Anomalous localized resonance using a folded geometry in three dimensions

If a body of dielectric material is coated by a plasmonic structure of negative dielectric material with nonzero loss parameter, then cloaking by anomalous localized resonance (CALR) may occur as the loss parameter tends to zero. It was proved in other papers by authors that if the coated structure is circular (2D) and dielectric constant of the shell is a negative constant (with loss parameter), then CALR occurs, and if the coated structure is spherical (3D), then CALR does not occur. The aim of this paper is to show that the CALR takes place if the spherical coated structure has a specially designed anisotropic dielectric tensor. The anisotropic dielectric tensor is designed by unfolding …

MR3061470 (Review) 35R30 35J25 Ammari, Habib (F-ENS-DAM); Garnier, Josselin (F-PARIS7-PMA); Sølna, Knut (1-CA3) Partial data resolving power of conductivity imaging from boundary measurements. (English summary) SIAM J. Math. Anal. 45 (2013), no. 3, 1704–1722.

A Quantitative Analysis of Metrics on Rn with Almost Constant Positive Scalar Curvature, with Applications to Fast Diffusion Flows

We prove a quantitative structure theorem for metrics on $\mathbf{R}^n$ that are conformal to the flat metric, have almost constant positive scalar curvature, and cannot concentrate more than one bubble. As an application of our result, we show a quantitative rate of convergence in relative entropy for a fast diffusion equation in $\mathbf{R}^n$ related to the Yamabe flow.

Symmetry of minimizers with a level surface parallel to the boundary

Wave propagation in non rectilinear waveguides

We present a mathematical framework for studying the problem of electromagnetic wave propagation in a 2-D or 3-D optical waveguide (optical fiber). We will consider both the case of a rectilinear waveguide and the one of a waveguide presenting imperfections, with applications to phenomenons of physical interest. Numerical examples will be given.

Hölder stability for Serrin’s overdetermined problem

In a bounded domain \(\varOmega \), we consider a positive solution of the problem \(\Delta u+f(u)=0\) in \(\varOmega \), \(u=0\) on \(\partial \varOmega \), where \(f:\mathbb {R}\rightarrow \mathbb {R}\) is a locally Lipschitz continuous function. Under sufficient conditions on \(\varOmega \) (for instance, if \(\varOmega \) is convex), we show that \(\partial \varOmega \) is contained in a spherical annulus of radii \(r_i 0\) and \(\tau \in (0,1]\). Here, \([u_\nu ]_{\partial \varOmega }\) is the Lipschitz seminorm on \(\partial \varOmega \) of the normal derivative of u. This result improves to Holder stability the logarithmic estimate obtained in Aftalion et al. (Adv Differ Equ 4:907–93…

Some overdetermined problems related to the anisotropic capacity

Abstract We characterize the Wulff shape of an anisotropic norm in terms of solutions to overdetermined problems for the Finsler p-capacity of a convex set Ω ⊂ R N , with 1 p N . In particular we show that if the Finsler p-capacitary potential u associated to Ω has two homothetic level sets then Ω is Wulff shape. Moreover, we show that the concavity exponent of u is q = − ( p − 1 ) / ( N − p ) if and only if Ω is Wulff shape.

On Serrin’s overdetermined problem in space forms

We consider Serrin’s overdetermined problem for the equation $$\Delta v + nK v = -\,1$$ in space forms, where K is the curvature of the space, and we prove a symmetry result by using a P-function approach. Our approach generalizes the one introduced by Weinberger to space forms and, as in the Euclidean case, it provides a short proof of the symmetry result which does not make use of the method of moving planes.

