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Hyeonbae Kang

Publications and source records attributed to Hyeonbae Kang.

At least 37 records · Page 2Linked to original sources
arXiv2020

Polarization tensor vanishing structure of general shape: Existence for small perturbations of balls

The polarization tensor is a geometric quantity associated with a domain. It is a signature of the small inclusion's existence inside a domain and used in the small volume expansion method to reconstruct small inclusions by boundary measurements. In this paper, we consider the question of the polarization tensor vanishing structure of general shape. The only known examples of the polarization tensor vanishing structure are concentric disks and balls. We prove, by the implicit function theorem on Banach spaces, that a small perturbation of a ball can be enclosed by a domain so that the resulting inclusion of the core-shell structure becomes polarization tensor vanishing. The boundary of the enclosing domain is given by a sphere perturbed by spherical harmonics of degree zero and two. This is a continuation of the earlier work \cite{KLS2D} for two dimensions.

math.AP↗
arXiv2020

Neutral inclusions, weakly neutral inclusions, and an over-determined problem for confocal ellipsoids

An inclusion is said to be neutral to uniform fields if upon insertion into a homogenous medium with a uniform field it does not perturb the uniform field at all. It is said to be weakly neutral if it perturbs the uniform field mildly. Such inclusions are of interest in relation to invisibility cloaking and effective medium theory. There have been some attempts lately to construct or to show existence of such inclusions in the form of core-shell structure or a single inclusion with the imperfect bonding parameter attached to its boundary. The purpose of this paper is to review recent progress in such attempts. We also discuss about the over-determined problem for confocal ellipsoids which is closely related with the neutral inclusion, and its equivalent formulation in terms of Newtonian potentials. The main body of this paper consists of reviews on known results, but some new results are also included.

math.AP↗
arXiv2019

Quantitative estimates for enhancement of the field excited by an emitter due to presence of two closely located spherical inclusions

A field in a homogeneous medium can be amplified or enhanced by inserting closely located perfectly conducting inclusions into the medium. In this paper precise quantitative estimates for such enhancement are derived when the given field is the one excited by an emitter of a dipole type and inclusions are spheres of the same radii in three dimensions. Derived estimates reveal the difference, as well as the similarity, between enhancement of the field excited by the emitter and that of the smooth back-ground field. In particular, an estimate shows that when the enhancement occurs, the factor of enhancement is $(\sqrtε|\log ε|)^{-1}$, which is different from that for the smooth background field, which is known to be $(ε|\log ε|)^{-1}$ ($ε$ is the distance between two inclusions).

math.AP↗
arXiv2019

Convergence rate for eigenvalues of the elastic Neumann--Poincaré operator on smooth and real analytic boundaries in two dimensions

The elastic Neumann--Poincaré operator is a boundary integral operator associated with the Lamé system of linear elasticity. It is known that if the boundary of a planar domain is smooth enough, it has eigenvalues converging to two different points determined by Lamé parameters. We show that eigenvalues converge at a polynomial rate on smooth boundaries and the convergence rate is determined by smoothness of the boundary. We also show that they converge at an exponential rate if the boundary of the domain is real analytic.

math.SP↗
arXiv2018

Quantitative estimates of the field excited by an emitter in a narrow region between two circular inclusions

A field excited by an emitter can be enhanced due to presence of closely located inclusions. In this paper we consider such field enhancement when inclusions are disks of the same radii, and the emitter is of dipole type and located in the narrow region between two inclusions. We derive quantitatively precise estimates of the field enhancement in the narrow region. The estimates reveal that the field is enhanced by a factor of $ε^{-1/2}$ in most area, where $ε$ is the distance between two inclusions. This factor is the same as that of gradient blow-up when there is a smooth back-ground field, not a field excited by an emitter. The method of deriving estimates shows clearly that enhancement is due to potential gap between two inclusions.

