TOHOKU UNIVERSITY Researchers - Shigeru Sakaguchi

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Shigeru Sakaguchi

Section

Admission Center

Job title

Program-Specific Professor

Degree

Doctor of Science (Tokyo Metropolitan University)

researchmap

https://researchmap.jp/sigersak2012415

J-GLOBAL ID

200901042737782460

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 current research topics are the following.

(1) Stationary level surfaces of solutions of diffusion equations: To know the shapes of graphs of functions, one may begin by investigating their level surfaces. An isothermic surface of the solution of the heat equation is called stationary if its temperature depends only on time. The existence of a stationary isothermic surface is deeply related to the symmetry of the heat conductor. The right helicoid, the circular cylinder, the sphere and the plane are examples of stationary isothermic surfaces in Euclidean 3-space. The characterization of the circular cylinder, the sphere and the plane by using stationary isothermic surfaces in Euclidean 3-space is almost completed, and in particular, similar good characterization of the right helicoid is wanted.

(2) Problems of partial differential equations on composite materials: Recently, we considered the heat diffusion over composite media and we got a characterization of planar layers by using either stationary isothermic surfaces or surfaces with the constant flow property among multi-layered heat conductors. In particular, we are interested in problems dealing with composite materials.

(3) Interaction between diffusion and geometry of domain: The shape of the heat conductor is deeply related to the initial heat diffusion. Diffusion equations we consider are the heat equation, the porous medium type equation, and their related equations.

(4) Shapes of solutions of elliptic equations: In general, solutions of elliptic equations describe steady states after a sufficiently long time. Liouville-type theorems characterize hyperplanes as graphs of entire solutions with some restriction. Overdetermined boundary value problems characterize some symmetrical domains. Isoperimetric inequalities accompanied by boundary value problems characterize shapes of the solutions which give the equalities.

(5) The point of view of inverse problems: Partial differential equations appear in models describing natural phenomena. There are many interesting problems which characterize some geometry in some reasonable way from the point of view of inverse problems.

Research History 10

2025/04 - Present

Tohoku University　Admissions Center　Program-Specific Professor
2022/04 - Present

Tohoku University　Professor Emeritus
2024/04 - 2025/03

Tohoku University　Institute for Excellence in Higher Education　Program-Specific Professor
2022/04 - 2024/03

Tsuda University　Part-time lecturer
2012/04 - 2022/03

Tohoku University　Professor
2008/04 - 2012/03

Hiroshima University　Professor
2002/02 - 2008/03

Ehime University　Professor
1993/04 - 2002/01

Ehime University　Associate Professor
1989/04 - 1993/03

Tokyo Institute of Technology　Research Associate
1988/04 - 1989/03

Numazu College of Technology　Lecturer

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Education 4

Tokyo Metropolitan University　Graduate School of Science

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

- 1986
Tokyo Institute of Technology　School of Science

- 1979
Tokyo Institute of Technology　Faculty of Science

- 1979

Committee Memberships 2

日本数学会　代議員

2007 - 2007
日本数学会　`数学' 編集委員

2003 - 2007

Professional Memberships 2

American Mathematical Society

- 2021/12
Mathematical Society of Japan

Research Interests 5

composite materials
elliptic and parabolic equations
point of view of inverse problems
geometric properties of solutions
theory of partial differential equations

Research Areas 1

Natural sciences / Basic analysis / Partial differential equations

Awards 1

2012 Analysis Prize

2012/09　The Mathematical Society of Japan　Geometry on the domain via the isothermic set for diffusion equations

Papers 63

The stationary critical points of the fractional heat flow Peer-reviewed

Nicola De Nitti, Shigeru Sakaguchi

Journal of Differential Equations　455　113996-113996　2026/02
Publisher: Elsevier BV
DOI： 10.1016/j.jde.2025.113996 　

ISSN： 0022-0396
Non-isoparametric Serrin domains of $\mathbb{S}^3$ with connected toric boundary

