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.

We consider Laplacian eigenfunctions on a domain S2 subset of Rd. 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 Rd, where among the first n eigenfunctions only similar to n1/d have a mean value different from zero. We prove that this rate is sharp in any smooth domain, up to a logarithmic factor: in any smooth domain S2, among the first n Dirichlet eigenfunctions at least (log n)-1/2 n1/d have a nonzero mean.

We prove optimal convergence rates for eigenvalues and eigenvectors of the graph Laplacian on Poisson point clouds. Our results are valid down to the critical percolation threshold, yielding error estimates for relatively sparse graphs.

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 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.

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.

We prove the existence of global minimizers to the double minimization problem where P(E) denotes the perimeter of the set E, Wp is the p-Wasserstein distance between Borel probability measures, and λ>0 is arbitrary. The result holds in all space dimensions, for all p∈[1,∞), and for all positive λ. This answers a question of Buttazzo, Carlier, and Laborde.

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.

We offer a replacement for [ 2 , Theorem 3.6], due to an error in the proof.

For a smooth bounded domain $$G\subset {{\mathbb {R}}}^3$$ , we consider maps $$n:{\mathbb {R}}^3\setminus G\rightarrow {\mathbb {S}}^2$$ 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 $$|x|\rightarrow \infty $$ . 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.

This chapter is a review of recent results on a variational model in the context of the gradient theory for fluid–fluid phase transitions with small-scale heterogeneities. We present a Γ-convergence result that identifies an anisotropic limiting surface energy and investigate some of its properties.

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.

We consider a homogenization problem associated with quasi-crystalline multiple integrals of the form \begin{equation*} \begin{aligned} u_\varepsilon\in L^p(\Omega;\mathbb{R}^d) \mapsto \int_\Omega f_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 $f_R$.

We prove a quantitative stability result for the Heisenberg-Pauli-Weyl inequality. This yields next and next-to-next order correction terms, sharpening the inequality in all dimensions.

We complete the analysis initiated in Dabade et al. (J Nonlinear Sci 21:415–460, 2018) 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 use a number of well-developed techniques in energy-driven pattern formation.

We carry out an asymptotic analysis of a variational problem relevant in the studies of nematic liquid crystalline films when one elastic constant dominates over the others, namely, $ \inf E_\varepsilon(u)$, where $E_\varepsilon(u) := \frac{1}{2}\int_\Omega \left\{\varepsilon\,|\nabla u|^2 + \frac{1}{\varepsilon} \,(|u|^2 - 1)^2 + L \,(\mathrm{div}\, u)^2\right\} \,dx. $ Here $u: \Omega \to \mathbb{R}^2$ is a vector field, $0 < \varepsilon \ll 1 $ is a small parameter, and $L > 0$ is a fixed constant, independent of $\varepsilon$. We identify a candidate for the $\Gamma$-limit $E_0$, which is a sum of a bulk term penalizing divergence and an Aviles--Giga-type wall energy involving the cube of the jump in the tangential component of the $\mathbb{S}^1$-valued nematic director. We establish the lower bound and provide the recovery sequence for this candidate within a restricted class. Then we consider a set of variational problems for $E_0$ arising from various choices of domain geometry and boundary conditions. We demonstrate that the criticality conditions for $E_0$ can be expressed as a pair of scalar conservation laws that share characteristics. We use the method of characteristics to analytically construct critical points of $E_0$ that we observe numerically.

We study the micromagnetics of soft cubic ferromagnets with large magnetostriction, with the goal of understanding the microstructure and behavior of recently reported single-crystal Galfenol samples [Chopra and Wuttig in Nature 521(7552):340–343, 2015]. First, taking the no-exchange formulation of the micromagnetics energy [De Simone and James in J Mech Phys Solids 50(2):283–320, 2002], we construct minimizing sequences that yield local average magnetization and strain curves matching the experimental findings of Chopra and Wuttig (2015). Then, reintroducing a sharp-interface version of the exchange energy [Choksi and Kohn in Commun Pure Appl Math 51(3):259–289, 1998], we construct normal and zig-zag Landau states; within the parameter regime of Galfenol, we show that the latter achieves lower-energy scaling via equipartition of energy between the \(90^\circ \) wall energy, \(180^\circ \) wall energy and the anisotropy energy. This forms the first step in adapting the program of Kohn and Müller [Philos Mag A 66(5):697–715, 1992] to explain why certain magnetic microstructures are observed over others.

Applying the Lyapunov–Schmidt reduction approach introduced by Mielke and Schneider in their analysis of the fourth-order scalar Swift–Hohenberg equation, we carry out a rigorous small-amplitude stability analysis of Turing patterns for the canonical second-order system of reaction–diffusion equations given by the Brusselator model. Our results confirm that stability is accurately predicted in the small-amplitude limit by the formal Ginzburg–Landau amplitude equations, rigorously validating the standard weakly unstable approximation and the Eckhaus criterion.

We prove the existence of non-constant time periodic vortex solutions to the Gross-Pitaevskii equations for small but \textit{fixed} $\varepsilon > 0.$ The vortices of these solutions follow periodic orbits to the point vortex system of ordinary differential equations \textit{for all time}. The construction uses two approaches-- constrained minimization techniques adapted from \cite{GS} and topological minimax techniques adapted from \cite{LinMinMax}, applied to a formulation of the problem within a rotational ansatz.