math.AP↗
arXiv2018

Spectral structure of the Neumann--Poincaré operator on tori

We address the question whether there is a three-dimensional bounded domain such that the Neumann--Poincaré operator defined on its boundary has infinitely many negative eigenvalues. It is proved in this paper that tori have such a property. It is done by decomposing the Neumann--Poincaré operator on tori into infinitely many self-adjoint compact operators on a Hilbert space defined on the circle using the toroidal coordinate system and the Fourier basis, and then by proving that the numerical range of infinitely many operators in the decomposition has both positive and negative values.

math.SP↗
arXiv2018

Precise estimates of the field excited by an emitter in presence of closely located inclusions of a bow-tie shape

This paper studies in a quantitatively precise manner the field enhancement due to presence of an emitter of the dipole type near the bow-tie structure of perfectly conducting inclusions in the two-dimensional space. We put special emphasis on field enhancement near vertices of the bow-tie structure, and derive upper and lower bounds of the gradient blow-up there. All three different kinds of symmetries are considered by varying locations and directions of the emitter, and a different estimate is derived for each case.

math.AP↗
arXiv2018

Existence of coated inclusions of general shape weakly neutral to multiple fields in two dimensions

A two dimensional inclusion of core-shell structure is neutral to multiple uniform fields if and only if the core and the shell are concentric disks, provided that the conductivity of the matrix is isotropic. An inclusion is said to be neutral if upon its insertion the uniform field is not perturbed at all. In this paper we consider inclusions of core-shell structure of general shape which are weakly neutral to multiple uniform fields. An inclusion is said to be weakly neutral if the field perturbation is mild. We show, by an implicit function theorem, that if the core is a small perturbation of a disk then we can coat it by a shell so that the resulting structure becomes weakly neutral to multiple uniform fields.

math.AP↗
arXiv2018

Surface Riesz transforms and spectral property of elastic Neumann--Poincaé operators on less smooth domains in three dimensions

It is known that the Neumann--Poincaré operator for the Lamé system of linear elasticity is polynomially compact and, as a consequence, that its spectrum consists of three non-empty sequences of eigenvalues accumulating to certain numbers determined by Lamé parameters, if the boundary of the domain where the operator is defined is $C^\infty$-smooth. We extend this result to less smooth boundaries, namely, $C^{1, α}$-smooth boundaries for some $α> 0$. The results are obtained by proving certain identities for surface Riesz transforms, which are singular integral operators of nonconvolution type, defined by the matrix tensor on a given surface.

math.FA↗
arXiv2018

The first Hadamard variation of Neumann-Poincaré eigenvalues on the sphere

The Neumann-Poincaré operator on the sphere has $\frac{1}{2(2k+1)}$, $k=0,1,2,\ldots$, as its eigenvalues and the corresponding multiplicity is $2k+1$. We consider the bifurcation of eigenvalues under deformation of domains, and show that Frechét derivative of the sum of the bifurcations is zero. We then discuss the connection of this result with some conjectures regarding the Neumann-Poincaré operator.

math.SP↗
arXiv2018

Construction of weakly neutral inclusions of general shape by imperfect interfaces

Upon insertion of an inclusion into a medium with the uniform field, if the field is not perturbed at all outside the inclusion, then it is called a neutral inclusion. It is called a weakly neutral inclusion if the field is perturbed weakly. The inclusions neutral to multiple uniform fields are of circular shape if the medium is isotropic, and any other shape cannot be neutral. We consider in this paper the problem of constructing inclusions of general shape which are weakly neutral to multiple fields in two dimensions. We show that a simply connected domain satisfying a certain geometric condition can be realized as a weakly neutral inclusion to multiple fields by introducing an imperfect interface parameter on the boundary. The geometric condition on the domain and the imperfect interface parameter are determined by the first coefficient of the conformal mapping from the exterior of the unit disk onto the exterior of the domain. We provide some numerical examples to compare field perturbations by weakly neutral inclusions and perfectly bonding interfaces. They clearly show that the field perturbation by weakly neutral inclusions is much weaker.