Andrea Bisterzo, Shigeru Sakaguchi

arXiv:2511.16531v1　2025/11
Interaction Between Initial Behavior of Temperature and the Mean Curvature of the Interface in Two-Phase Heat Conductors Peer-reviewed

Shigeru Sakaguchi

The Journal of Geometric Analysis　35　(8)　2025/06/11
Publisher: Springer Science and Business Media LLC
DOI： 10.1007/s12220-025-02058-5 　

ISSN： 1050-6926

eISSN： 1559-002X

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Abstract 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.
Symmetry results for some overdetermined obstacle problems Peer-reviewed

Nicola De Nitti, Shigeru Sakaguchi

Proc. Amer. Math. Soc.　153　2919-2932　2025/05/20
Two extremum problems for Neumann eigenvalues Peer-reviewed

Lorenzo Cavallia, Kei Funano, Antoine Henrot, Antoine Lemenant, Ilaria Lucardesi, Shigeru Sakaguchi

J. Anal. Math.　155　657-697　2025/03/24
A characterization of a hyperplane in two-phase heat conductors Peer-reviewed

Lorenzo Cavallina, Shigeru Sakaguchi, Seiichi Udagawa

Communications in Analysis and Geometry　31　(7)　1867-1888　2024/08/10
Publisher: International Press of Boston
DOI： 10.4310/cag.2023.v31.n7.a9 　

ISSN： 1019-8385

eISSN： 1944-9992
A symmetry theorem in two-phase heat conductors Invited Peer-reviewed

Hyeonbae Kang, Shigeru Sakaguchi

Mathematics in Engineering　5　(3)　1-7　2022/11
Publisher: American Institute of Mathematical Sciences (AIMS)
DOI： 10.3934/mine.2023061 　

ISSN： 2640-3501

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<p lang="fr"><abstract><p>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, \alpha} $, 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.</p></abstract></p>
Patterns with prescribed numbers of critical points on topological tori Peer-reviewed

Putri Zahra Kamalia, Shigeru Sakaguchi

Complex Variables and Elliptic Equations　67　(10)　2382-2396　2022/10/03
Publisher: Informa UK Limited
DOI： 10.1080/17476933.2021.1924157 　

ISSN： 1747-6933

eISSN： 1747-6941
Existence of weakly neutral coated inclusions of general shape in two dimensions Peer-reviewed

Hyeonbae Kang, Xiaofei Li, Shigeru Sakaguchi

Applicable Analysis　101　(4)　1330-1353　2022/03/04
Publisher: Informa UK Limited
DOI： 10.1080/00036811.2020.1781821 　

ISSN： 0003-6811

eISSN： 1563-504X
The principal eigenfunction of the Dirichlet Laplacian with prescribed numbers of critical points on the upper half of a topological torus Peer-reviewed

Putri Zahra Kamalia, Shigeru Sakaguchi

Journal of Mathematical Analysis and Applications　509　(2)　125972　2022/01
Large time behavior of temperature in two-phase heat conductors Peer-reviewed

Hyeonbae Kang, Shigeru Sakaguchi

Journal of Differential Equations　303　268-276　2021/12
Polarization tensor vanishing structure of general shape: Existence for small perturbations of balls Peer-reviewed

Hyeonbae Kang, Xiaofei Li, Shigeru Sakaguchi

Asymptotic Analysis　125　(1-2)　101-132　2021/09
The double queen Dido's problem Peer-reviewed

Lorenzo Cavallina, Antoine Henrot, Shigeru Sakaguchi

Journal of Geometric Analysis　31　(8)　7750-7772　2021/08
Publisher: Springer Science and Business Media LLC
DOI： 10.1007/s12220-020-00549-1 　

ISSN： 1050-6926

eISSN： 1559-002X
Neutral inclusions, weakly neutral inclusions, and an over-determined problem for confocal ellipsoids Peer-reviewed

