Searcharxiv⌕ Search

arXiv subjects

Jeffrey J. Langford

Publications and source records attributed to Jeffrey J. Langford.

10 recordsLinked to original sources
arXiv2026

Exit Times for Brownian Motion and Location Detection

In 2023, Wyman and Xi asked ``Can you hear your location on a manifold?'' We pose a related question, asking if probabilistic data can be used to recover location within a manifold. More precisely, we ask if location can be recovered from the distribution of exit times of Brownian sample paths. We show that exit time moments determine location up to symmetry for a variety of two-dimensional domains. Beginning with (convex) domains whose boundaries are conic curves, we show that in elliptic, parabolic, and hyperbolic domains, the first two exit time moments are sufficient to detect location up to symmetry. We establish similar results in unbounded wedge domains and equilateral triangular domains. We show that the full sequence of exit time moments determines location in rectangular domains. We end with two location detection results on general planar domains. The key tool driving our work is a ``Poisson hierarchy,'' which establishes a connection between exit time moments of Brownian motion and solutions to a family of PDE problems.

math.AP↗
arXiv2024

Eigenvalue Estimates for $p$-Laplace Problems on Domains Expressed in Fermi Coordinates

We prove explicit and sharp eigenvalue estimates for Neumann $p$-Laplace eigenvalues in domains that admit a representation in Fermi coordinates. More precisely, if $γ$ denotes a non-closed curve in $\mathbb{R}^2$ symmetric with respect to the $y$-axis, let $D\subset \mathbb{R}^2$ denote the domain of points that lie on one side of $γ$ and within a prescribed distance $δ(s)$ from $γ(s)$ (here $s$ denotes the arc length parameter for $γ$). Write $μ_1^{odd}(D)$ for the lowest nonzero eigenvalue of the Neumann $p$-Laplacian with an eigenfunction that is odd with respect to the $y$-axis. For all $p>1$, we provide a lower bound on $μ_1^{odd}(D)$ when the distance function $δ$ and the signed curvature $k$ of $γ$ satisfy certain geometric constraints. In the linear case ($p=2$), we establish sufficient conditions to guarantee $μ_1^{odd}(D)=μ_1(D)$. We finally study the asymptotics of $μ_1(D)$ as the distance function tends to zero. We show that in the limit, the eigenvalues converge to the lowest nonzero eigenvalue of a weighted one-dimensional Neumann $p$-Laplace problem.

math.AP↗
arXiv2023

Maximizing the second Robin eigenvalue of simply connected curved membranes

The second eigenvalue of the Robin Laplacian is shown to be maximal for a spherical cap among simply connected Jordan domains on the 2-sphere, for substantial intervals of positive and negative Robin parameters and areas. Geodesic disks in the hyperbolic plane similarly maximize the eigenvalue on a natural interval of negative Robin parameters. These theorems extend work of Freitas and Laugesen from the Euclidean case (zero curvature) and the authors' hyperbolic and spherical results for Neumann eigenvalues (zero Robin parameter). Complicating the picture is the numerically observed fact that the second Robin eigenfunction on a large spherical cap is purely radial, with no angular dependence, when the Robin parameter lies in a certain negative interval depending on the cap aperture.

math.SP↗
arXiv2021

A Sharp Isoperimetric Inequality for the Second Eigenvalue of the Robin Plate

Among all $C^{\infty}$ bounded domains with equal volume, we show that the second eigenvalue of the Robin plate is uniquely maximized by an open ball, so long as the Robin parameter lies within a particular range of negative values. Our methodology combines recent techniques introduced by Freitas and Laugesen to study the second eigenvalue of the Robin membrane problem and techniques employed by Chasman to study the free plate problem. In particular, we choose eigenfunctions of the ball as trial functions in the Rayleigh quotient for a general domain; such eigenfunctions are comprised of ultraspherical Bessel and modified Bessel functions. Much of our work hinges on developing an understanding of delicate properties of these special functions, which may be of independent interest.

math.AP↗
arXiv2021

Extremizing Temperature Functions of Rods with Robin Boundary Conditions

We compare the solutions of two one-dimensional Poisson problems on an interval with Robin boundary conditions, one with given data, and one where the data has been symmetrized. When the Robin parameter is positive and the symmetrization is symmetric decreasing rearrangement, we prove that the solution to the symmetrized problem has larger increasing convex means. When the Robin parameter equals zero (so that we have Neumann boundary conditions) and the symmetrization is decreasing rearrangement, we similarly show that the solution to the symmetrized problem has larger convex means.

math.AP↗
arXiv2021

PDE Comparison Principles for Robin Problems

We compare the solutions of two Poisson problems in a spherical shell with Robin boundary conditions, one with given data, and one where the data has been cap symmetrized. When the Robin parameters are nonnegative, we show that the solution to the symmetrized problem has larger (increasing) convex means. We prove similar results on balls. We also prove a comparison principle on generalized cylinders with mixed boundary conditions (Neumann and Robin).

math.AP↗
arXiv2019

Improved Bounds for Hermite-Hadamard Inequalities in Higher Dimensions

Let $Ω\subset \mathbb{R}^n$ be a convex domain and let $f:Ω\rightarrow \mathbb{R}$ be a positive, subharmonic function (i.e. $Δf \geq 0$). Then $$ \frac{1}{|Ω|} \int_Ω{f dx} \leq \frac{c_n}{ |\partial Ω| } \int_{\partial Ω}{ f dσ},$$ where $c_n \leq 2n^{3/2}$. This inequality was previously only known for convex functions with a much larger constant. We also show that the optimal constant satisfies $c_n \geq n-1$. As a byproduct, we establish a sharp geometric inequality for two convex domains where one contains the other $ Ω_2 \subset Ω_1 \subset \mathbb{R}^n$: $$ \frac{|\partial Ω_1|}{|Ω_1|} \frac{| Ω_2|}{|\partial Ω_2|} \leq n.$$

math.CA↗
arXiv2017

Exit Time Moments and Eigenvalue Estimates

We study estimates involving the principal Dirichlet eigenvalue associated to a smoothly bounded domain in a complete Riemannian manifold and L1-norms of exit time moments of Brownian motion. Our results generalize a classical inequality of Polya.

math.SP↗