$ W_p(f(x)dx, g(x) dx) \cdot \mathcal{H}^1 \left\{x \in M: f(x) = g(x) \right\} \gtrsim_{M,p} \frac{\|f-g\|_{L^1(M)}^{1+1/p}}{\|f-g\|_{L^{\infty}(M)}}.$$
英文摘要：
A central result of Sturm-Liouville theory (also called the Sturm-Hurwitz Theorem) states that if $ϕ_k$ 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 $$ \# \left\{x \in I:\sum_{k \geq n}{ a_k ϕ_k(x)} = 0 \right\} \geq n-1.$$ 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 $(ϕ_k)$ denote the sequence of eigenfunctions of a uniformly elliptic operator $-\mbox{div}(a(\cdot) \nabla)$ (with Dirichlet or Neumann boundary conditions). Then, for any linear combination of eigenfunctions above a certain index $n$, $$ f = \sum_{k \geq n}{a_k ϕ_k} ~ \mbox{we have} \quad \mathcal{H}^1 \left\{ x: f(x) = 0\right\} \gtrsim_{} \frac{\sqrt{n}}{\sqrt{\log{n}}} \log \left(n \frac{\|f\|_{L^2(M)}}{\|f\|_{L^1(M)}} \right)^{-1/2} \frac{\|f\|_{L^1(M)}}{\| f \|_{L^{\infty}(M)}} .$$ 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 $1 \leq p <\infty$, $$ W_p(f(x)dx, g(x) dx) \cdot \mathcal{H}^1 \left\{x \in M: f(x) = g(x) \right\} \gtrsim_{M,p} \frac{\|f-g\|_{L^1(M)}^{1+1/p}}{\|f-g\|_{L^{\infty}(M)}}.$$
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