Solutions of elliptic equations with a level surface parallel to the boundary: stability of the radial configuration

A positive solution of a homogeneous Dirichlet boundary value problem or initial-value problems for certain elliptic or parabolic equations must be radially symmetric and monotone in the radial direction if just one of its level surfaces is parallel to the boundary of the domain. Here, for the elliptic case, we prove the stability counterpart of that result. We show that if the solution is almost constant on a surface at a fixed distance from the boundary, then the domain is almost radially symmetric, in the sense that is contained in and contains two concentric balls $${B_{{r_e}}}$$ and $${B_{{r_i}}}$$ , with the difference r e -r i (linearly) controlled by a suitable norm of the deviation…

Wave Propagation in a 3-D Optical Waveguide

In this paper we study the problem of wave propagation in a 3-D optical fiber. The goal is to obtain a solution for the time-harmonic field caused by a source in a cylindrically symmetric waveguide. The geometry of the problem, corresponding to an open waveguide, makes the problem challenging. To solve it, we construct a transform theory which is a nontrivial generalization of a method for solving a 2-D version of this problem given by Magnanini and Santosa.\cite{MS} The extension to 3-D is made complicated by the fact that the resulting eigenvalue problem defining the transform kernel is singular both at the origin and at infinity. The singularities require the investigation of the behavio…

Wulff shape characterizations in overdetermined anisotropic elliptic problems

We study some overdetermined problems for possibly anisotropic degenerate elliptic PDEs, including the well-known Serrin's overdetermined problem, and we prove the corresponding Wulff shape characterizations by using some integral identities and just one pointwise inequality. Our techniques provide a somehow unified approach to this variety of problems.

Stress concentration for closely located inclusions in nonlinear perfect conductivity problems

We study the stress concentration, which is the gradient of the solution, when two smooth inclusions are closely located in a possibly anisotropic medium. The governing equation may be degenerate of $p-$Laplace type, with $1<p \leq N$. We prove optimal $L^\infty$ estimates for the blow-up of the gradient of the solution as the distance between the inclusions tends to zero.

Spectral theory of a Neumann-Poincaré-type operator and analysis of cloaking by anomalous localized resonance II

If a body of dielectric material is coated by a plasmonic structure of negative dielectric constant with nonzero loss parameter, then cloaking by anomalous localized resonance (CALR) may occur as the loss parameter tends to zero. The aim of this paper is to investigate this phenomenon in two and three dimensions when the coated structure is radial, and the core, shell and matrix are isotropic materials. In two dimensions, we show that if the real part of the permittivity of the shell is $-1$ (under the assumption that the permittivity of the background is $1$), then CALR takes place. If it is different from $-1$, then CALR does not occur. In three dimensions, we show that CALR does not occu…

An overdetermined problem for the anisotropic capacity

We consider an overdetermined problem for the Finsler Laplacian in the exterior of a convex domain in \({\mathbb {R}}^{N}\), establishing a symmetry result for the anisotropic capacitary potential. Our result extends the one of Reichel (Arch Ration Mech Anal 137(4):381–394, 1997), where the usual Newtonian capacity is considered, giving rise to an overdetermined problem for the standard Laplace equation. Here, we replace the usual Euclidean norm of the gradient with an arbitrary norm H. The resulting symmetry of the solution is that of the so-called Wulff shape (a ball in the dual norm \(H_0\)).

Symmetry of minimizers with a level surface parallel to the boundary

We consider the functional $$I_\Omega(v) = \int_\Omega [f(|Dv|) - v] dx,$$ where $\Omega$ is a bounded domain and $f$ is a convex function. Under general assumptions on $f$, G. Crasta [Cr1] has shown that if $I_\Omega$ admits a minimizer in $W_0^{1,1}(\Omega)$ depending only on the distance from the boundary of $\Omega$, then $\Omega$ must be a ball. With some restrictions on $f$, we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss how these results extend to more general settings, in particular to functionals that are not differenti…

A weak comparison principle for solutions of very degenerate elliptic equations

We prove a comparison principle for weak solutions of elliptic quasilinear equations in divergence form whose ellipticity constants degenerate at every point where \(\nabla u\in K\), where \(K\subset \mathbb{R }^N\) is a Borel set containing the origin.

Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvature

We prove that the boundary of a (not necessarily connected) bounded smooth set with constant nonlocal mean curvature is a sphere. More generally, and in contrast with what happens in the classical case, we show that the Lipschitz constant of the nonlocal mean curvature of such a boundary controls its $C^2$-distance from a single sphere. The corresponding stability inequality is obtained with a sharp decay rate.

