Mishio Kawashita

Eshelby Tensor of a Polygonal Inclusion and its Special Properties

Nozaki and Taya (ASME J. Appl. Mech. 64 (1997) 495–502) analyzed the elastic field in a convex polygonal inclusion in an infinite body. By numerical analysis, they found that, when the shape of the inclusion is a regular polygon, “the strain at the center of inclusion” and “the strain energy per unit volume of inclusion” have strange and remarkable properties: these values are the same as those of a circular inclusion and are invariant for inclusions orientation if the shape of the inclusion is not a square. In this paper, we first derive a simple, exact expression of the Eshelby tensor for an arbitrary polygonal inclusion. Using the expression, we then show a mathematical explanation why these special properties appear.

The enclosure method for the heat equation

This paper shows how the enclosure method which was originally introduced for elliptic equations can be applied to inverse initial boundary value problems for parabolic equations. For the purpose a prototype of inverse initial boundary value problems whose governing equation is the heat equation is considered. An explicit method to extract an approximation of the value of the support function at a given direction, of unknown discontinuity embedded in a heat conductive body, from the temperature, for a suitable heat flux on the lateral boundary for a fixed observation time is given.

On the reconstruction of inclusions in a heat conductive body from dynamical boundary data over a finite time interval

The enclosure method was originally introduced for inverse problems of concerning non destructive evaluation governed by elliptic equations. It was developed as one of useful approach in inverse problems and applied for various equations. In this article, an application of the enclosure method to an inverse initial boundary value problem for a parabolic equation with a discontinuous coefficients is given. A simple method to extract the depth of unknown inclusions in a heat conductive body from a single set of the temperature and heat flux on the boundary observed over a finite time interval is introduced. Other related results with infinitely many data are also reported. One of them gives the minimum radius of the open ball centered at a given point that contains the inclusions. The formula for the minimum radius is newly discovered.

Relation Between Scattering Theories of the Wilcox and Lax–Phillips Types and a Concrete Construction of the Translation Representation

Abstract In this article we extract characteristic points from scattering theories of the Lax–Phillips type and the Wilcox one, and in view of those points we show that the both settings can be translated into each other. Furthermore, we construct concrete translation representations of the Lax–Phillips type for the elastic equation in the half-space, in order to show the good selection from admitted constructions in the abstract arguments.

An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method

This paper studies a prototype of inverse initial boundary value problems whose governing equation is the heat equation in three dimensions. An unknown discontinuity embedded in a three-dimensional heat conductive body is considered. A {\it single} set of the temperature and heat flux on the lateral boundary for a fixed observation time is given as an observation datum. It is shown that this datum yields the minimum length of broken paths that start at a given point outside the body, go to a point on the boundary of the unknown discontinuity and return to a point on the boundary of the body under some conditions on the input heat flux, the unknown discontinuity and the body. This is new information obtained by using enclosure method.

Asymptotic behavior of the solutions for the Laplace equation with a large spectral parameter and the inhomogeneous Robin type conditions

Reduced problems are elliptic problems with a large parameter (as the spectral parameter) given by the Laplace transform of time dependent problems. In this paper, asymptotic behavior of the solutions of the reduced problem for the classical heat equation in bounded domains with the inhomogeneous Robin type conditions is discussed. The boundary of the domain consists of two disjoint surfaces, outside one and inside one. When there are inhomogeneous Robin type data at both boundaries, it is shown that asymptotics of the value of the solution with respect to the large parameter at a given point inside the domain is closely connected to the distance from the point to the both boundaries. It is also shown that if the inside boundary is strictly convex and the data therein vanish, then the asymptotics is different from the previous one. The method for the proof employs a representation of the solution via single layer potentials. It is based on some non trivial estimates on the integral kernels of related integral equations which are previously established and used in studying an inverse problem for the heat equation via the enclosure method.

Scattering theory for the elastic wave equation in perturbed half-spaces

In this paper we consider the linear elastic wave equation with the free boundary condition (the Neumann condition), and formulate a scattering theory of the Lax and Phillips type and a representation of the scattering kernel. We are interested in surface waves (the Rayleigh wave, etc.) connected closely with situations of boundaries, and make the formulations intending to extract this connection. The half-space is selected as the free space, and making dents on the boundary is considered as a perturbation from the flat one. Since the lacuna property for the solutions in the outgoing and incoming spaces does not hold because of the existence of the surface waves, instead of it, certain decay estimates for the free space solutions and a weak version of the Morawetz arguments are used to formulate the scattering theory. We construct the representation of the scattering kernel with outgoing scattered plane waves. In this step, again because of the existence of the surface waves, we need to introduce new outgoing and incoming conditions for the time dependent solutions to ensure uniqueness of the solutions. This introduction is essential to show the representation by reasoning similar to the case of the reduced wave equation.

On finding a buried obstacle in a layered medium via the time domain enclosure method

An inverse obstacle problem for the wave equation in a two layered medium is considered. It is assumed that the unknown obstacle is penetrable and embedded in the lower half-space. The wave as a solution of the wave equation is generated by an initial data whose support is in the upper half-space and observed at the same place as the support over a finite time interval. From the observed wave an indicator function in the time domain enclosure method is constructed. It is shown that, one can find some information about the geometry of the obstacle together with the qualitative property in the asymptotic behavior of the indicator function.