Yong-Gwan Ji, Hyeonbae Kang, Xiaofei Li, Shigeru Sakaguchi

`` Geometric Properties of Parabolic and Elliptic PDE's ", Springer INdAM Series,　47　151-181　2021/06
Two-Phase Heat Conductors with a Surface of the Constant Flow Property Peer-reviewed

Lorenzo Cavallina, Rolando Magnanini, Shigeru Sakaguchi

Journal of Geometric Analysis　31　(1)　312-345　2021/01
Publisher: Springer Science and Business Media LLC
DOI： 10.1007/s12220-019-00262-8 　

ISSN： 1050-6926

eISSN： 1559-002X
A construction of patterns with many critical points on topological tori Peer-reviewed

Putri Zahra Kamalia, Shigeru Sakaguchi

Nonlinear Differential Equations and Applications NoDEA　27　(4)　27:39　2020/07
Some characterizations of parallel hyperplanes in multi-layered heat conductors Peer-reviewed

Shigeru Sakaguchi

Journal de Mathématiques Pures et Appliquées　140　185-210　2020/06
Two-phase heat conductors with a stationary isothermic surface and their related elliptic overdetermined problems Invited Peer-reviewed

Shigeru Sakaguchi

RIMS Kôkyûroku Bessatsu　B80　113-132　2020/04
A simple proof of a strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation Peer-reviewed

Masaki Ohnuma, Shigeru Sakaguchi

Nonlinear Analysis Theory, Methods and Applications　181　180-188　2019/04
Stationary isothermic surfaces in Euclidean 3-space Peer-reviewed

Rolando Magnanini, Daniel Peralta-Salas, Shigeru Sakaguchi

MATHEMATISCHE ANNALEN　364　(1-2)　97-124　2016/02

DOI： 10.1007/s00208-015-1212-1 　

ISSN： 0025-5831

eISSN： 1432-1807
SOLUTIONS OF ELLIPTIC EQUATIONS WITH A LEVEL SURFACE PARALLEL TO THE BOUNDARY: STABILITY OF THE RADIAL CONFIGURATION Peer-reviewed

Giulio Ciraolo, Rolando Magnanini, Shigeru Sakaguchi

JOURNAL D ANALYSE MATHEMATIQUE　128　(1)　337-353　2016/02

DOI： 10.1007/s11854-016-0011-2 　

ISSN： 0021-7670

eISSN： 1565-8538
Two-phase heat conductors with a stationary isothermic surface Invited Peer-reviewed

Shigeru Sakaguchi

Rendiconti dell'Istituto di Matematica dell'Universita di Trieste　48　(1)　167-187　2016
Publisher: EUT Edizioni Universita di Trieste
DOI： 10.13137/2464-8728/13155 　

ISSN： 0049-4704
Symmetry Problems on Stationary Isothermic Surfaces in Euclidean Spaces Peer-reviewed

Shigeru Sakaguchi

GEOMETRIC PROPERTIES FOR PARABOLIC AND ELLIPTIC PDE'S　176　231-239　2016

DOI： 10.1007/978-3-319-41538-3_13 　

ISSN： 2194-1009
An over-determined boundary value problem arising from neutrally coated inclusions in three dimensions Peer-reviewed

Hyeonbae Kang, Hyundae Lee, Shigeru Sakaguchi

ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA-CLASSE DI SCIENZE　16　(4)　1193-1208　2016

ISSN： 0391-173X

eISSN： 2036-2145
When does the heat equation have a solution with a sequence of similar level sets? Peer-reviewed

Tatsuki Kawakami, Shigeru Sakaguchi

ANNALI DI MATEMATICA PURA ED APPLICATA　194　(6)　1595-1605　2015/12

DOI： 10.1007/s10231-014-0435-1 　

ISSN： 0373-3114

eISSN： 1618-1891
Symmetry of minimizers with a level surface parallel to the boundary Peer-reviewed

