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Shigeru Sakaguchi

Publications and source records attributed to Shigeru Sakaguchi.

35 records · Page 2Linked to original sources
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.

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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.

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arXiv2018

A simple proof of a strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation

A strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation is considered. The difficulties of the problem come from the fact that this nonlinear equation is non-uniformly elliptic, does not depend on the value of unknown functions, depends on spatial variables and solutions are semicontinuous. Our simple proof of the strong comparison principle consists only of three ingredients, the definition of viscosity solutions, the inf and sup convolutions of functions, and the theory of classical solutions of quasilinear elliptic equations. Once we have the strong comparison principle, we can prove a weak comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation in a bounded domain.

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arXiv2018

Two-phase heat conductors with a stationary isothermic surface and their related elliptic overdetermined problems

We consider a two-phase heat conductor in two dimensions consisting of a core and a shell with different constant conductivities. When the medium outside the two-phase conductor has a possibly different conductivity, we consider the Cauchy problem in two dimensions where initially the conductor has temperature 0 and the outside medium has temperature 1. It is shown that, if there is a stationary isothermic surface in the shell near the boundary, then the structure of the conductor must be circular. Moreover, as by-products of the method of the proof, we mention other proofs of all the previous results of the author in $N(\ge 2)$ dimensions and two theorems on their related two-phase elliptic overdetermined problems.

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arXiv2016

Two-phase heat conductors with a stationary isothermic surface

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. Suppose that, initially, the conductor has temperature 0 and, at all times, its boundary is kept at temperature 1. It is shown that, if there is a stationary isothermic surface in the shell near the boundary, then the structure of the conductor must be spherical. Also, when the medium outside the two-phase conductor has a possibly different conductivity, we consider the Cauchy problem with $N \ge 3$ and the initial condition where the conductor has temperature 0 and the outside medium has temperature 1. Then we show that almost the same proposition holds true.

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arXiv2016

Symmetry problems on stationary isothermic surfaces in Euclidean spaces

Let $S$ be a smooth hypersurface properly embedded in $\mathbb R^N$ with $N \geq 3$ and consider its tubular neighborhood $\mathcal N$. We show that, if a heat flow over $\mathcal N$ with appropriate initial and boundary conditions has $S$ as a stationary isothermic surface, then $S$ must have some sort of symmetry.

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arXiv2015

Stationary isothermic surfaces in Euclidean 3-space

Let $Ω$ be a domain in $\mathbb R^3$ with $\partialΩ= \partial\left(\mathbb R^3\setminus \overlineΩ\right)$, where $\partialΩ$ is unbounded and connected, and let $u$ be the solution of the Cauchy problem for the heat equation $\partial_t u= Δu$ over $\mathbb R^3,$ where the initial data is the characteristic function of the set $Ω^c = \mathbb R^3\setminus Ω$. We show that, if there exists a stationary isothermic surface $Γ$ of $u$ with $Γ\cap \partialΩ= \varnothing$, then both $\partialΩ$ and $Γ$ must be either parallel planes or co-axial circular cylinders . This theorem completes the classification of stationary isothermic surfaces in the case that $Γ\cap\partialΩ=\varnothing$ and $\partialΩ$ is unbounded. To prove this result, we establish a similar theorem for {\it uniformly dense domains } in $\mathbb R^3$, a notion that was introduced by Magnanini, Prajapat \& Sakaguchi in \cite{MPS2006tams}. In the proof, we use methods from the theory of surfaces with constant mean curvature, combined with a careful analysis of certain asymptotic expansions and a surprising connection with the theory of transnormal functions.

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arXiv2015

An over-determined boundary value problem arising from neutrally coated inclusions in three dimensions

We consider the neutral inclusion problem in three dimensions which is to prove if a coated structure consisting of a core and a shell is neutral to all uniform fields, then the core and the shell must be concentric balls if the matrix is isotropic and confocal ellipsoids if the matrix is anisotropic. We first derive an over-determined boundary value problem in the shell of the neutral inclusion, and then prove in the isotropic case that if the over- determined problem admits a solution, then the core and the shell must be concentric balls. As a consequence it is proved that the structure is neutral to all uniform fields if and only if it consists of concentric balls provided that the coefficient of the core is larger than that of the shell.

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arXiv2014

Symmetry of minimizers with a level surface parallel to the boundary

We consider the functional $$I_Ω(v) = \int_Ω[f(|Dv|) - v] dx,$$ where $Ω$ is a bounded domain and $f$ is a convex function. Under general assumptions on $f$, G. Crasta [Cr1] has shown that if $I_Ω$ admits a minimizer in $W_0^{1,1}(Ω)$ depending only on the distance from the boundary of $Ω$, then $Ω$ must be a ball. With some restrictions on $f$, we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss how these results extend to more general settings, in particular to functionals that are not differentiable and to solutions of fully nonlinear elliptic and parabolic equations.

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arXiv2014

Interaction between fast diffusion and geometry of domain

Let $Ω$ be a domain in $\mathbb R^N$, where $N \ge 2$ and $\partialΩ$ is not necessarily bounded. We consider two fast diffusion equations $\partial_t u= \mbox{div}(|\nabla u|^{p-2}{\nabla u})$ and $\partial_t u= Δu^{m}$, where $1 \frac {(N+1)(2-p)}{2p}$ or $α> \frac {(N+1)(1-m)}{4}$. Then, we derive asymptotic estimates for the integral of $u^α$ over $B$ for short times in terms of principal curvatures of $\partialΩ$ at the point, which tells us about the interaction between fast diffusion and geometry of domain.