MR3063539 (Review) 35R30 35J05 35P25 65N21 Bao, Gang [Bao, Gang1] (PRC-ZHJ); Lin, Junshan (1-MN-MA) Near-field imaging of the surface displacement on an infinite ground plane. (English summary) Inverse Probl. Imaging 7 (2013), no. 2, 377–396.

Helmholtz equation in unbounded domains: some convergence results for a constrained optimization problem

We consider a constrained optimization problem arising from the study of the Helmholtz equation in unbounded domains. The optimization problem provides an approximation of the solution in a bounded computational domain. In this paper we prove some estimates on the rate of convergence to the exact solution.

A note on an overdetermined problem for the capacitary potential

We consider an overdetermined problem arising in potential theory for the capacitary potential and we prove a radial symmetry result.

A Viscosity Equation for Minimizers of a Class of Very Degenerate Elliptic Functionals

We consider the functional $$J(v) = \int_\varOmega\bigl[f\bigl(|\nabla v|\bigr) - v\bigr] dx, $$ where Ω is a bounded domain and f:[0,+∞)→ℝ is a convex function vanishing for s∈[0,σ], with σ>0. We prove that a minimizer u of J satisfies an equation of the form $$\min\bigl(F\bigl(\nabla u, D^2 u\bigr), |\nabla u|-\sigma\bigr)=0 $$ in the viscosity sense.

A computational method for the Helmholtz equation in unbounded domains based on the minimization of an integral functional

Abstract We study a new approach to the problem of transparent boundary conditions for the Helmholtz equation in unbounded domains. Our approach is based on the minimization of an integral functional arising from a volume integral formulation of the radiation condition. The index of refraction does not need to be constant at infinity and may have some angular dependency as well as perturbations. We prove analytical results on the convergence of the approximate solution. Numerical examples for different shapes of the artificial boundary and for non-constant indexes of refraction will be presented.

A radiation condition for the 2-D Helmholtz equation in stratified media

We study the 2-D Helmholtz equation in perturbed stratified media, allowing the existence of guided waves. Our assumptions on the perturbing and source terms are not too restrictive. We prove two results. Firstly, we introduce a Sommerfeld-Rellich radiation condition and prove the uniqueness of the solution for the studied equation. Then, by careful asymptotic estimates, we prove the existence of a bounded solution satisfying our radiation condition.

A Remark on an Overdetermined Problem in Riemannian Geometry

Let (M, g) be a Riemannian manifold with a distinguished point O and assume that the geodesic distance d from O is an isoparametric function. Let \(\varOmega \subset M\) be a bounded domain, with \(O \in \varOmega \), and consider the problem \(\varDelta _p u = -1\ \mathrm{in}\ \varOmega \) with \(u=0\ \mathrm{on}\ \partial \varOmega \), where \(\varDelta _p\) is the p-Laplacian of g. We prove that if the normal derivative \(\partial _{\nu }u\) of u along the boundary of \(\varOmega \) is a function of d satisfying suitable conditions, then \(\varOmega \) must be a geodesic ball. In particular, our result applies to open balls of \(\mathbb {R}^n\) equipped with a rotationally symmetric metr…

Symmetry for positive critical points of Caffarelli–Kohn–Nirenberg inequalities

Abstract We consider positive critical points of Caffarelli–Kohn–Nirenberg inequalities and prove a Liouville type result which allows us to give a complete classification of the solutions in a certain range of parameters, providing a symmetry result for positive solutions. The governing operator is a weighted p -Laplace operator, which we consider for a general p ∈ ( 1 , d ) . For p = 2 , the symmetry breaking region for extremals of Caffarelli–Kohn–Nirenberg inequalities was completely characterized in Dolbeault et al. (2016). Our results extend this result to a general p and are optimal in some cases.

Spectral analysis of the Neumann-Poincaré operator and characterization of the stress concentration in anti-plane elasticity

When holes or hard elastic inclusions are closely located, stress which is the gradient of the solution to the anti-plane elasticity equation can be arbitrarily large as the distance between two inclusions tends to zero. It is important to precisely characterize the blow-up of the gradient of such an equation. In this paper we show that the blow-up of the gradient can be characterized by a singular function defined by the single layer potential of an eigenfunction corresponding to the eigenvalue 1/2 of a Neumann–Poincare type operator defined on the boundaries of the inclusions. By comparing the singular function with the one corresponding to two disks osculating to the inclusions, we quant…

A rigidity problem on the round sphere

We consider a class of overdetermined problems in rotationally symmetric spaces, which reduce to the classical Serrin's overdetermined problem in the case of the Euclidean space. We prove some general integral identities for rotationally symmetric spaces which imply a rigidity result in the case of the round sphere.

