arXiv subjects

Robert V. Kohn

Publications and source records attributed to Robert V. Kohn.

At least 19 recordsLinked to original sources

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

A nonlinear homogenization-based perspective on the soft modes and effective energies of some conformal metamaterials

There is a growing mechanics literature concerning the macroscopic properties of mechanism-based mechanical metamaterials. This amounts mathematically to a homogenization problem involving nonlinear elasticity. A key goal is to identify the "soft modes" of the metamaterial. We achieve this goal using methods from homogenization for some specific 2D examples -- including discrete models of the Rotating Squares metamaterial and the Kagome metamaterial -- whose soft modes are compressive conformal maps. The innovation behind this achievement is a new technique for bounding the effective energy from below, which takes advantage of the metamaterial's structure and symmetry.

math.AP↗

arXiv2025

The effective energy of a lattice metamaterial

We study the sense in which the continuum limit of a broad class of discrete materials with periodic structures can be viewed as a nonlinear elastic material. While we are not the first to consider this question, our treatment is more general and more physical than those in the literature. Indeed, it applies to a broad class of systems, including ones that possess mechanisms; and we discuss how the degeneracy that plagues prior work in this area can be avoided by penalizing change of orientation. A key motivation for this work is its relevance to mechanism-based mechanical metamaterials. Such systems often have ``soft modes'', achieved in typical examples by modulating mechanisms. Our results permit the following more general definition of a soft mode: it is a macroscopic deformation whose effective energy vanishes -- in other words, one whose spatially-averaged elastic energy tends to zero in the continuum limit.

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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

A PDE-Based Analysis of the Symmetric Two-Armed Bernoulli Bandit

This work addresses a version of the two-armed Bernoulli bandit problem where the sum of the means of the arms is one (the symmetric two-armed Bernoulli bandit). In a regime where the gap between these means goes to zero as the number of prediction periods approaches infinity, i.e., the difficulty of detecting the gap increases as the sample size increases, we obtain the leading order terms of the minmax optimal regret and pseudoregret for this problem by associating each of them with a solution of a linear heat equation. Our results improve upon the previously known results; specifically, we explicitly compute these leading order terms in three different scaling regimes for the gap. Additionally, we obtain new non-asymptotic bounds for any given time horizon. Although optimal player strategies are not known for more general bandit problems, there is significant interest in considering how regret accumulates under specific player strategies, even when they are not known to be optimal. We expect that the methods of this paper should be useful in settings of that type.

cs.LG↗

arXiv2023

Some results on the Guest-Hutchinson modes and periodic mechanisms of the Kagome lattice metamaterial

Lattice materials are interesting mechanical metamaterials, and their mechanical properties are often related to the presence of mechanisms. The existence of periodic mechanisms can be indicated by the presence of Guest-Hutchinson (GH) modes, since GH modes are sometimes infinitesimal versions of periodic mechanisms. However, not every GH mode comes from a periodic mechanism. This paper focuses on: (1) clarifying the relationship between GH modes and periodic mechanisms; and (2) answering the question: which GH modes come from periodic mechanisms? We focus primarily on a special lattice system, the Kagome lattice. Our results include explicit formulas for all two-periodic mechanisms of the Kagome lattice, and a necessary condition for a GH mode to come from a periodic mechanism in general. We apply our necessary condition to the two-periodic GH modes, and also to some special GH modes found by Fleck and Hutchinson using Bloch-type boundary conditions on the unit cell of the Kagome lattice.

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arXiv2023

A New Approach to Drifting Games, Based on Asymptotically Optimal Potentials

We develop a new approach to drifting games, a class of two-person games with many applications to boosting and online learning settings. Our approach involves (a) guessing an asymptotically optimal potential by solving an associated partial differential equation (PDE); then (b) justifying the guess, by proving upper and lower bounds on the final-time loss whose difference scales like a negative power of the number of time steps. The proofs of our potential-based upper bounds are elementary, using little more than Taylor expansion. The proofs of our potential-based lower bounds are also elementary, combining Taylor expansion with probabilistic or combinatorial arguments. Not only is our approach more elementary, but we give new potentials and derive corresponding upper and lower bounds that match each other in the asymptotic regime.

cs.LG↗

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.