math.AP↗
arXiv2017

Quantitative characterization of stress concentration in the presence of closely spaced hard inclusions in two-dimensional linear elasticity

In the region between close-to-touching hard inclusions, the stress may be arbitrarily large as the inclusions get closer. The stress is represented by the gradient of a solution to the Lamé system of linear elasticity. We consider the problem of characterizing the gradient blow-up of the solution in the narrow region between two inclusions and estimating its magnitude. We introduce singular functions which are constructed in terms of nuclei of strain and hence are solutions of the Lamé system, and then show that the singular behavior of the gradient in the narrow region can be precisely captured by singular functions. As a consequence of the characterization, we are able to regain the existing upper bound on the blow-up rate of the gradient, namely, $ε^{-1/2}$ where $ε$ is the distance between two inclusions. We then show that it is in fact an optimal bound by showing that there are cases where $ε^{-1/2}$ is also a lower bound. This work is the first to completely reveal the singular nature of the gradient blow-up in the context of the Lamé system with hard inclusions. The singular functions introduced in this paper play essential roles to overcome the difficulties in applying the methods of previous works. Main tools of this paper are the layer potential techniques and the variational principle. The variational principle can be applied because the singular functions of this paper are solutions of the Lamé system.

math.AP↗
arXiv2017

Optimal estimates of the field enhancement in presence of a bow-tie structure of perfectly conducting inclusions in two dimensions

This paper deals with the field enhancement, that is, the gradient blow-up, due to presence of a bow-tie structure of perfectly conducting inclusions in two dimensions. The bow-tie structure consists of two disjoint bounded domains which have corners with possibly different aperture angles. The domains are parts of cones near the vertices, and they are nearly touching to each other. We characterize the field enhancement using explicit functions and, as consequences, derive optimal estimates of the gradient in terms of the distance between two inclusions and aperture angles of the corners. The estimates show that the field is enhanced beyond the corner singularities due to the interaction between two inclusions.

math.AP↗
arXiv2017

A proof of the Flaherty-Keller formula on the effective property of densely packed elastic composites

We prove in a mathematically rigorous way the asymptotic formula of Flaherty and Keller on the effective property of densely packed periodic elastic composites with hard inclusions. The proof is based on the primal-dual variational principle, where the upper bound is derived by using the Keller-type test functions and the lower bound by singular functions made of nuclei of strain. Singular functions are solutions of the Lamé system and capture precisely singular behavior of the stress in the narrow region between two adjacent hard inclusions.

math.AP↗
arXiv2017

Elastic Neumann-Poincaré operators on three dimensional smooth domains: Polynomial compactness and spectral structure

We prove that the elastic Neumann--Poincaré operator defined on the smooth boundary of a bounded domain in three dimensions, which is known to be non-compact, is in fact polynomially compact. As a consequence, we prove that the spectrum of the elastic Neumann-Poincaré operator consists of three non-empty sequences of eigenvalues accumulating to certain numbers determined by Lamé parameters. These results are proved using the surface Riesz transform, calculus of pseudo-differential operators and the spectral mapping theorem.

math.SP↗
arXiv2016

Cloaking by anomalous localized resonance for linear elasticity on a coated structure

We investigate anomalous localized resonance on the circular coated structure and cloaking related to it in the context of elasto-static systems. The structure consists of the circular core with constant Lamé parameters and the circular shell of negative Lamé parameters proportional to those of the core. We show that the eigenvalues of the Neumann-Poincaré operator corresponding to the structure converges to certain non-zero numbers determined by Lamé parameters and derive precise asymptotics of the convergence. We then show with estimates that cloaking by anomalous localized resonance takes place if and only if the dipole type source lies inside critical radii determined by the radii of the core and the shell.

math.SP↗