Giulio Ciraolo, Rolando Magnanini, Shigeru Sakaguchi

JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY　17　(11)　2789-2804　2015

DOI： 10.4171/JEMS/571 　

ISSN： 1435-9855
INTERACTION BETWEEN FAST DIFFUSION AND GEOMETRY OF DOMAIN Peer-reviewed

Shigeru Sakaguchi

KODAI MATHEMATICAL JOURNAL　37　(3)　680-701　2014/10

DOI： 10.2996/kmj/1414674616 　

ISSN： 0386-5991
Matzoh ball soup revisited: the boundary regularity issue Peer-reviewed

Rolando Magnanini, Shigeru Sakaguchi

MATHEMATICAL METHODS IN THE APPLIED SCIENCES　36　(15)　2023-2032　2013/10

DOI： 10.1002/mma.1551 　

ISSN： 0170-4214
Stationary level surfaces and Liouville-type theorems characterizing hyperplanes Peer-reviewed

Shigeru Sakaguchi

Springer INdAM Series　2　269-282　2013
Publisher: Springer International Publishing
DOI： 10.1007/978-88-470-2841-8_17 　

ISSN： 2281-5198 2281-518X
Interaction between nonlinear diffusion and geometry of domain Peer-reviewed

Rolando Magnanini, Shigeru Sakaguchi

JOURNAL OF DIFFERENTIAL EQUATIONS　252　(1)　236-257　2012/01

DOI： 10.1016/j.jde.2011.08.017 　

ISSN： 0022-0396
Optimization problems on general classes of rearrangements Peer-reviewed

F. Cuccu, G. Porru, S. Sakaguchi

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS　74　(16)　5554-5565　2011/11

DOI： 10.1016/j.na.2011.05.039 　

ISSN： 0362-546X

eISSN： 1873-5215
Some fully nonlinear elliptic boundary value problems with ellipsoidal free boundaries Peer-reviewed

Cristian Enache, Shigeru Sakaguchi

MATHEMATISCHE NACHRICHTEN　284　(14-15)　1872-1879　2011/10

DOI： 10.1002/mana.200810170 　

ISSN： 0025-584X
A LIOUVILLE-TYPE THEOREM FOR SOME WEINGARTEN HYPERSURFACES Peer-reviewed

Shigeru Sakaguchi

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S　4　(4)　887-895　2011/08

DOI： 10.3934/dcdss.2011.4.887 　

ISSN： 1937-1632

eISSN： 1937-1179
Nonlinear diffusion with a bounded stationary level surface Peer-reviewed

Rolando Magnanini, Shigeru Sakaguchi

ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE　27　(3)　937-952　2010/05

DOI： 10.1016/j.anihpc.2009.12.001 　

ISSN： 0294-1449
Stationary isothermic surfaces and some characterizations of the hyperplane in the N-dimensional Euclidean space Peer-reviewed

Rolando Magnanini, Shigeru Sakaguchi

JOURNAL OF DIFFERENTIAL EQUATIONS　248　(5)　1112-1119　2010/03

DOI： 10.1016/j.jde.2009.11.017 　

ISSN： 0022-0396
Polygonal heat conductors with a stationary hot spot Peer-reviewed

Rolando Magnanini, Shigeru Sakaguchi

JOURNAL D ANALYSE MATHEMATIQUE　105　1-18　2008/01

DOI： 10.1007/s11854-008-0029-1 　

ISSN： 0021-7670
A linear-quadratic control problem with discretionary stopping Peer-reviewed

Shigeaki Koike, Hiroaki Morimoto, Shigeru Sakaguchi

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B　8　(2)　261-277　2007/09

ISSN： 1531-3492
Stationary isothermic surfaces for unbounded domains Peer-reviewed

Rolando Magnanini, Shigeru Sakaguchi

INDIANA UNIVERSITY MATHEMATICS JOURNAL　56　(6)　2723-2738　2007

DOI： 10.1512/iumj.2007.56.3150 　

ISSN： 0022-2518
Interaction between degenerate diffusion and shape of domain Peer-reviewed

