Browsing by Author "Steinerberger, Stefan"

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    A Variation on the Donsker-Varadhan Inequality for the Principial Eigenvalue
    (2017-04-23) Lu, Jianfeng; Steinerberger, Stefan
    The purpose of this short note is to give a variation on the classical Donsker-Varadhan inequality, which bounds the first eigenvalue of a second-order elliptic operator on a bounded domain $\Omega$ by the largest mean first exit time of the associated drift-diffusion process via $$\lambda_1 \geq \frac{1}{\sup_{x \in \Omega} \mathbb{E}_x \tau_{\Omega^c}}.$$ Instead of looking at the mean of the first exist time, we study quantiles: let $d_{p, \partial \Omega}:\Omega \rightarrow \mathbb{R}_{\geq 0}$ be the smallest time $t$ such that the likelihood of exiting within that time is $p$, then $$\lambda_1 \geq \frac{\log{(1/p)}}{\sup_{x \in \Omega} d_{p,\partial \Omega}(x)}.$$ Moreover, as $p \rightarrow 0$, this lower bound converges to $\lambda_1$.
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    Non-Convex Planar Harmonic Maps
    Kovalsky, Shahar Z; Aigerman, Noam; Daubechies, Ingrid; Kazhdan, Michael; Lu, Jianfeng; Steinerberger, Stefan
    We formulate a novel characterization of a family of invertible maps between two-dimensional domains. Our work follows two classic results: The Rad\'o-Kneser-Choquet (RKC) theorem, which establishes the invertibility of harmonic maps into a convex planer domain; and Tutte's embedding theorem for planar graphs - RKC's discrete counterpart - which proves the invertibility of piecewise linear maps of triangulated domains satisfying a discrete-harmonic principle, into a convex planar polygon. In both theorems, the convexity of the target domain is essential for ensuring invertibility. We extend these characterizations, in both the continuous and discrete cases, by replacing convexity with a less restrictive condition. In the continuous case, Alessandrini and Nesi provide a characterization of invertible harmonic maps into non-convex domains with a smooth boundary by adding additional conditions on orientation preservation along the boundary. We extend their results by defining a condition on the normal derivatives along the boundary, which we call the cone condition; this condition is tractable and geometrically intuitive, encoding a weak notion of local invertibility. The cone condition enables us to extend Alessandrini and Nesi to the case of harmonic maps into non-convex domains with a piecewise-smooth boundary. In the discrete case, we use an analog of the cone condition to characterize invertible discrete-harmonic piecewise-linear maps of triangulations. This gives an analog of our continuous results and characterizes invertible discrete-harmonic maps in terms of the orientation of triangles incident on the boundary.
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    Riesz Energy on the Torus: Regularity of Minimizers
    (2017-11-30) Lu, Jianfeng; Steinerberger, Stefan
    We study sets of $N$ points on the $d-$dimensional torus $\mathbb{T}^d$ minimizing interaction functionals of the type \[ \sum_{i, j =1 \atop i \neq j}^{N}{ f(x_i - x_j)}. \] The main result states that for a class of functions $f$ that behave like Riesz energies $f(x) \sim \|x\|^{-s}$ for $0< s < d$, the minimizing configuration of points has optimal regularity w.r.t. a Fourier-analytic regularity measure that arises in the study of irregularities of distribution. A particular consequence is that they are optimal quadrature points in the space of trigonometric polynomials up to a certain degree. The proof extends to other settings and also covers less singular functions such as $f(x) = \exp\bigl(- N^{\frac{2}{d}} \|x\|^2 \bigr)$.