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→ 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.

We study the sense in which the continuum limit of a broad class of discrete materials with periodic structure 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.

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.

Motivated by the physics literature on "photonic doping" of scatterers made from "epsilon-near-zero" (ENZ) materials, we consider how the scattering of time-harmonic TM electromagnetic waves by a cylindrical ENZ region & omega;xR$\Omega \times \mathbb {R}$ is affected by the presence of a "dopant" D & SUB;& omega;$D \subset \Omega$ in which the dielectric permittivity is not near zero. Mathematically, this reduces to analysis of a 2D Helmholtz equation div(a(x) backward difference u)+k2u=f$\mathrm{div}\, (a(x)\nabla u) + k<^>2 u = f$ with a piecewise-constant, complex valued coefficient a that is nearly infinite (say a=1 & delta;$a = \frac{1}{\delta }$ with & delta;& AP;0$\delta \approx 0$) in & omega; set minus D over bar $\Omega \setminus \overline{D}$. We show (under suitable hypotheses) that the solution u depends analytically on & delta; 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 & delta; 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 the dielectric permittivity is merely small. Equally important: our results include a PDE characterization of the leading-order electric field in the ENZ region as & delta;& RARR;0$\delta \rightarrow 0$, whereas the existing literature on photonic doping provides only the leading-order magnetic field.

We study certain “geometric-invariant resonant cavities” introduced by Liberal, Mahmoud, and Engheta in a 2016 Nature Communications paper. They are cylindrical devices modeled using the transverse magnetic reduction of Maxwell’s equations, so the mathematics is two-dimensional. The cross-section consists of a dielectric inclusion surrounded by an “epsilon-near-zero” (ENZ) shell. When the shell has just 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 ∇·( ε ^-1(x,ω ) ∇ u ) + ω ^2 μ u = 0 , where ε (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 ε depends on ω . Use of an ENZ material in the shell means that ε (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 parameter of the Lorentz model is sufficiently small. Our analysis is perturbative in character, using the implicit function theorem despite the apparently singular form of the PDE. While the existence of the resonance depends only on the area of the ENZ shell, its quality (that is, the rate at which the resonance decays) depends on the shape of the shell. It is therefore natural to 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. Finally, we discuss the conjecture that our relaxed problem amounts to considering homogenization-like limits of nearly optimal designs.

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.

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 ≤ 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. © 2022 Wiley Periodicals LLC.

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.

In materials that undergo martensitic phase transformation, macroscopic loading often leads to the creation and/or rearrangement of elastic domains. This paper considers an example involving a single-crystal slab made from two martensite variants. When the slab is made to bend, the two variants form a characteristic microstructure that we like to call “twinning with variable volume fraction.” Two 1996 papers by Chopra et. al. explored this example using bars made from InTl, providing considerable detail about the microstructures they observed. Here we offer an energy-minimization-based model that is motivated by their account. It uses geometrically linear elasticity, and treats the phase boundaries as sharp interfaces. For simplicity, rather than model the experimental forces and boundary conditions exactly, we consider certain Dirichlet or Neumann boundary conditions whose effect is to require bending. This leads to certain nonlinear (and nonconvex) variational problems that represent the minimization of elastic plus surface energy (and the work done by the load, in the case of a Neumann boundary condition). Our results identify how the minimum value of each variational problem scales with respect to the surface energy density. The results are established by proving upper and lower bounds that scale the same way. The upper bounds are ansatz-based, providing full details about some (nearly) optimal microstructures. The lower bounds are ansatz-free, so they explain why no other arrangement of the two phases could be significantly better.

In materials that undergo martensitic phase transformation, distinct elastic phases often form layered microstructures — a phenomenon known as twinning. In some settings the volume fractions of the phases vary macroscopically; this has been seen, in particular, in experiments involving the bending of a bar. We study a two-dimensional (2D) model problem of this type, involving two geometrically nonlinear phases with a single rank-one connection. We adopt a variational perspective, focusing on the minimization of elastic plus surface energy. To get started, we show that twinning with variable volume fraction must occur when bending is imposed by a Dirichlet-type boundary condition. We then turn to paper’s main goal, which is to determine how the minimum energy scales with respect to the surface energy density and the transformation strain. Our analysis combines ansatz-based upper bounds with ansatz-free lower bounds. For the upper bounds we consider two very different candidates for the microstructure: one that involves self-similar refinement of its length scale near the boundary, and another based on piecewise-linear approximation with a single length scale. Our lower bounds adapt methods previously introduced by Chan and Conti to address a problem involving twinning with constant volume fraction. The energy minimization problem considered in this paper is not intended to model twinning with variable volume fraction involving two martensite variants; rather, it provides a convenient starting point for the development of a mathematical toolkit for the study of twinning with variable volume fraction.