MR2954311 (Review) 35R30 35J05 76Nxx Elschner, J. (D-WIAS); Hsiao, G. C. [Hsiao, George C.] (1-DE); Rathsfeld, A. [Rathsfeld, Andreas] (D-WIAS) Reconstruction of elastic obstacles from the far-field data of scattered acoustic waves. (English, Georgian summaries) Mem. Differential Equations Math. Phys. 53 (2011), 63–97.

A RADIATION CONDITION FOR UNIQUENESS IN A WAVE PROPAGATION PROBLEM FOR 2-D OPEN WAVEGUIDES

We study the uniqueness of solutions of Helmholtz equation for a problem that concerns wave propagation in waveguides. The classical radiation condition does not apply to our problem because the inhomogeneity of the index of refraction extends to infinity in one direction. Also, because of the presence of a waveguide, some waves propagate in one direction with different propagation constants and without decaying in amplitude. Our main result provides an explicit condition for uniqueness which takes into account the physically significant components, corresponding to guided and non-guided waves; this condition reduces to the classical Sommerfeld-Rellich condition in the relevant cases. Final…

Gradient estimates for the perfect conductivity problem in anisotropic media

Abstract We study the perfect conductivity problem when two perfectly conducting inclusions are closely located to each other in an anisotropic background medium. We establish optimal upper and lower gradient bounds for the solution in any dimension which characterize the singular behavior of the electric field as the distance between the inclusions goes to zero.

Classification and non-existence results for weak solutions to quasilinear elliptic equations with Neumann or Robin boundary conditions

Abstract We classify positive solutions to a class of quasilinear equations with Neumann or Robin boundary conditions in convex domains. Our main tool is an integral formula involving the trace of some relevant quantities for the problem. Under a suitable condition on the nonlinearity, a relevant consequence of our results is that we can extend to weak solutions a celebrated result obtained for stable solutions by Casten and Holland and by Matano.

Spectral theory of a Neumann-Poincare-type operator and analysis of cloaking due to anomalous localized resonance

The aim of this paper is to give a mathematical justification of cloaking due to anomalous localized resonance (CALR). We consider the dielectric problem with a source term in a structure with a layer of plasmonic material. Using layer potentials and symmetrization techniques, we give a necessary and sufficient condition on the fixed source term for electromagnetic power dissipation to blow up as the loss parameter of the plasmonic material goes to zero. This condition is written in terms of the Newtonian potential of the source term. In the case of concentric disks, we make the condition even more explicit. Using the condition, we are able to show that for any source supported outside a cr…

Analytical results for 2-D non-rectilinear waveguides based on a Green's function

We consider the problem of wave propagation for a 2-D rectilinear optical waveguide which presents some perturbation. We construct a mathematical framework to study such a problem and prove the existence of a solution for the case of small imperfections. Our results are based on the knowledge of a Green's function for the rectilinear case.

A method of variation of boundaries for waveguide grating couplers

We describe a method for calculating the solution of the electromagnetic field in a non-rectilinear open waveguide by using a series expansion, starting from the field of a rectilinear waveguide. Our approach is based on a method of variation of boundaries. We prove that the obtained series expansion converges and we provide a radiation condition at infinity in such a way that the problem has a unique solution. Our approach can model several kinds of optical devices which are used in optical integrated circuits. Numerical examples will be shown for the case of finite aperiodic waveguide grating couplers.

