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Raghavendra Venkatraman

Publications and source records attributed to Raghavendra Venkatraman.

At least 19 recordsLinked to original sources
arXiv2026

An indefinite Coulomb interaction from the Steklov spectrum of perforated manifolds

We establish optimal convergence rates for Steklov eigenvalues and harmonically extended eigenfunctions toward their weighted Laplace--Beltrami counterparts on a closed manifold perforated by many small geodesic balls. The holes have radii that scale critically with respect to their spacing in the sense that the boundary area of each hole balances with the weighted volume of its Voronoi cell. We then derive a higher-order expansion of the Steklov eigenvalues; in dimensions two and three, we identify two correction scales. The expansion is governed by an indefinite Coulomb-type energy of the discrepancy between the boundary and bulk measures, mediated by the reduced Green function of the limiting operator. The proof relies on sharp estimates for certain auxiliary functions that we introduce in order to quantify the discrepancy measure between the surface measure on the holes and their background density. Our paper serves to bridge an emerging literature in spectral geometry with one on systems of points interacting via Coulomb-type energies.

math.AP↗
arXiv2026

Analytic spectral perturbation theory for a high-contrast Maxwell operator

We study analytic spectral perturbation theory for the time-harmonic Maxwell operator in a perfectly electrically conducting cavity containing a high-contrast core--shell structure. The dielectric permittivity equals $1$ in a bounded inclusion and a small complex parameter $δ$ in the surrounding shell. The limit $δ\to 0$ corresponds to an infinite-contrast regime and leads to a degenerate Maxwell system. Despite this degeneracy, we develop a detailed spectral theory for the limiting problem for general Lipschitz inclusions and shells. Using a novel operator-theoretic reformulation, we prove complex-analytic dependence of the spectrum on $δ$ in a neighborhood of $δ= 0$. When the inclusion is a ball, we analyze the asymptotic expansion of eigenvalues and identify conditions under which the leading-order term is independent of the geometry of the surrounding shell. We also construct examples of resonances for which the leading-order asymptotics depend sensitively on the shell geometry, even in this symmetric setting. These results clarify the mechanisms underlying geometry-invariance of resonances in high-contrast Maxwell systems and explain their robustness under small complex perturbations.

math.AP↗
arXiv2025

Convergence of Empirical Measures for i.i.d. samples in $W^{-α, p}$

Given $N$ i.i.d. samples from a probability measure $μ$ on $\mathbf{R}^d$, we study the rate of convergence of the empirical measure $μ_N \to μ$ in the negative Sobolev space $W^{-α, p}$. When $W^{-α, p}$ contains point measures (i.e. when $αp > (p-1)d$), we show $\mathbf{E} \| μ_N - μ\|_{W^{-α, p}}^p \leq C_d / N^{p/2}$ for an explicit dimensional constant $C_d$, and obtain a Gaussian tail bound. When $0 < αp \leq d(p-1)$, we prove a similar result for Gaussian regularizations.

math.PR↗
arXiv2025

Quantitative homogenization and large-scale regularity of Poisson point clouds

We prove quantitative homogenization results for harmonic functions on supercritical continuum percolation clusters--that is, Poisson point clouds with edges connecting points which are closer than some fixed distance. We show that, on large scales, harmonic functions resemble harmonic functions in Euclidean space with sharp quantitative bounds on their difference. In particular, for every point cloud which is supercritical (meaning that the intensity of the Poisson process is larger than the critical parameter which guarantees the existence of an infinite connected component), we obtain optimal corrector bounds, homogenization error estimates and large-scale regularity results.

math.PR↗
arXiv2025

Interaction energies in paranematic colloids

We consider a system of colloidal particles embedded in a paranematic -- an isotropic phase of a nematogenic medium above the temperature of the nematic-to-isotropic transition. In this state, the nematic order is induced by the boundary conditions in a narrow band around each particle and it decays exponentially in the bulk. We develop rigorous asymptotics of the linearization of the appropriate variational model that allow us to describe weak far-field interactions between the colloidal particles in two dimensional paranematic suspensions. We demonstrate analytically that decay rates of solutions to the full nonlinear and linear problems are similar and verify numerically that the interactions between the particles in these problems have similar dependence on the distance between the particles. Finally, we perform Monte-Carlo simulations for a system of colloidal particles in a paranematic and describe the statistical properties of this system.

