Abstract : The Neumann—Poincaré operator （abbreviated by NP） is a boundary integral linear operator known as one of the important tools associated with boundary value problems in the field of partial differential equations. The detailed properties of NP operators can be comprehended as governing dynamics of many physical systems. Especially, the NP spectrum controls some physical systems （Electro dynamics, elastic systems and etc.）.
Our purpose here is to discuss the spectral structure of NP operators and their applications to physical systems.
N.B. We would like to hold 4 sessions at this minisymposium.
00268 (1/4) : 1E @G701 [Chair: Yoshihisa Miyanishi]
[03110] Homogenization and the spectrum of the Neumann Poincaré operator
Format : Talk at Waseda University
Author(s) :
Eric Bonnetier (Institut Fourier, Université Grenoble Alpes)
Abstract : Resonances of metallic nano-particles has been an active topic of investigation in the last decade, as this phenomenon allows localization of strong electro-magnetic fields in very small regions of space, an interesting feature for many applications. Asymptotically as the size of the particles tends to 0, the resonant frequencies are related to the spectral properties of the Neumann-Poincaré operator (NP).
In this talk, we discuss the spectrum of that integral operator, when one considers a periodic distribution of inclusions made of metamaterials in a dielectric background medium. The underlying question, is what can happen when many resonant particles interact ?
We show that under the assumptions that the inclusions are fully embedded in the periodicity cells, the spectra $\sigma_\varepsilon$ of the NP operators for a collection of period $\varepsilon$ converge to a limiting set composed of 2 parts : the union of the Bloch spectra of NP operators defined over periodicity cells with quasi-periodic boundary conditions and a boundary spectrum associated with eigenfunctions which spend a not too small part of their energy near
the boundary.
This is joint work with Charles Dapogny and Faouzi Triki.
[00406] From condensed matter theory to sub-wavelength physics
Format : Talk at Waseda University
Author(s) :
Habib Ammari (ETH Zurich)
Abstract : The ability to manipulate and control waves at scales much smaller than their wavelengths is revolutionizing nanotechnology. The speaker will present a mathematical framework for this emerging field of physics and elucidate its duality with condensed matter theory.
[00777] Eigenvalues of zero order pseudodifferential operators and applications to Neumann-Poincare
Format : Online Talk on Zoom
Author(s) :
Grigori Rozenblioum (Chalmers University of Technology)
Abstract : The NP operator for 3D elasticity is a zero order pseudodifferential operator, polynomially compact for a homogeneous material . For such operators we study behavior of eigenvalues converging to the points of essential spectrum and find their relation with the geometry of the body. For a nonhomogeneous material the essential spectrum fills intervals, we study eigenvalues converging to the tips of the essential spectrum.
[00564] On a uniqueness property of harmonic functions
Format : Online Talk on Zoom
Author(s) :
Dmitry Khavinson (University of South Florida)
Abstract : This is not a new result, yet the paper was dedicated to Walter Hayman and the main question , raised there is still unanswered. The paper was the joint work with late Harold S. Shapiro. We shall discuss the problem of uniqueness for functions u harmonic in a domain G in R^n and vanishing on some parts of the intersection V {not necessarily connected} of G with a line m. The question originated more than two decades ago with N. Nadirashvili {private communication}. For example, let G be a spherical shell, i.e., the region between two concentric spheres, and m is a line through the origin. Does u vanish on both segments along which m intersects G if it does so on one of them? To illustrate the cunning depth of the question note that if you let G to be the annulus with a sector cut out, the function u= arg z in the plane does vanish on the positive part of the real axis, but not on the whole intersection. What happens if G is a spherical shell but m does NOT pass through the center? What if we replace harmonic functions by polyharmonic functions, or, more generally, solutions of analytic elliptic equations, or even worse, by linear combinations of Riesz potentials that satisfy no PDE altogether? The answers are by no means obvious and, in many cases, may be judged as surprising.
[00367] Factorization of Neumann-Poincare operator
Format : Talk at Waseda University
Author(s) :
Mihai Putinar (University of California)
Abstract : It is well known that the Neumann-Poincare operator (double layer potential) is symmetrizable.
