KiHyun Yun

Estimates for Electric Fields Blown Up between Closely Adjacent Conductors with Arbitrary Shape

It may be well known in practice that high stress concentrations occur in fiber‐reinforced composites. There have been several works by analysis to estimate for the stresses between closed spaced fibers. However, the known results on stiff fibers have until now been restricted to the particular case of circular cross‐sections. Thus, we extend the blow‐up results on the stresses specialized only for disks to the general case of arbitrary shapes. Moreover, we prove that the blow‐up rate of the general case is exactly the same as that of disks. Nevertheless, from the viewpoint of methodology, the technique we use is significantly different from the previous one restricted to the case of disks. Referring to antiplane shear problems, these works are reduced to the gradient estimates for the solution to the conductivity problem containing two closely spaced conductors which represent the cross‐sections of fibers. We establish a novel representation for the solution on conductors by a probability function. Based...

Blow-up of Electric Fields between Closely Spaced Spherical Perfect Conductors

The electric field increases toward infinity in the narrow region between closely adjacent perfect conductors as they approach each other. Much attention has been devoted to the blow-up estimate, especially in two dimensions, for the practical relevance to high stress concentration in fiber-reinforced elastic composites. In this paper, we establish optimal estimates for the electric field associated with the distance between two spherical conductors in n-dimensional spaces for n ≥ 2. The novelty of these estimates is that they explicitly describe the dependency of the blow-up rate on the geometric parameters: the radii of the conductors.

Boundary determination of conductivities and Riemannian metrics via local Dirichlet-to-Neumann operator

We consider the inverse problem to identify an anisotropic conductivity from the Dirichlet-to-Neumann (DtN) map. We first find an explicit reconstruction of the boundary value of less regular anisotropic (transversally isotropic) conductivities and their derivatives. Based on the reconstruction formula, we prove Holder stability, up to isometry, of the inverse problem using a local DtN map.

Stability of the Scattering from a Large Electromagnetic Cavity in Two Dimensions

This work is concerned with a time harmonic scattering problem of electromagnetic waves from a two-dimensional open cavity embedded in the infinite ground plane. Because of the highly oscillatory nature of the solution for large or deep cavity, the model scattering problem is challenging both mathematically and computationally. A variational formulation reduces the scattering problem into a bounded domain (the cavity) problem. The stability of the solution is established for the bounded domain problem in the energy space. Moreover, our stability estimates provide the explicit dependence on the high wave number and the depth of the cavity.

Characterization of the Electric Field Concentration between Two Adjacent Spherical Perfect Conductors

When two perfectly conducting inclusions are located close to each other, the electric field concentrates in a narrow region in between two inclusions and becomes arbitrarily large as the distance between two inclusions tends to zero. The purpose of this paper is to derive an asymptotic formula of the concentration which completely characterizes the singular behavior of the electric field when inclusions are balls of the same radii in three dimensions.

On the stability of an inverse problem for the wave equation

Consider the inverse problem of determining the potential q from the Neumann to Dirichlet mapq of the wave equation utt − �u + qu = 0i n� × (0 ,T) with u(x,0) = ut (x,0) = 0. In this paper, a nearly Lipschitz-type stability estimate is established for the inverse problem: for any small �> 0, there exists β0 > 0 such that � q1 − q2� L∞(�) C� � q1 − � q2 � 1−� ∗ whenq1 − q2� H β (R n ) M for some β> β 0. Here, �·� ∗ represents the operator norm.

Recovery of an inhomogeneity in an elliptic equation

We consider the inverse problem to identify an unknown domain D entering an elliptic equation Δu-χ(D)u = 0 in Ω. We show that solutions of the differential equation can be represented as solutions of an integral equation using the volume potential. We then prove the global uniqueness of the inverse problem within the class of two- or three-dimensional balls. Based on the representation we propose a numerical algorithm to detect the unknown domain D and show some results of numerical experiments.

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.