math.AP↗
arXiv2025

Minimax Rates for the Estimation of Eigenpairs of Weighted Laplace-Beltrami Operators on Manifolds

We study the problem of estimating eigenpairs of elliptic differential operators from samples of a distribution $ρ$ supported on a manifold $M$. The operators discussed in the paper are relevant in unsupervised learning and in particular are obtained by taking suitable scaling limits of widely used graph Laplacians over data clouds. We study the minimax risk for this eigenpair estimation problem and explore the rates of approximation that can be achieved by commonly used graph Laplacians built from random data. More concretely, assuming that $ρ$ belongs to a certain family of distributions with controlled second derivatives, and assuming that the $d$-dimensional manifold $M$ where $ρ$ is supported has bounded geometry, we prove that the statistical minimax rate for approximating eigenvalues and eigenvectors in the $H^1(M)$-sense is $n^{-2/(d+4)}$, a rate that matches the minimax rate for a closely related density estimation problem. We then revisit the literature studying Laplacians over proximity graphs in the large data limit and prove that, under slightly stronger regularity assumptions on the data generating model, eigenpairs of graph Laplacians induce manifold agnostic estimators with an error of approximation that, up to logarithmic corrections, matches our lower bounds. Our analysis allows us to expand the existing literature on graph-based learning in at least two significant ways: 1) we consider stronger norms to measure the error of approximation than the ones that had been analyzed in the past; 2) our rates of convergence are uniform over a family of smooth distributions and do not just apply to densities with special symmetries, and, as a consequence of our lower bounds, are essentially sharp when the connectivity of the graph is sufficiently high.

stat.ML↗
arXiv2025

Dirichlet eigenfunctions with nonzero mean value

We consider Laplacian eigenfunctions on a domain $Ω\subset \mathbb{R}^d$. Under Neumann boundary conditions, the first eigenfunction is constant and the others have mean value 0. The situation is different for Dirichlet boundary conditions: on `generic' domains, one would expect that every eigenfunction has nonzero mean value. The other extreme is the ball in $\mathbb{R}^d$, where among the first $n$ eigenfunctions only $\sim n^{1/d}$ have a mean value different from zero. We prove that this rate is sharp in \textit{any} smooth domain, up to a logarithmic factor: in any smooth domain~$Ω$, among the first $n$ Dirichlet eigenfunctions at least $ (\log{n})^{-1/2} \cdot n^{1/d} $ have a nonzero mean.

math.AP↗
arXiv2025

Interaction energies in nematic liquid crystal suspensions

We establish, as $ρ\to 0$, an asymptotic expansion for the minimal Dirichlet energy of $\mathbb S^2$-valued maps outside a finite number of three-dimensional particles of size $ρ$ with fixed centers $x_j\in\mathbb{R}^3$, under general anchoring conditions at the particle boundaries. Up to a scaling factor, this expansion is of the form \begin{align*} E_ρ= \sum_j μ_j -4πρ\sum_{i\neq j} \frac{\langle v_i,v_j\rangle}{|x_i-x_j|} +o(ρ)\,, \end{align*} where $μ_j$ is the minimal energy after zooming in at scale $ρ$ around each particle, and $v_j\in\mathbb{R}^3$ is a torque determined by the far-field behavior of the corresponding single-particle minimizer. The above expansion highlights Coulomb-like interactions between the particle centers. This agrees with the \textit{electrostatics analogy} commonly used in the physics literature for colloid interactions in nematic liquid crystal. That analogy was pioneered by Brochard and de Gennes in 1970, based on a formal linearization argument. We obtain here for the first time a precise estimate of the energy error introduced by this linearization procedure.