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arXiv2020

A PDE Approach to the Prediction of a Binary Sequence with Advice from Two History-Dependent Experts

The prediction of a binary sequence is a classic example of online machine learning. We like to call it the 'stock prediction problem,' viewing the sequence as the price history of a stock that goes up or down one unit at each time step. In this problem, an investor has access to the predictions of two or more 'experts,' and strives to minimize her final-time regret with respect to the best-performing expert. Probability plays no role; rather, the market is assumed to be adversarial. We consider the case when there are two history-dependent experts, whose predictions are determined by the d most recent stock moves. Focusing on an appropriate continuum limit and using methods from optimal control, graph theory, and partial differential equations, we discuss strategies for the investor and the adversarial market, and we determine associated upper and lower bounds for the investor's final-time regret. When d is less than 4 our upper and lower bounds coalesce, so the proposed strategies are asymptotically optimal. Compared to other recent applications of partial differential equations to prediction, ours has a new element: there are two timescales, since the recent history changes at every step whereas regret accumulates more slowly.

math.OC↗

arXiv2020

New Potential-Based Bounds for Prediction with Expert Advice

This work addresses the classic machine learning problem of online prediction with expert advice. We consider the finite-horizon version of this zero-sum, two-person game. Using verification arguments from optimal control theory, we view the task of finding better lower and upper bounds on the value of the game (regret) as the problem of finding better sub- and supersolutions of certain partial differential equations (PDEs). These sub- and supersolutions serve as the potentials for player and adversary strategies, which lead to the corresponding bounds. To get explicit bounds, we use closed-form solutions of specific PDEs. Our bounds hold for any given number of experts and horizon; in certain regimes (which we identify) they improve upon the previous state of the art. For two and three experts, our bounds provide the optimal leading order term.

cs.LG↗

arXiv2020

New Potential-Based Bounds for the Geometric-Stopping Version of Prediction with Expert Advice

This work addresses the classic machine learning problem of online prediction with expert advice. A new potential-based framework for the fixed horizon version of this problem has been recently developed using verification arguments from optimal control theory. This paper extends this framework to the random (geometric) stopping version. To obtain explicit bounds, we construct potentials for the geometric version from potentials used for the fixed horizon version of the problem. This construction leads to new explicit lower and upper bounds associated with specific adversary and player strategies. While there are several known lower bounds in the fixed horizon setting, our lower bounds appear to be the first such results in the geometric stopping setting with an arbitrary number of experts. Our framework also leads in some cases to improved upper bounds. For two and three experts, our bounds are optimal to leading order.

cs.LG↗

arXiv2019

Prediction with Expert Advice: a PDE Perspective

This work addresses a classic problem of online prediction with expert advice. We assume an adversarial opponent, and we consider both the finite-horizon and random-stopping versions of this zero-sum, two-person game. Focusing on an appropriate continuum limit and using methods from optimal control, we characterize the value of the game as the viscosity solution of a certain nonlinear partial differential equation. The analysis also reveals the predictor's and the opponent's minimax optimal strategies. Our work provides, in particular, a continuum perspective on recent work of Gravin, Peres, and Sivan (Proc SODA 2016). Our techniques are similar to those of Kohn and Serfaty (Comm Pure Appl Math 2010), where scaling limits of some two-person games led to elliptic or parabolic PDEs.

math.AP↗

arXiv2017

Wrinkling of a thin circular sheet bonded to a spherical substrate

We consider a disk-shaped thin elastic sheet bonded to a compliant sphere. (Our sheet can slip along the sphere; the bonding controls only its normal displacement.) If the bonding is stiff (but not too stiff), the geometry of the sphere makes the sheet wrinkle to avoid azimuthal compression. The total energy of this system is the elastic energy of the sheet plus a (Winkler-type) substrate energy. Treating the thickness of the sheet $h$ as a small parameter, we determine the leading-order behavior of the energy as $h$ tends to zero, and give (almost matching) upper and lower bounds for the next-order correction. Our analysis of the leading-order behavior determines the macroscopic deformation of the sheet; in particular it determines the extent of the wrinkled region, and predicts the (nontrivial) radial strain of the sheet. The leading-order behavior also provides insight about the length scale of the wrinkling, showing that it must be approximately independent of the distance $r$ from the center of the sheet (so that the number of wrinkles must increase with $r$). Our results on the next-order correction provide insight about how the wrinkling pattern should vary with $r$. Roughly speaking, they suggest that the length scale of wrinkling should {\it not} be exactly constant -- rather, it should vary a bit, so that the number of wrinkles at radius $r$ can be approximately piecewise constant in its dependence on $r$, taking values that are integer multiples of $h^{-α}$ with $α\approx 1/2$.