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.

Calculus of Variations touches several interrelated areas. In this workshop we covered several topics, such as minimal submanifolds, mean curvature and related flows, free boundary problems, variational models of interacting dislocations, defects in physical systems, phase transitions, etc.

Crystal growth is inhibited by the presence of impurities. Cabrera and Vermilyea introduced a model in 1958, in which the impurities are modeled as immobile stoppers. The quantitative consequences of this model have mainly been explored for the special case where the stoppers are immobile and arranged in a periodic array. Here we use numerical simulation to explore what happens when the stopper locations are randomly distributed and the stoppers have finite lifetimes. As this problem has just two nondimensional parameters, namely, nondimensionalized versions of the mean stopper distance and the mean stopper lifetime, we are able to explore a large region of the parameter space using simulation. The stopper density is measured by the percolation parameter, a nondimensionalized inverse distance between stoppers, zeta. Our results show that when the stopper density is relatively small (zeta below about 0.8), the macroscopic velocity of the step is roughly the same for randomly located stoppers as for a periodic array of stoppers. Moreover, in this regime the average velocity is almost independent of the stopper lifetime. For large stopper densities (more precisely, when the percolation parameter zeta is above about 0.8), the situation is entirely different. For periodically placed immobile stoppers, the average velocity drops sharply to 0 at zeta = 1. For randomly located immobile stoppers, by contrast, the average velocity remains positive for zeta well above 1, and it approaches 0 gradually rather than abruptly. For randomly located stoppers with finite lifetimes, the average velocity has a nonzero asymptote for large zeta; thus, for large stopper densities, the average velocity depends mainly on the mean stopper lifetime. In this regime, the inhibition kinetics predicted by our model resemble those of the Bliznakov kink blocking mechanism.

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.

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 et al. (in: Proceedings of the twenty-seventh annual ACM-SIAM symposium on discrete algorithms, SODA ’16, (Philadelphia, PA, USA), Society for Industrial and Applied Mathematics, 2016). Our techniques are similar to those of Kohn and Serfaty (Commun Pure Appl Math 63(10):1298–1350, 2010), where scaling limits of some two-person games led to elliptic or parabolic PDEs.

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.

Mathematics is the language of science, and partial differential equations are a crucial component: they provide the language we use to describe—and the tools we use to understand—phenomena in many areas including geometry, engineering, and physics.

The Calculus of Variations is at once a classical subject, and a very modern one. Its scope encompasses a broad range of topics in geometric analysis, and deep questions about PDE. New frontiers are constantly emerging, where problems from mechanics, physics, and other applications introduce new challenges. The 2018 Calculus of Variations workshop reflected this breadth and diversity.

We derive a one-dimensional variational problem representing the elastic energy of a rod with misfit, starting from a nonlinear, three-dimensional elastic energy with nontrivial preferred strain. Our approach to dimension reduction is to find a Gamma-limit as the thickness of the rod tends to 0. The limiting energy is a quadratic function of the rates at which the rod bends and twists, and we give explicit expressions for the preferred curvature and twist in the special case of isotropic elastic moduli.

Recent experiments by Chopin and Kudrolli (Phys Rev Lett 111:174302, 2013) showed that a thin elastic ribbon, when twisted into a helicoid, may wrinkle in the center. We study this from the perspective of elastic energy minimization, building on recent work by Chopin et al. (J Elast 119(1–2):137–189, 2015) in which they derive a modified von Kármán functional and solve the relaxed problem. Our main contribution is to show matching upper and lower bounds for the minimum energy in the small-thickness limit. Along the way, we show that the displacements must be small where we expect that the ribbon is helicoidal, and we estimate the wavelength of the wrinkles.