math.AP↗
arXiv2024

Transverse Magnetic ENZ Resonators: Robustness and Optimal Shape Design

We study certain "geometric-invariant resonant cavitie"' introduced by Liberal et. al in a 2016 Nature Comm. paper, modeled using the transverse magnetic reduction of Maxwell's equations. The cross-section consists of a dielectric inclusion surrounded by an "epsilon-near-zero" (ENZ) shell. When the shell has the right area, its interaction with the inclusion produces a resonance. Mathematically, the resonance is a nontrivial solution of a 2D divergence-form Helmoltz equation $\nabla \cdot \left(\varepsilon^{-1}(x,ω) \nabla u \right) + ω^2 μu = 0$, where $\varepsilon(x,ω)$ is the (complex-valued) dielectric permittivity, $ω$ is the frequency, $μ$ is the magnetic permeability, and a homogeneous Neumann condition is imposed at the outer boundary of the shell. This is a nonlinear eigenvalue problem, since $\varepsilon$ depends on $ω$. Use of an ENZ material in the shell means that $\varepsilon(x,ω)$ is nearly zero there, so the PDE is rather singular. Working with a Lorentz model for the dispersion of the ENZ material, we put the discussion of Liberal et.~al.~on a sound foundation by proving the existence of the anticipated resonance when the loss is sufficiently small. Our analysis is perturbative in character despite the apparently singular form of the PDE. While the existence of the resonance depends only on the area of the ENZ shell, the rate at which it decays depends on the shape of the shell. We consider an associated optimal design problem: what shape shell gives the slowest-decaying resonance? We prove that if the dielectric inclusion is a ball then the optimal shell is a concentric annulus. For an inclusion of any shape, we study a convex relaxation of the design problem using tools from convex duality, and discuss the conjecture that our relaxed problem amounts to considering homogenization-like limits of nearly optimal designs.

math.AP↗
arXiv2023

Asymptotic expansion of the spectrum for periodic Schrödinger operators

We prove an asymptotic expansion for the eigenvalues and eigenfunctions of Schrödinger-type operator with a confining potential and with principle part a periodic elliptic operator in divergence form. We compare the spectrum to the homogenized operator and characterize the corrections up to arbitrarily high order.

math.AP↗
arXiv2023

Far-Field Expansions for Harmonic Maps and the Electrostatics Analogy in Nematic Suspensions

For a smooth bounded domain $G\subset\mathbb{R}^3$ we consider maps $n\colon\mathbb R^3\setminus G\to\mathbb S^2$ minimizing the energy $E(n)=\int_{\mathbb R^3\setminus G}|\nabla n|^2 +F_s(n_{\lfloor\partial G})$ among $\mathbb S^2$-valued map such that $n(x)\approx n_0$ as $|x|\to\infty$. This is a model for a particle $G$ immersed in nematic liquid crystal. The surface energy $F_s$ describes the anchoring properties of the particle, and can be quite general. We prove that such minimizing map $n$ has an asymptotic expansion in powers of $1/r$. Further, we show that the leading order $1/r$ term is uniquely determined by the far-field condition $n_0$ for almost all $n_0\in\mathbb S^2$, by relating it to the gradient of the minimal energy with respect to $n_0$. We derive various consequences of this relation in physically motivated situations: when the orientation of the particle $G$ is stable relative to a prescribed far-field alignment $n_0$; and when the particle $G$ has some rotational symmetries. In particular, these corollaries justify some approximations that can be found in the physics literature to describe nematic suspensions via a so-called electrostatics analogy.

math.AP↗
arXiv2022

Least Wasserstein distance between disjoint shapes with perimeter regularization

We prove the existence of global minimizers to the double minimization problem \[ \inf\Big\{ P(E) + λW_p(\mathcal{L}^n \lfloor \, E,\mathcal{L}^n \lfloor\, F) \colon |E \cap F| = 0, \, |E| = |F| = 1\Big\}, \] where $P(E)$ denotes the perimeter of the set $E$, $W_p$ is the $p$-Wasserstein distance between Borel probability measures, and $λ> 0$ is arbitrary. The result holds in all space dimensions, for all $p \in [1,\infty),$ and for all positive $λ$. This answers a question of Buttazzo, Carlier, and Laborde.