R. Magnanini, S. Sakaguchi

PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS　137　(2)　373-388　2007

DOI： 10.1017/S0308210505001071 　

ISSN： 0308-2105
Stationary isothermic surfaces and uniformly dense domains Peer-reviewed

R. Magnanini, J. Prajapat, S. Sakaguchi

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY　358　(11)　4821-4841　2006/11

DOI： 10.1090/S0002-9947-06-04145-6 　

ISSN： 0002-9947
Behavior of spatial critical points and zeros of solutions of diffusion equations

S. Sakaguchi

American Mathematical Society Translations Series 2　215　15-31　2005
On heat conductors with a stationary hot spot Peer-reviewed

Rolando Magnanini, Shigeru Sakaguchi

ANNALI DI MATEMATICA PURA ED APPLICATA　183　(1)　1-23　2004/03

DOI： 10.1007/s10231-003-0077-1 　

ISSN： 0373-3114
On stationary hot spots and isothermic surfaces

R Magnanini, S Sakaguchi

PROGRESS IN ANALYSIS, VOLS I AND II　877-881　2003
Matzoh ball soup: Heat conductors with a stationary isothermic surface Peer-reviewed

R Magnanini, S Sakaguchi

ANNALS OF MATHEMATICS　156　(3)　931-946　2002/11

ISSN： 0003-486X
Stationary critical points of the heat flow in the plane Peer-reviewed

R Magnanini, S Sakaguchi

JOURNAL D ANALYSE MATHEMATIQUE　88　383-396　2002

ISSN： 0021-7670
Regularity of the interfaces with sign changes of solutions of the one-dimensional porous medium equation Peer-reviewed

S Sakaguchi

JOURNAL OF DIFFERENTIAL EQUATIONS　178　(1)　1-59　2002/01

DOI： 10.1006/jdeq.2000.4002 　

ISSN： 0022-0396
拡散方程式の解の空間臨界点と零点の挙動 Invited Peer-reviewed

坂口茂

数学　54　(3)　249-264　2002
Publisher: The Mathematical Society of Japan
DOI： 10.11429/sugaku1947.54.249 　

ISSN： 0039-470X
Stationary critical points of the heat flow in spaces of constant curvature Peer-reviewed

S Sakaguchi

JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES　63　(2)　400-412　2001/04

ISSN： 0024-6107
Spatial critical points not moving along the heat flow II: The centrosymmetric case Peer-reviewed

R Magnanini, S Sakaguchi

MATHEMATISCHE ZEITSCHRIFT　230　(4)　695-712　1999/04

ISSN： 0025-5874
When are the spatial level surfaces of solutions of diffusion equations invariant with respect to the time variable? Peer-reviewed

S Sakaguchi

JOURNAL D ANALYSE MATHEMATIQUE　78　219-243　1999

ISSN： 0021-7670
On a class of L¹-illposed quasilinear parabolic equations Peer-reviewed

S. Sakaguchi, T. Suzuki

Advances in Differential Equations　4　(5)　671-696　1999
Nonexistence of solutions for a degenerate parabolic equation describing imperfect ignition Peer-reviewed

S Sakaguchi, T Suzuki

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS　31　(5-6)　665-669　1998/03

ISSN： 0362-546X
Interior imperfect ignition cannot occur on a set of positive measure Peer-reviewed

S Sakaguchi, T Suzuki

ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS　142　(2)　143-153　1998

ISSN： 0003-9527
Spatial critical points of nonnegative solutions of the evolution p-Laplacian equation : The fast diffusion case Peer-reviewed

S. Sakaguchi

Differential and Integral Equations　10　(6)　1049-1063　1997
The spatial critical points not moving along the heat flow Peer-reviewed