math.AP↗

arXiv2015

The coarsening of folds in hanging drapes

We consider the elastic energy of a hanging drape -- a thin elastic sheet, pulled down by the force of gravity, with fine-scale folding at the top that achieves approximately uniform confinement. This example of energy-driven pattern formation in a thin elastic sheet is of particular interest because the length scale of folding varies with height. We focus on how the minimum elastic energy depends on the physical parameters. As the sheet thickness vanishes, the limiting energy is due to the gravitational force and is relatively easy to understand. Our main accomplishment is to identify the "scaling law" of the correction due to positive thickness. We do this by (i) proving an upper bound, by considering the energies of several constructions and taking the best; (ii) proving an ansatz-free lower bound, which agrees with the upper bound up to a parameter-independent prefactor. The coarsening of folds in hanging drapes has also been considered in the recent physics literature, using a self-similar construction whose basic cell has been called a "wrinklon." Our results complement and extend that work, by showing that self-similar coarsening achieves the optimal scaling law in a certain parameter regime, and by showing that other constructions (involving lateral spreading of the sheet) do better in other regions of parameter space. Our analysis uses a geometrically linear Föppl-von Kármán model for the elastic energy, and is restricted to the case when Poisson's ratio is zero.

math.AP↗

arXiv2014

Convergence of a crystalline algorithm for the heat equation in one dimension and for the motion of a graph by weighted curvature

Motion by (weighted) mean curvature is a geometric evolution law for surfaces, representing steepest descent with respect to (an)isotropic surface energy. It has been proposed that this motion could be computed by solving the analogous evolution law using a "crystalline" approximation to the surface energy. We present the first convergence analysis for a numerical scheme of this type. Our treatment is restricted to one dimensional surfaces (curves in the plane) which are graphs. In this context, the scheme amounts to a new algorithm for solving quasilinear parabolic equations in one space dimension.

math.NA↗

arXiv2013

Blister patterns and energy minimization in compressed thin films on compliant substrates

This paper is motivated by the complex blister patterns sometimes seen in thin elastic films on thick, compliant substrates. These patterns are often induced by an elastic misfit which compresses the film. Blistering permits the film to expand locally, reducing the elastic energy of the system. It is natural to ask: what is the minimum elastic energy achievable by blistering on a fixed area fraction of the substrate? This is a variational problem involving both the {\it elastic deformation} of the film and substrate and the {\it geometry} of the blistered region. It involves three small parameters: the {\it nondimensionalized thickness} of the film, the {\it compliance ratio} of the film/substrate pair and the {\it mismatch strain}. In formulating the problem, we use a small-slope (Föppl-von Kármán) approximation for the elastic energy of the film, and a local approximation for the elastic energy of the substrate. For a 1D version of the problem, we obtain "matching" upper and lower bounds on the minimum energy, in the sense that both bounds have the same scaling behavior with respect to the small parameters. For a 2D version of the problem, our results are less complete. Our upper and lower bounds only "match" in their scaling with respect to the nondimensionalized thickness, not in the dependence on the compliance ratio and the mismatch strain. The upper bound considers a 2D lattice of blisters, and uses ideas from the literature on the folding or "crumpling" of a confined elastic sheet. Our main 2D result is that in a certain parameter regime, the elastic energy of this lattice is significantly lower than that of a few large blisters.

math.AP↗

arXiv2012

A variational perspective on cloaking by anomalous localized resonance

A body of literature has developed concerning "cloaking by anomalous localized resonance". The mathematical heart of the matter involves the behavior of a divergence-form elliptic equation in the plane, $\nabla\cdot (a(x)\nabla u(x)) = f(x)$. The complex-valued coefficient has a matrix-shell-core geometry, with real part equal to 1 in the matrix and the core, and -1 in the shell; one is interested in understanding the resonant behavior of the solution as the imaginary part of $a(x)$ decreases to zero (so that ellipticity is lost). Most analytical work in this area has relied on separation of variables, and has therefore been restricted to radial geometries. We introduce a new approach based on a pair of dual variational principles, and apply it to some non-radial examples. In our examples, as in the radial setting, the spatial location of the source $f$ plays a crucial role in determining whether or not resonance occurs.

math.AP↗

arXiv2012

Wrinkles as a relaxation of compressive stresses in an annular thin film

It is well known that an elastic sheet loaded in tension will wrinkle and that the length scale of the wrinkles tends to zero with vanishing thickness of the sheet [Cerda and Mahadevan, Phys. Rev. Lett. 90, 074302 (2003)]. We give the first mathematically rigorous analysis of such a problem. Since our methods require an explicit understanding of the underlying (convex) relaxed problem, we focus on the wrinkling of an annular sheet loaded in the radial direction [Davidovitch et al., PNAS 108 (2011), no. 45]. Our main achievement is identification of the scaling law of the minimum energy as the thickness of the sheet tends to zero. This requires proving an upper bound and a lower bound that scale the same way. We prove both bounds first in a simplified Kirchhoff-Love setting and then in the nonlinear three-dimensional setting. To obtain the optimal upper bound, we need to adjust a naive construction (one family of wrinkles superimposed on a planar deformation) by introducing a cascade of wrinkles. The lower bound is more subtle, since it must be ansatz-free.

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