Perfectly Matched Layers (PML) has become a very common method for the numerical approximation of wave and wave-like equations on unbounded domains. This technique allows one to obtain accurate solutions while working on a finite computational domain, and the technique is relatively simple to implement. Results concerning the accuracy of the PML method have been obtained, but mostly with regard to problems at a fixed frequency. In this paper we provide very explicit time-domain bounds on the accuracy of PML for the inhomogeneous two-dimensional wave equation with a particular type of forcing term, and illustrate our conclusions with some numerical examples.

We study scattering for the linear Helmholtz operator in two dimensions and develop a technique, which can be used to ascertain scattering of a given incident wave from very regular inhomogeneities. This technique is then applied to a number of interesting examples.

We formulate a problem that can be viewed as a natural variation of the so-called Pompeiu or Schiffer problem in the context of scattering of plane waves for the Linear Helmholtz equation. For the two dimensional version of this variation, we establish conditions on the wave numbers and incident directions, that ensure a non-vanishing scattered field.

We study non-scattering phenomena associated with the time-harmonic Helmholtz equation in two dimensions. For very general classes of star-shaped domains, we show that there are at most finitely many wavenumbers such that Herglotz incident waves with a fixed density function are non-scattering.

This paper concerns the scattering problem for a nonlinear medium of compact support, $D$, with second-harmonic generation. Such a medium, when probed with monochromatic light beams at frequency $\omega$, generates additional waves at frequency $2\omega$. The response of the medium is governed by a system of two coupled semilinear partial differential equations for the electric fields at frequency $\omega$ and $2\omega$. We investigate whether there are situations in which the generated $2\omega$ wave is localized inside $D$, that is, the nonlinear interaction of the medium with the probing wave is invisible to an outside observer. This leads to the analysis of a semilinear elliptic system formulated in $D$ with non-standard boundary conditions. The analysis presented here sets up a mathematical framework needed to investigate a multitude of questions related to nonlinear scattering with second-harmonic generation.

This paper concerns the analysis of a passive, broadband approximate cloaking scheme for the Helmholtz equation in R-d for d = 2 or d = 3. Using ideas from transformation optics, we construct an approximate cloak by "blowing up" a small ball of radius E > 0 to one of radius 1. In the anisotropic cloaking layer resulting from the "blow-up" change of variables, we incorporate a Drude-Lorentz-type model for the index of refraction, and we assume that the cloaked object is a soft (perfectly conducting) obstacle. We first show that (for any fixed E) there are no real transmission eigenvalues associated with the inhomogeneity representing the cloak, which implies that the cloaking devices we have created will not yield perfect cloaking at any frequency, even for a single incident time harmonic wave. Secondly, we establish estimates on the scattered field due to an arbitrary time harmonic incident wave. These estimates show that, as E approaches 0, the L2-norm of the scattered field outside the cloak, and its far field pattern, approach 0 uniformly over any bounded band of frequencies. In other words: our scheme leads to broadband approximate cloaking for arbitrary incident time harmonic waves. (c) 2023 Published by Elsevier Masson SAS.

We derive exact reconstruction methods for cracks consisting of unions of Lipschitz hypersurfaces in the context of Calder\'on's inverse conductivity problem. Our first method obtains upper bounds for the unknown cracks, bounds that can be shrunk to obtain the exact crack locations upon verifying certain operator inequalities for differences of the local Neumann -to -Dirichlet maps. This method can simultaneously handle perfectly insulating and perfectly conducting cracks, and it appears to be the first rigorous reconstruction method capable of this. Our second method assumes that only perfectly insulating cracks or only perfectly conducting cracks are present. Once more using operator inequalities, this method generates approximate cracks that are guaranteed to be subsets of the unknown cracks that are being reconstructed.

In this paper we examine necessary conditions for an anisotropic inhomogeneous medium to be non-scattering at a single wave number and for a single incident field. These conditions are expressed in terms of the regularity of the boundary of the inhomogeneity. We assume that the coefficients, characterizing the constitutive material properties of the medium, are sufficiently smooth, and the incident wave is appropriately non-degenerate. Our analysis utilizes the Hodograph transform as well as regularity results for nonlinear elliptic partial differential equations. Our approach requires that the boundary a-priori is of class $C^{1,α}$ for some $0<α<1$.

In this paper we examine necessary conditions for an inhomogeneity to be non-scattering, or equivalently, by negation, sufficient conditions for it to be scattering. These conditions are formulated in terms of the regularity of the boundary of the inhomogeneity. We examine broad classes of incident waves in both two and three dimensions. Our analysis is greatly influenced by the analysis carried out by Williams [28] in order to establish that a domain, which does not possess the Pompeiu Property, has a real analytic boundary. That analysis, as well as ours, relies crucially on classical free boundary regularity results due to Kinderlehrer and Nirenberg [18], and Caffarelli [6].

In this article, we study the impact of a change in the type of boundary conditions of an elliptic boundary value problem. In the context of the conductivity equation we consider a reference problem with mixed homogeneous Dirichlet and Neumann boundary conditions. Two different perturbed versions of this “background” situation are investigated, when (i) The homogeneous Neumann boundary condition is replaced by a homogeneous Dirichlet boundary condition on a “small” subset ωε of the Neumann boundary; and when (ii) The homogeneous Dirichlet boundary condition is replaced by a homogeneous Neumann boundary condition on a “small” subset ωε of the Dirichlet boundary. The relevant quantity that measures the “smallness” of the subset ωε differs in the two cases: while it is the harmonic capacity of ωε in the former case, we introduce a notion of “Neumann capacity” to handle the latter. In the first part of this work we derive representation formulas that catch the structure of the first non trivial term in the asymptotic expansion of the voltage potential, for a general ωε, under the sole assumption that it is “small” in the appropriate sense. In the second part, we explicitly calculate the first non trivial term in the asymptotic expansion of the voltage potential, in the particular geometric situation where the subset ωε is a vanishing surfacic ball.