R Magnanini, S Sakaguchi

JOURNAL D ANALYSE MATHEMATIQUE　71　237-261　1997

ISSN： 0021-7670
The number of peaks of nonnegative solutions to some nonlinear degenerate parabolic equations Peer-reviewed

S Sakaguchi

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS　203　(1)　78-103　1996/10

DOI： 10.1006/jmaa.1996.0368 　

ISSN： 0022-247X
MOVEMENT OF HOT-SPOTS OVER UNBOUNDED-DOMAINS IN RN Peer-reviewed

S JIMBO, S SAKAGUCHI

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS　182　(3)　810-835　1994/03

ISSN： 0022-247X
Critical points of solutions to the obstacle problem in the plane Peer-reviewed

S. Sakaguchi

Ann. Scuola Norm. Sup. Pisa Ser IV　21　(2)　157-173　1994
UNIQUENESS OF THE CRITICAL-POINT OF THE SOLUTIONS TO SOME SEMILINEAR ELLIPTIC BOUNDARY-VALUE-PROBLEMS IN R2 Peer-reviewed

S SAKAGUCHI

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY　319　(1)　179-190　1990/05

ISSN： 0002-9947
UNIQUENESS OF CRITICAL-POINT OF THE SOLUTION TO THE PRESCRIBED CONSTANT MEAN-CURVATURE EQUATION OVER CONVEX DOMAIN IN R2 Peer-reviewed

S SAKAGUCHI

RECENT TOPICS IN NONLINEAR PDE IV　160　129-151　1989
Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems Peer-reviewed

S. Sakaguchi

Ann. Scuola Norm. Sup. Pisa Ser IV　14　(3)　403-421　1987
COINCIDENCE SETS IN THE OBSTACLE PROBLEM FOR THE P-HARMONIC OPERATOR Peer-reviewed

S SAKAGUCHI

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY　95　(3)　382-386　1985/11

ISSN： 0002-9939
Star shaped coincidence sets in the obstacle problem Peer-reviewed

S. Sakaguchi

Ann. Scuola Norm. Sup. Pisa Ser IV　11　(1)　123-128　1984

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Misc. 7

関数の形 Invited

坂口茂

数理科学　53　(10)　35-39　2015/10
Special Issue on "Geometric Properties for Parabolic and Elliptic PDE's" Preface

Shigeru Sakaguchi

KODAI MATHEMATICAL JOURNAL　37　(3)　2014/10

ISSN： 0386-5991
非線形拡散, 領域の幾何, および Liouville 型定理 (非線形拡散の数理)

坂口茂

数理解析研究所講究録　1810　139-152　2012/10
Publisher: 京都大学
ISSN： 1880-2818
PREFACE: GEOMETRIC PROPERTIES FOR PARABOLIC AND ELLIPTIC PDE'S

Filippo Gazzola, Rolando Magnanini, Shigeru Sakaguchi

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S　4　(4)　I-II　2011/08

DOI： 10.3934/dcdss.2011.4.4i 　

ISSN： 1937-1632

eISSN： 1937-1179
Stationary isothermic surfaces and a characterization of the spherical cylinder (Problems in the Calculus of Variations and Related Topics)

Sakaguchi Shigeru

RIMS Kokyuroku　1628　48-57　2009/02
Publisher: Kyoto University
ISSN： 1880-2818
微分方程式における自明解 Invited

坂口茂

数理科学　43　(11)　22-26　2005/11
Spatial critical points not moving along the heat flow II: The centrosymmetric case (vol 230, pg 695, 1999)

R Magnanini, S Sakaguchi

MATHEMATISCHE ZEITSCHRIFT　232　(2)　389-389　1999/10

ISSN： 0025-5874

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Books and Other Publications 1

Geometry of solutions of partial differential equations

SAKAGUCHI Shigeru

Saiensu-sha Co., Ltd. Publishers　2017/03

Research Projects 26

Geometric analysis of partial differential equations and inverse problems

SAKAGUCHI Shigeru

Offer Organization: Japan Society for the Promotion of Science

System: Grant-in-Aid for Scientific Research (C)

