arXiv subjects

Shigeru Sakaguchi

Publications and source records attributed to Shigeru Sakaguchi.

At least 19 recordsLinked to original sources

arXiv2026

Ellipsoidal characterization of neutral inclusions for imperfect bonding of high-conductivity type

This paper concerns neutral inclusions for imperfect bonding of high-conductivity type. An inclusion, which is a bounded domain, is said to be of imperfect bonding of high-conductivity type if the flux is discontinuous along its boundary while the potential is continuous. The inclusion is neutral to a uniform field if the presence of the inclusion does not perturb the field outside the inclusion. It is known that ellipses and ellipsoids can be neutral to all uniform fields by introducing a proper imperfect bonding coefficient on boundaries. The purpose of this paper is to prove the converse. We prove that if an inclusion of imperfect bonding of high-conductivity type is neutral to all uniform fields, then it is an ellipse or an ellipsoid. The neutrality condition is given by existence of the solution to a certain differential equation on the boundary surface and the main result is proved by converting the neutrality condition into an algebraic boundary identity characterizing ellipses and ellipsoids.

math.AP↗

arXiv2025

Non-isoparametric Serrin domains of $\mathbb{S}^3$ with connected toric boundary

We investigate the overdetermined torsion problem $\begin{cases} -\Delta u = 1 & \text{in}\ \Omega\\ u=0 & \text{on}\ \partial \Omega\\ \frac{\partial u}{\partial \nu}=\text{const.} & \text{on}\ \partial \Omega, \end{cases}$ where $\Omega$ is a smooth Riemannian domain. Domains admitting a solution to this problem are called \textit{Serrin domains}, after the celebrated work of Serrin \cite{Se71}, where is proved that in $\mathbb{R}^n$ such domains are geodesic balls. In the present paper we establish the existence of two distinct types of Serrin domains of $\mathbb{S}^3$, respectively of small and large volume, each of whose boundary is connected and is neither isometric to a geodesic sphere nor to a Clifford torus. These domains arise as nontrivial perturbations of some classical symmetric solutions to the same problem. Our approach relies on an implicit construction based on the Crandall-Rabinowitz bifurcation theorem, which allows us to detect branches of non-radial solutions bifurcating from a family of radial ones. The resulting examples highlight new geometric configurations of the torsion problem in the three-dimensional sphere, providing another proof of the fact that the rigidity of Serrin-type results can fail in the presence of curvature.

math.AP↗

arXiv2024

Interaction between initial behavior of temperature and the mean curvature of the interface in two-phase heat conductors

We consider the Cauchy problem for the heat diffusion equation in the whole Euclidean space consisting of two media locally with different constant conductivities, where initially one medium has temperature 0 and the other has temperature 1. Under the assumption that a part of the interface between two media with different constant conductivities is of class $C^2$ in a neighborhood of a point $x$ on it, we extract the mean curvature of the interface at $x$ from the initial behavior of temperature at $x$. This result is purely local in space. As a corollary, when the whole Euclidean space consists of two media globally with different constant conductivities, it is shown that if a connected component $\Gamma$ of the interface is of class $C^2$ and is stationary isothermic, then the mean curvature of $\Gamma$ must be constant. Moreover, we apply this result to some overdetermined problems for two-phase heat conductors and obtain some symmetry theorems which relax considerably the regularity assumptions of some previous results.

math.AP↗

arXiv2023

Two extremum problems for Neumann eigenvalues

Neumann eigenvalues being non-decreasing with respect to domain inclusion, it makes sense to study the two shape optimization problems $\min\{μ_k(Ω):Ω\mbox{ convex},Ω\subset D, \}$ (for a given box $D$) and $\max\{μ_k(Ω):Ω\mbox{ convex},ω\subset Ω, \}$ (for a given obstacle $ω$). In this paper, we study existence of a solution for these two problems in two dimensions and we give some qualitative properties. We also introduce the notion of {\it self-domains} that are domains solutions of these extremal problems for themselves and give examples of the disk and the square. A few numerical simulations are also presented.

math.SP↗

arXiv2023

Symmetry results for some overdetermined obstacle problems

We establish symmetry results for two categories of overdetermined obstacle problems: a Serrin-type problem and a two-phase problem under the overdetermination that the interface serves as a level surface of the solution. The first proof avoids the method of moving planes; instead, it leverages the comparison principle and the fact that the solution of the obstacle problem is superharmonic within its domain. The second proof utilizes a suitable version of Serrin's method of moving planes.

math.AP↗

arXiv2022

The stationary critical points of the fractional heat flow

We study the spatial critical points of the solutions $u=u(x,t)$ of the fractional heat equation. For the Cauchy problem, we show that the origin $0$ satisfies $\nabla_x u(0,t) = 0$ for $t>0$ if and only if the initial data satisfy a balance law of the form $\int_{\mathbb{S}^{N-1}} ωu_0(rω) \, \mathrm{d} ω=0$ for a. e. $r \ge 0$. Moreover, for the Dirichlet initial-boundary value problem, we prove two symmetry results: $Ω$ is a ball centered at the origin if and only if $\nabla_x u(0,t) = 0$ for $t>0$ provided that the initial data satisfies the above-mentioned balance law; $Ω$ is centrosymmetric if and only if $\nabla_x u(0,t) = 0$ for $t>0$ provided that the initial data is centrosymmetric. These results extend some theorems obtained by Magnanini and Sakaguchi in 1997-1999 for the (local) heat equation to the fractional context. These extensions are nontrivial because of the nonlocal nature of the fractional Laplacian. Among others, the proof of the characterization of a ball in the Dirichlet initial-boundary value problem for the fractional heat flow not only works for the classical heat flow but also gives a new insight into the problem.

