# SAKAGUCHI Shigeru

Last updated: Jul 10, 2014 at 13:16

Name SAKAGUCHI Shigeru Tohoku University Graduate School of Information Sciences Professor Doctor of Science(Tokyo Metropolitan University)

## Profile

The main purpose is to know geometric properties of solutions of partial differential equations. Since solutions are functions, it is natural to want to know their shapes and geometric properties. The main research topics are the following.

(1) Stationary isothermic surfaces and stationary critical points

To know the shapes of graphs of functions, one may begin by investigating their level surfaces and critical points. In particular, an isothermic surface (a critical point) of the solution of the heat equation is called stationary if its temperature depends only on time (if it is invariant with time). The existence of a stationary isothermic surface (or a stationary critical point) is deeply related to the symmetry of the heat conductor. The right helicoid is an interesting example of stationary isothermic surfaces in the three-dimensional Euclidean space.

(2) Interaction between diffusion and geometry of domain

We consider linear and nonlinear diffusion equations (the heat equation, the porous medium equation, etc.). In the problem where the initial value equals zero and the boundary value equals 1, the short-time behavior of solutions is deeply related to the curvatures of the boundary.

(3) Shape of solutions of linear and nonlinear elliptic equations

In general, solutions of elliptic equations describe steady states after a sufficiently long time. We consider linear and nonlinear elliptic equations. Liouville-type theorems characterize hyperplanes as graphs of entire solutions with some reasonable restriction. Overdetermined boundary value problems characterize balls, ellipsoids, or some symmetrical domains in general.

(4) The point of view of inverse problems

Partial differential equations appear in models describing natural phenomena. It is an interesting problem that characterizes some geometry in some reasonable way from the point of view of inverse problems.

## Research Areas

Apr 2012
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Today
Professor, Tohoku University

Apr 2008
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Mar 2012
Professor, Hiroshima University

Feb 2002
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Mar 2008
Professor, Ehime University

Apr 1993
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Jan 2002
Associate Professor, Ehime University

Apr 1989
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Mar 1993
Research Associate, Tokyo Institute of Technology

## Education

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1986
Mathematics, Graduate School, Division of Natural Science, Tokyo Metropolitan University

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1979
Faculty of Science, Tokyo Institute of Technology

## Committee Memberships

2003
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2007
Mathematical Society of Japan

2007
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2007
Mathematical Society of Japan

## Awards & Honors

Sep 2012
Geometry on the domain via the isothermic set for diffusion equations, 2012 Analysis Prize, The Mathematical Society of Japan

## Published Papers

Stationary isothermic surfaces in Euclidean 3-space
R. Magnanini, D. Peralta-Salas, and S. Sakaguchi
arXiv:1407.2419v1      Jul 2014
Interaction between fast diffusion and geometry of domain
S. Sakaguchi
Kodai Math. J. to appear(arXiv:1404.4915v2)      May 2014   [Refereed]
When does the heat equation have a solution with a sequence of similar level sets?
T. Kawakami and S. Sakaguchi
Annali di Matematica Pura ed Applicata, to appear(arXiv:1403.6203v2)      Jun 2014   [Refereed]
Solutions of elliptic equations with a level surface parallel to the boundary: stability of the radial configuration
G. Ciraolo, R. Magnanini and S. Sakaguchi
J. Analyse Math.   to appear(arXiv:1307.1257v1)    2014   [Refereed]
Stationary level surfaces and Liouville-type theorems characterizing hyperplanes
S. Sakaguchi
`` Geometric Properties of Parabolic and Elliptic PDE's ", Springer INdAM Series   2 269-282   2013   [Refereed]

## Research Grants & Projects

Geometry of solutions of partial differential equations and the inverse problems accompanied by it
Japan Society for the Promotion of Science: Grant-in-Aid for Scientific Research (B)
Project Year: Apr 2014 - Mar 2018    Investigator(s): SAKAGUCHI Shigeru
Search for new isoperimetric inequalities relating to elliptic equations
Japan Society for the Promotion of Science: Grant-in-Aid for challenging Exploratory Research
Project Year: Apr 2013 - Mar 2016    Investigator(s): SAKAGUCHI Shigeru
Diffusion and geometry of domain
Japan Society for the Promotion of Science: Grant-in-Aid for Scientific Research (B)
Project Year: Apr 2008 - Mar 2013    Investigator(s): SAKAGUCHI Shigeru
Symmetry and geometric properties of solutions of nonlinear parabolic initial-boundary value problems
Japan Society for the Promotion of Science: Grant-in-Aid for JSPS Fellows
Project Year: Oct 2006 - Sep 2008    Investigator(s): SAKAGUCHI Shigeru
Behavior of spatial critical points and level surfaces of solutions of partial differential equations and shapes of the solutions
Japan Society for the Promotion of Science: Grant-in-Aid for Scientific Research (B)
Project Year: Apr 2003 - Mar 2007    Investigator(s): SAKAGUCHI Shigeru