2022/04 - 2026/03
Analysis of elliptic operators and its applications to Geometric Function Theory

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B)

Institution: Tohoku University

2017/04/01 - 2022/03/31
Geometry of partial differential equations and inverse problems Competitive

SAKAGUCHI Shigeru

Offer Organization: Japan Society for the Promotion of Science

System: Grant-in-Aid for Scientific Research (B)

2018/04 - 2022/03
Transmission problems in composite media and overdetermined problems with transmission conditions Competitive

SAKAGUCHI Shigeru

Offer Organization: Japan Society for the Promotion of Science

System: Grant-in-Aid for Challenging Exploratory Research

2016/04 - 2019/03
Geometry of solutions of partial differential equations and the inverse problems accompanied by it Competitive

SAKAGUCHI Shigeru

Offer Organization: Japan Society for the Promotion of Science

System: Grant-in-Aid for Scientific Research (B)

2014/04 - 2018/03
Analysis of the nonlinear elliptic eigenvalue problems and inverse problems

SHIBATA TETSUTARO, SAKAGUCHI SHIGERU, TANAKA KAZUNAGA, KURATA KAZUHIRO

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: Hiroshima University

2013/04/01 - 2017/03/31

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In this study, we consider the inverse and direct bifurcation problems of nonlinear eigenvalue problems. For the direct problems, we establish the precise asymptotic formulas for the eigenvalue problems which have biological and physical background. For the inverse bifurcation problems, we consider the typical inverse problem for elliptic equations to understand well the structure of inverse problems. In particular, we concentrate on the study of the global structure of bifurcation curves for some nonlinear ordinary differential equations. We apply these precise asymptotic properties to the typical inverse bifurcation problem and obtained some fundamental and new results in this direction.
Search for new isoperimetric inequalities relating to elliptic equations Competitive

SAKAGUCHI Shigeru

Offer Organization: Japan Society for the Promotion of Science

System: Grant-in-Aid for challenging Exploratory Research

2013/04 - 2016/03
Diffusion and geometry of domain Competitive

SAKAGUCHI Shigeru

Offer Organization: Japan Society for the Promotion of Science

System: Grant-in-Aid for Scientific Research (B)

Category: Grant-in-Aid for Scientific Research (B)

2008/04 - 2013/03

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Partial differential equations describing diffusion phenomena have been widely considered. To know the relationship between the behavior of solutions and the geometry of domain, we showed both the relationship between the initial behavior and the curvatures of the boundary and that between the existence of a stationary level surface with time and the symmetry of domain. In particular, we obtained characterizations of the sphere, the hyperplane, and the circular cylinder involving a stationary level surface. These yielded a new development of inverse problems determining the geometry of domain. Also, as a by-product, we obtained Liouville-type theorems for viscosity solutions of fully nonlinear elliptic partial differential equations describing an important class of Weingarten hypersurfaces.
Asymptotic analysis and inverse problems of the nonlinear elliptic eigenvalue problems

SHIBATA Tetsutaro, SAKAGUCHI Shigeru, TANAKA Kazunaga, KURATA Kazuhiro

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: Hiroshima University

2009 - 2012

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In this study, we investigated the precise asymptotic properties of the eigenfunctions and eigenvalues of nonlinear elliptic equations, and studied the inverse eigenvalue problems associated with thedirect problems. We clarify the global and local structures of the bifurcation curves for the equations with several types of nonlinear terms. As for inverse problems, we mainly studied the inverse bifurcation problems for logistic type equations. By using the theory of ordinary differential equations and the asymptotic expansion formulas for bifurcation curves, we determined the unknown nonlinear terms by the asymptotic behaviors of the bifurcation curves.
Synthetic study of nonlinear evolution equation and its related topics