We will discuss a factorization of this singular integral operator which explains this essential spectral feature.
[00452] The quasi-static plasmonic problem for polyhedra
Format : Talk at Waseda University
Author(s) :
Karl-Mikael Perfekt (NTNU)
Abstract : I will present a characterization of the essential spectrum of the plasmonic problem for polyhedra in $\mathbb{R}^3$. The description is particularly simple for convex polyhedra and relative permittivities $\varepsilon < -1$. The results are obtained through detailed analysis of the double layer potential for polyhedral cones and polyhedra.
Based on joint work with Marta de León-Contreras.
[00364] Surface localized resonances and applications
Format : Talk at Waseda University
Author(s) :
Hongyu Liu (City University of Hong Kong)
Abstract : In this talk, I shall discuss our recent discoveries on certain novel surface localized resonances which were inspired by the surface plasmon resonance. These localized resonances generate a variety of interesting applications.
[01244] Spectral structure of the Neumann-Operator on thin domains
Format : Talk at Waseda University
Author(s) :
Hyeonbae Kang (Inha University)
Abstract : The Neumann-Poincare operator on thin domains, such as thin rectangles, thin prolate spheroids, flat oblate ellipsoids, exhibits interesting spectral structure. In this talk we review recent development on this topic.
[00730] A unified approach to the field concentration problem
Format : Talk at Waseda University
Author(s) :
Sanghyeon Yu (Korea University)
Abstract : Composite materials shows the high field concentration when the inclusions have geometric singularities in their boundaries. This phenomenon has many practical applications in imaging, spectroscopy, and meta-materials. In this talk, we discuss a new way of tackling the field concentration problem via the spectral analysis of the Neumann-Poincare operator. We focus on two kinds of important singularities: nearly touching surfaces and high curvature points.
00268 (4/4) : 2E @G701 [Chair: Yoshihisa Miyanishi]
[00354] Spectral properties of the Neumann–Poincaré operator on rotationally symmetric domains
Format : Talk at Waseda University
Author(s) :
Yong-Gwan Ji (Korea Institute for Advanced Study)
Hyeonbae Kang (Inha University)
Abstract : In this talk, we will discuss the spectral properties of the Neumann-Poincaré operator when domains have rotational symmetry. We prove that if a domain $\Omega$, in two dimensions, has rotational symmetry then NP spectrum on $\Omega$ contains NP spectrum on $D$ which generates rotationally symmetric domain $\Omega$ by $m$-th root transformation.
[01245] Vector field decomposition and eigenvalues of elastic Neumann-Poincaré operators
Format : Talk at Waseda University
Author(s) :
Shota Fukushima (Inha University)
Yong-Gwan Ji (Korea Institute for Advanced Study)
Hyeonbae Kang (Inha University)
Abstract : We show that all vector fields restricted to a surface is decomposed into three components and each component is characterized by the divergence-free or rotation-free harmonic extension to inside or outside of the domain. These three components correspond to three accumulation points of the eigenvalues of the elastic Neumann-Poincaré operator, which is a singular integral operator on the boundary.
[01265] Essential spectrum of elastic Neumann-Poincar\'e operators with a corner
Format : Talk at Waseda University
Author(s) :
Daisuke Kawagoe (Kyoto University)
Abstract : The elastic Neumann--Poincar\'e operator is a boundary integral operator naturally appearing when we solve the Lam\’e system in a bounded domain. For the two-dimensional case, if the boundary is smooth, then its spectrum consists of two sequences of eigenvalues with two accumulation points. In this talk, we consider the situation where the planar domain is smooth except at a corner and show that the essential spectrum appears around the above accumulation points.
[00662] Fundamental solutions in Colombeau algebra
Format : Talk at Waseda University
Author(s) :
Nobuto Yoneyama (Shinshu university)
Yoshihisa Miyanishi (Shinshu University)
Abstract : The notion of fundamental solutions {abbreviated by FS} is introduced in Colombeau algebra. Then we can construct a little more generalized FS even for Lewy-type equation whereas there are no FS in the sense of distributions.