A central ingredient of cloaking-by-mapping is the diffeomorphisn which transforms an annulus with a small hole into an annulus with a finite size hole, while being the identity on the outer boundary of the annulus. The resulting meta-material is anisotropic, which makes it difficult to manufacture. The problem of minimizing anisotropy among radial transformations has been studied in [4]. In this work, as in [4], we formulate the problem of minimizing anisotropy as an energy minimization problem. Our main goal is to provide strong evidence for the conjecture that for cloaks with circular boundaries, non-radial transformations do not lead to lower degree of anisotropy. In the final section, we consider cloaks with non-circular boundaries and show that in this case, non-radial cloaks may be advantageous, when it comes to minimizing anisotropy.

In this paper we introduce an approach to establish finiteness results for the set of wave numbers that may lead to vanishing scattering effects. We use this approach to establish two results concerning the two dimensional Helmholtz equation in the context of a penetrable obstacle and (1) incident plane waves as well as (2) incident Herglotz waves. For a smooth, strictly convex, bounded domain, we show that there are at most finitely many positive wave numbers at which a plane wave with a fixed incident direction is nonscattering. For a disk there exist densities such that the corresponding incident Herglotz waves are nonscattering for infinitely many positive wave numbers. Here we show that any small perturbation of the disk to a proper ellipse will lead to at most finitely many such wave numbers.

The lectures about small electromagnetic inhomogeneities will cover internal inhomogeneities (see for example [1],[9],[10] and [11]) as well as more recent material about ”boundary” inhomogeneities [7]. I shall derive the first non-trivial term in an asymptotic (Rayleigh) expansion and contrast that with the small amplitude asymptotic expansion (Born approximation). Using such asymptotic formulas, I will discuss numerical methods to approximately determine the total volume of the inhomogeneities as well as their individual locations, based on electrostatic boundary measurements [8], [10]. In the second lecture I shall discuss the issue of uniformity of the asymptotic formulas, and some consequences of such uniformity [12],[14]. At this point I shall introduce so-called cloaking by mapping techniques (transformation optics) [5],[13]. In particular I will show how uniform estimates enable one to demonstrate the viability of cloaking-bymapping schemes (electromagnetic invisibility shields).

Asymptotic approximations of voltage potentials in the presence of diametrically small inhomogeneities are well studied. In particular it is known that one may construct approximations that are accurate to any order (in the diameter) uniformly in the conductivity of the inhomogeneity. The correspon ding problem for thin inhomogeneities is not so well understood, in particular as concerns uniformity of the approximations. If the conductivity degenerates to 0 or goes to infinity as the width of the inhomogeneity goes to zero, the voltage potential may converge to different limiting solutions, and so the construction of uniform approximations is not straightforward. For the case of thin two dimensional inhomogeneities with closed mid-curves such approximations were constructed and rigorously verified in (Chinese Annals of Mathematics, Series B 38 (2017) 293–344). The analysis relied heavily on the regularity of the approximate solutions. In this two part paper we continue this line of research, by showing that the same approximations remain valid, even when the mid-curve is open, and the corresponding approximate solutions have singularities at the endpoints of the curve.

In this second part of a two part paper, we establish a local, uniform energy-approximation estimate for the solutions to a simplified model of thin inhomogeneities with open mid-curves. This local result plays a crucial role in the proof of the global, uniform approximation results established in the first part of this paper (Asymptotic Analysis (2019)). For more details about the model we also refer the reader to (Chinese Annals of Mathematics, Series B 38 (2017) 293–344).

This is a survey on approximate cloaking using transformation optics on acoustic and electromagnetic waves. Both the time-harmonic and the time regimes are discussed.

Asymptotic expansions of the voltage potential in terms of the “radius” of a diametrically small (or several diametrically small) material inhomogeneity(ies) are by now quite well-known. Such asymptotic expansions for diametrically small inhomogeneities are uniform with respect to the conductivity of the inhomogeneities. In contrast, thin inhomogeneities, whose limit set is a smooth, codimension 1 manifold, σ, are examples of inhomogeneities for which the convergence to the background potential, or the standard expansion cannot be valid uniformly with respect to the conductivity, a , of the inhomogeneity. Indeed, by taking a close to 0 or to infinity, one obtains either a nearly homogeneous Neumann condition or nearly constant Dirichlet condition at the boundary of the inhomogeneity, and this difference in boundary condition is retained in the limit. The purpose of this paper is to find a “simple” replacement for the background potential, with the following properties: (1) This replacement may be (simply) calculated from the limiting domain Ωσ, the boundary data on the boundary of Ω, and the right-hand side. (2) This replacement depends on the thickness of the inhomogeneity and the conductivity, a , through its boundary conditions on σ. (3) The difference between this replacement and the true voltage potential converges to 0 uniformly in a , as the inhomogeneity thickness tends to 0.

These institutes are important components of the US mathematical sciences infrastructure.The institutes serve as national resources for advancing research, increasing the impact of the mathematical sciences, engaging with scientific opportunities in other fields, enabling the mathematical sciences to respond to national needs, and expanding the US talent base engaged in mathematical and statistical research.Recently the NSF has decided to

Cascades dans la dynamique des feuilletages mesurésNous étudions le comportement des feuilletages mesurés harmoniques sur les surfaces de Riemann compactes. Quand les périodes relatives varient, on peut observer des cascades dans la dynamique d'un tel feuilletage. Dans le cas du genre 2, on montre que le lieu de bifurcation résultant d'une telle variation est un sous-ensemble dénombrable et fermé de R, qui se plonge dans^.