math.AP↗
arXiv2022

Complex Analytic Dependence on the Dielectric Permittivity in ENZ Materials: The Photonic Doping Example

Motivated by the physics literature on "photonic doping" of scatterers made from "epsilon-near-zero" (ENZ) matrials, we consider how the scattering of time-harmonic TM electromagnetic waves by a cylindrical ENZ region $Ω\times \mathbb{R}$ is affected by the presence of a "dopant" $D \subset Ω$ in which the dielectric permittivity is not near zero. Mathematically, this reduces to analysis of a 2D Helmholtz equation $\mathrm{div}\, (a(x)\nabla u) + k^2 u = f$ with a piecewise-constant, complex valued coefficient $a$ that is nearly infinite (say $a = \frac{1}δ$ with $δ\approx 0$) in $Ω\setminus \bar{D}.$ We show (under suitable hypotheses) that the solution $u$ depends analytically on $δ$ near $0$, and we give a simple PDE characterization of the terms in its Taylor expansion. For the application to photonic doping, it is the leading-order corrections in $δ$ that are most interesting: they explain why photonic doping is only mildly affected by the presence of losses, and why it is seen even at frequencies where dielectric permittivity is merely small. Equally important: our results include a PDE characterization of the leading-order electric field in the ENZ region as $δ\to 0$, whereas the existing literature on photonic doping provides only the leading-order magnetic field.

math.AP↗
arXiv2021

Anisotropic Surface Tensions for Phase Transitions in Periodic Media

This paper establishes bounds on the homogenized surface tension for a heterogeneous Allen-Cahn energy functional in a periodic medium. The approach is based on relating the homogenized energy to a purely geometric variational problem involving the large scale behaviour of the signed distance function to a hyperplane in periodic media. Motivated by this, a homogenization result for the signed distance function to a hyperplane in both periodic and almost periodic media is proven.

math.AP↗
arXiv2020

Bounds on the energy of a soft cubic ferromagnet with large magnetostriction

We complete the analysis initiated in [5] on the micromagnetics of cubic ferromagnets in which the role of magnetostriction is significant. We prove ansatz-free lower bounds for the scaling of the total micromagnetic energy including magnetostriction contribution, for a two-dimensional sample. This corresponds to the micromagnetic energy-per-unit-length of an infinitely thick sample. A consequence of our analysis is an explanation of the multi-scale zig-zag Landau state patterns recently reported in single crystal Galfenol disks from an energetic viewpoint. Our proofs combines a number of well-developed techniques in energy-driven pattern formation with a recent regularity result for the kinetic formulation of the Eikonal equation of Ghiraldin and Lamy.

math.AP↗
arXiv2020

Homogenization of quasi-crystalline functionals via two-scale-cut-and-project convergence

We consider a homogenization problem associated with quasi-crystalline multiple integrals of the form \begin{equation*} \begin{aligned} u_\varepsilon\in L^p(Ω;\mathbb{R}^d) \mapsto \int_Ωf_R\Big(x,\frac{x}{\varepsilon}, u_\varepsilon(x)\Big)\, dx, \end{aligned} \end{equation*} where $u_\varepsilon$ is subject to constant-coefficient linear partial differential constraints. The quasi-crystalline structure of the underlying composite is encoded in the dependence on the second variable of the Lagrangian, $f_R$, and is modeled via the cut-and-project scheme that interprets the heterogeneous microstructure to be homogenized as an irrational subspace of a higher-dimensional space. A key step in our analysis is the characterization of the quasi-crystalline two-scale limits of sequences of the vector fields $u_\varepsilon$ that are in the kernel of a given constant-coefficient linear partial differential operator, $\mathcal{A}$, that is, $\mathcal{A} u _\varepsilon =0$. Our results provide a generalization of related ones in the literature concerning the ${\rm \mathcal{A} =curl } $ case to more general differential operators $\mathcal{A}$ with constant coefficients, and without coercivity assumptions on the Lagrangian $f_R$.

math.AP↗