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arXiv2013

Solutions of elliptic equations with a level surface parallel to the boundary: stability of the radial configuration

Positive solutions of homogeneous Dirichlet boundary value problems or initial-value problems for certain elliptic or parabolic equations must be radially symmetric and monotone in the radial direction if just one of their level surfaces is parallel to the boundary of the domain. Here, for the elliptic case, we prove the stability counterpart of that result. In fact, we show that if the solution is almost constant on a surface at a fixed distance from the boundary, then the domain is almost radially symmetric, in the sense that is contained in and contains two concentric balls $B_{r_e}$ and $B_{r_i}$, with the difference $r_e-r_i$ (linearly) controlled by a suitable norm of the deviation of the solution from a constant. The proof relies on and enhances arguments developed in a paper by Aftalion, Busca and Reichel.

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arXiv2012

Stationary level surfaces and Liouville-type theorems characterizing hyperplanes

We consider an entire graph $S$ in $\mathbb R^{N+1}$ of a continuous real function $f$ over $\mathbb R^{N}$ with $N\ge 1$. Let $Ω$ be an unbounded domain in $\mathbb R^{N+1}$ with boundary $S$. Consider nonlinear diffusion equations of the form $\partial_t U= Δϕ(U)$ containing the heat equation. Let $U$ be the solution of either the initial-boundary value problem over $Ω$ where the initial value equals zero and the boundary value equals 1, or the Cauchy problem where the initial data is the characteristic function of the set $\mathbb R^{N+1}\setminus Ω$. The problem we consider is to characterize $S$ in such a way that there exists a stationary level surface of $U$ in $Ω$. We introduce a new class $\mathcal A$ of entire graphs $S$ and, by using the sliding method, we show that $S\in\mathcal A$ must be a hyperplane if there exists a stationary level surface of $U$ in $Ω$. This is an improvement of the previous result. Next, we consider the heat equation in particular and we introduce the class $\mathcal B$ of entire graphs $S$ of functions $f$ such that each ${|f(x)-f(y)|: |x-y| \le 1}$ is bounded. With the help of the theory of viscosity solutions, we show that $S \in \mathcal B$ must be a hyperplane if there exists a stationary isothermic surface of $U$ in $Ω$. This is a considerable improvement of the previous result. Related to the problem, we consider a class $\mathcal W$ of Weingarten hypersurfaces in $\mathbb R^{N+1}$ with $N \ge 1$. Then we show that, if $S$ belongs to $\mathcal W$ in the viscosity sense and $S$ satisfies some natural geometric condition, then $S \in \mathcal B$ must be a hyperplane. This is also a considerable improvement of the previous result.

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arXiv2011

Interaction between nonlinear diffusion and geometry of domain

Let $Ω$ be a domain in $\mathbb R^N$, where $N \ge 2$ and $\partialΩ$ is not necessarily bounded. We consider nonlinear diffusion equations of the form $\partial_t u= Δϕ(u)$. Let $u=u(x,t)$ be the solution of either the initial-boundary value problem over $Ω$, where the initial value equals zero and the boundary value equals 1, or the Cauchy problem where the initial data is the characteristic function of the set $\mathbb R^N\setminus Ω$. We consider an open ball $B$ in $Ω$ whose closure intersects $\partialΩ$ only at one point, and we derive asymptotic estimates for the content of substance in $B$ for short times in terms of geometry of $Ω$. Also, we obtain a characterization of the hyperplane involving a stationary level surface of $u$ by using the sliding method due to Berestycki, Caffarelli, and Nirenberg. These results tell us about interactions between nonlinear diffusion and geometry of domain.

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arXiv2011

Matzoh ball soup revisited: the boundary regularity issue

We consider nonlinear diffusion equations of the form $\partial_t u= Δϕ(u)$ in $\mathbb R^N$ with $N \ge 2.$ When $ϕ(s) \equiv s$, this is just the heat equation. Let $Ω$ be a domain in $\mathbb R^N$, where $\partialΩ$ is bounded and $\partialΩ= \partial (\mathbb R^N\setminus \bar Ω)$. We consider the initial-boundary value problem, where the initial value equals zero and the boundary value equals 1, and the Cauchy problem where the initial data is the characteristic function of the set $Ω^c = \mathbb R^N\setminus Ω$. We settle the boundary regularity issue for the characterization of the sphere as a stationary level surface of the solution $u:$ no regularity assumption is needed for $\partialΩ.$

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arXiv2009

Nonlinear diffusion with a bounded stationary level surface

We consider nonlinear diffusion of some substance in a container (not necessarily bounded) with bounded boundary of class C^2. Suppose that, initially, the container is empty and, at all times, the substance at its boundary is kept at density 1. We show that, if the container contains a proper C^2-subdomain on whose boundary the substance has constant density at each given time, then the boundary of the container must be a sphere. We also consider nonlinear diffusion in the whole R^N of some substance whose density is initially a characteristic function of the complement of a domain with bounded C^2 boundary, and obtain similar results. These results are also extended to the heat flow in the sphere S^N and the hyperbolic space H^N.

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arXiv2008

Stationary isothermic surfaces and some characterizations of the hyperplane in the N-dimensional Euclidean space

We consider an entire graph S of a continuous real function over (N-1)-dimensional Euclidean space with N larger than or equal to 3. Let D be a domain in N-dimensional Euclidean space with S as a boundary. Consider in D the heat flow with initial temperature 0 and boundary temperature 1. The problem we consider is to characterize S in such a way that there exists a stationary isothermic surface in D. We show that S must be a hyperplane under some general conditions on S. This is related to Liouville or Bernstein-type theorems for some elliptic Monge-Ampère-type equation.

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