math.AP↗

arXiv2022

A symmetry theorem in two-phase heat conductors

We consider the Cauchy problem for the heat diffusion equation in the whole Euclidean space consisting of two media with different constant conductivities, where initially one medium has temperature 0 and the other has temperature 1. Under the assumptions that one medium is bounded and the interface is of class $C^{2,α}$, we show that if the interface is stationary isothermic, then it must be a sphere. The method of moving planes due to Serrin is directly utilized to prove the result.

math.AP↗

arXiv2021

Large time behavior of temperature in two-phase heat conductors

We consider the Cauchy problem for the heat diffusion equation in the whole Euclidean space consisting of two media with different constant conductivities. The large time behavior of temperature, the solution of the problem, is studied when initially temperature is assigned to be 0 on one medium and $1$ on the other. We show that under a certain geometric condition of the configuration of the media, temperature is stabilized to a constant as time tends to infinity. We also show by examples that temperature in general oscillates and is not stabilized.

math.AP↗

arXiv2021

The principal eigenfunction of the Dirichlet Laplacian with prescribed numbers of critical points on the upper half of a topological torus

We consider the principal eigenvalue problem for the Laplace-Beltrami operator on the upper half of a topological torus under the Dirichlet boundary condition. We present a construction of the upper half of a topological torus that admits the principal eigenfunction having exact numbers of critical points. Furthermore, we manage to identify the locations of all the critical points of the principal eigenfunction explicitly.

math.AP↗

arXiv2020

Patterns with prescribed numbers of critical points on topological tori

We study the existence of critical points of stable stationary solutions to reaction-diffusion problems on topological tori. Stable nonconstant stationary solutions are often called patterns. We construct topological tori and patterns with prescribed numbers of critical points whose locations are explicit.

math.AP↗

arXiv2020

Some characterizations of parallel hyperplanes in multi-layered heat conductors

We consider the Cauchy problem for the heat diffusion equation in the whole space consisting of three layers with different constant conductivities, where initially the upper and middle layers have temperature 0 and the lower layer has temperature 1. Under some appropriate conditions, it is shown that, if either the interface between the lower layer and the middle layer is a stationary isothermic surface or there is a stationary isothermic surface in the middle layer near the lower layer, then the two interfaces must be parallel hyperplanes. Similar propositions hold true, either if a stationary isothermic surface is replaced by a surface with the constant flow property or if the Cauchy problem is replaced by an appropriate initial-boundary value problem.

math.AP↗

arXiv2020

The double queen Dido's problem

This paper deals with a variation of the classical isoperimetric problem in dimension $N\ge 2$ for a two-phase piecewise constant density whose discontinuity interface is a given hyperplane. We introduce a weighted perimeter functional with three different weights, one for the hyperplane and one for each of the two open half-spaces in which $\mathbb{R}^N$ gets partitioned. We then consider the problem of characterizing the sets $Ω$ that minimize this weighted perimeter functional under the additional constraint that the volumes of the portions of $Ω$ in the two half-spaces are given. It is shown that the problem admits two kinds of minimizers, which will be called type I and type II, respectively. These minimizers are made of the union of two spherical domes whose angle of incidence satisfies some kind of \textquotedblleft Snell's law\textquotedblright. Finally, we provide a complete classification of the minimizers depending on the various parameters of the problem.

math.DG↗

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

A construction of patterns with many critical points on topological tori

We consider reaction-diffusion equations on closed surfaces in $\mathbb R^3$ having genus $1$. Stable nonconstant stationary solutions are often called patterns. The purpose of this paper is to construct closed surfaces together with patterns having as many critical points as one wants.

math.AP↗

arXiv2019

A characterization of a hyperplane in two-phase heat conductors

We consider the Cauchy problem for the heat diffusion equation in the whole Euclidean space consisting of two media with different constant conductivities, where initially one has temperature 0 and the other has temperature 1. Suppose that the interface is uniformly of class $C^6$. We show that if the interface has a time-invariant constant temperature, then it must be a hyperplane.

math.AP↗

arXiv2019

Two-phase heat conductors with a surface of the constant flow property

We consider a two-phase heat conductor in $\mathbb R^N$ with $N \geq 2$ consisting of a core and a shell with different constant conductivities. We study the role played by radial symmetry for overdetermined problems of elliptic and parabolic type. First of all, with the aid of the implicit function theorem, we give a counterexample to radial symmetry for some two-phase elliptic overdetermined boundary value problems of Serrin-type. Afterwards, we consider the following setting for a two-phase parabolic overdetermined problem. We suppose that, initially, the conductor has temperature 0 and, at all times, its boundary is kept at temperature 1. A hypersurface in the domain has the constant flow property if at every of its points the heat flux across surface only depends on time. It is shown that the structure of the conductor must be spherical, if either there is a surface of the constant flow property in the shell near the boundary or a connected component of the boundary of the heat conductor is a surface of the constant flow property. Also, by assuming that the medium outside the conductor has a possibly different conductivity, we consider a Cauchy problem in which the conductor has initial inside temperature $0$ and outside temperature $1$. We then show that a quite similar symmetry result holds true.

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↗