Cloaking by anomalous localized resonance via spectral analysis of a Neumann-Poincaré operator

The classical notion of Neumann-Poincar´e (NP) operator appears naturally when we attempt to solve Dirichlet or Neumann boundary value problems using layer potentials. In this talk, we will review some properties of NP-operators and show that they can be used to give a mathematical analysis of cloaking by anomalous localized resonance (CALR). More precisely, we shall discuss the following problem. If a body of dielectric material is coated by a plasmonic structure of negative dielectric constant with nonzero loss parameter, then CALR may occur as the loss parameter tends to zero. Anomalous localized resonance is the phenomenon of field blow-up in a localized region and it may (and may not) …

Wave propagation in a 3-D optical waveguide II. Numerical results

A sharp quantitative version of Alexandrov's theorem via the method of moving planes

We prove the following quantitative version of the celebrated Soap Bubble Theorem of Alexandrov. Let $S$ be a $C^2$ closed embedded hypersurface of $\mathbb{R}^{n+1}$, $n\geq1$, and denote by $osc(H)$ the oscillation of its mean curvature. We prove that there exists a positive $\varepsilon$, depending on $n$ and upper bounds on the area and the $C^2$-regularity of $S$, such that if $osc(H) \leq \varepsilon$ then there exist two concentric balls $B_{r_i}$ and $B_{r_e}$ such that $S \subset \overline{B}_{r_e} \setminus B_{r_i}$ and $r_e -r_i \leq C \, osc(H)$, with $C$ depending only on $n$ and upper bounds on the surface area of $S$ and the $C^2$ regularity of $S$. Our approach is based on a…

The method of moving planes: a quantitative approach

We review classical results where the method of the moving planes has been used to prove symmetry properties for overdetermined PDE's boundary value problems (such as Serrin's overdetermined problem) and for rigidity problems in geometric analysis (like Alexandrov soap bubble Theorem), and we give an overview of some recent results related to quantitative studies of the method of moving planes, where quantitative approximate symmetry results are obtained.

On the shape of compact hypersurfaces with almost constant mean curvature

The distance of an almost constant mean curvature boundary from a finite family of disjoint tangent balls with equal radii is quantitatively controlled in terms of the oscillation of the scalar mean curvature. This result allows one to quantitatively describe the geometry of volume-constrained stationary sets in capillarity problems.

MR3007263 (Review) 78A40 82D10 Burlak, Gennadiy (MEX-UAEM-CAP); Rabinovich, Vladimir [Rabinovich, Vladimir S.] Time-frequency integrals and the stationary phase method in problems of waves propagation from moving sources. (English summary) SIGMA Symmetry Integrability Geom. Methods Appl. 8 (2012), Paper 096, 21 pp.

A spectral approach to a constrained optimization problem for the Helmholtz equation in unbounded domains

We study some convergence issues for a recent approach to the problem of transparent boundary conditions for the Helmholtz equation in unbounded domains (Ciraolo et al. in J Comput Phys 246:78–95, 2013) where the index of refraction is not required to be constant at infinity. The approach is based on the minimization of an integral functional, which arises from an integral formulation of the radiation condition at infinity. In this paper, we implement a Fourier–Chebyshev collocation method to study some convergence properties of the numerical algorithm; in particular, we give numerical evidence of some convergence estimates available in the literature (Ciraolo in Helmholtz equation in unbou…

MR3194685 (Review) Canuto, B. Stability results for the $N$-dimensional Schiffer conjecture via a perturbation method. Calc. Var. Partial Differential Equations 50 (2014), no. 1-2, 305–334.

Stability results for solutions of elliptic equations with a level surface parallel to the boundary

A note on Serrin's overdetermined problem

We consider the solution of the torsion problem $$−Δu = N \quad\mathrm{in}\quad Ω,\quad u = 0\quad\mathrm{on}\quad ∂Ω,$$ where Ω is a bounded domain in RN. ¶ Serrin's celebrated symmetry theorem states that, if the normal derivative uν is constant on ∂Ω, then Ω must be a ball. In [6], it has been conjectured that Serrin's theorem may be obtained by stability in the following way: first, for the solution u of the torsion problem prove the estimate $$r_e − r_i ≤ C_t\Bigl(\max_{\Gamma_t} u-\min_{\Gamma_t} u\Bigr)$$ for some constant Ct depending on t, where re and ri are the radii of an annulus containing ∂Ω and Γt is a surface parallel to ∂Ω at distance t and sufficiently close to ∂Ω secondly…