OTANI Mitsuharu, YAMADA Yoshio, TANAKA Kazunaga, NISHIHARA Kenji, ISHII Hitoshi, OZAWA Tohru, OGAWA Takayoshi, KENMOCHI Nobuyuki, KOIKE Shigeaki, SAKAGUCHI Shigeru, SUZUKI Takashi, HAYASHI Nakao, IDOGAWA Tomoyuki, ISHIWATA Michinori, AKAGI Gorou

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B)

Institution: Waseda University

2009 - 2012

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Various types of nonlinear PDEs (nonlinear elliptic equations, nonlinear diffusion equations, nonlinear wave equations, nonlinear Schrodinger equations) arising in physics and engineering were synthetically studied from the viewpoint of the theory of nonlinear evolution equations by using the techniques from the theory of nonlinear functional analysis, the theory of functions of a real variable, the theory of ordinary differential equations and the calculus of variations.
定曲率空間における不変な等温面と一様分離曲面 Competitive

坂口茂

System: 科学研究費補助金（萌芽研究）

2006/04 - 2009/03
Bifurcation structure of stationary solutions for a reaction-diffusion system with density-dependent diffusion

KAN-ON Yukio, YANAGI Shigenori, SAKAGUCHI Shigeru

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: Ehime University

2007 - 2009

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We consider a competition-diffusion system with the density-dependent diffusion, which describes the dynamics of the population density for a two competing species community, and we study the bifurcation structure of radially symmetric stationary solutions of the system for the case where the habitat of the community is the inside of a certain ball. At this time, the local bifurcation structure around the constant stationary solution can be determined by the value of the integral whose integrand is the cubic of the Bessel function of the first kind with the positive weight function. In this research, when the dimension of the habitat is not bigger than 3, we determine the sign of the value of the integral by employing the mathematical method and the numerical verification method. The result of this research is applicable for determining the local bifurcation structure around the constant stationary solution to not only the competition-diffusion system but also the reaction-diffusion system.
Symmetry and geometric properties of solutions of nonlinear parabolic initial-boundary value problems Competitive

SAKAGUCHI Shigeru

Offer Organization: Japan Society for the Promotion of Science

System: Grant-in-Aid for JSPS Fellows

2006/10 - 2008/09
Behavior of spatial critical points and level surfaces of solutions of partial differential equations and shapes of the solutions Competitive

SAKAGUCHI Shigeru

Offer Organization: Japan Society for the Promotion of Science

System: Grant-in-Aid for Scientific Research (B)

Category: Grant-in-Aid for Scientific Research (B)

Institution: Ehime University

2003/04 - 2007/03

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The main purpose of this project is to study the relationship between shape of solutions of partial differential equations (behavior of spatial critical points and level surfaces) and shape of domains. We obtain the following: 1. Consider the initial value problem for the heat equation in Euclidean space with initial data being the characteristic function of a domain Ω. We introduce the geometrical condition that Ω is uniformly dense in Γ, which is necessary for the solution to have a stationary isothermic surface Γ. The uniformly dense domains are classified. (Trans. Amer. Math. Soc., 358 (2006), 4821-4841 ) 2. Consider the initial-boundary value problem for linear and nonlinear diffusion equations in a bounded domain Ω in Euclidean space with zero initial data and with positive constant boundary value. Let B be a ball in Ω touching ∂ Ω only at one point. Then the asymptotic formula of the integral of the solution over B at the initial time involves the principal curvatures of ∂ Ω at the point. This fact explains the relationship between diffusion and the geometry of Ω. ( Proc. Royal Soc. Edinburgh Sect. A, 137 (2007), 373-388 ) 3. Consider the initial-Dirichlet problem for the heat equation with positive constant initial data in a domain Ω with unbounded boundary ∂ Ω in Euclidean space. Under various global assumptions on Ω, we prove that if the solution has a stationary isothermic surface, then ∂ Ω consists of hyperplanes. ( Indiana University Math. J., to appear ) 4. Consider the initial-Dirichlet problem for the heat equation with positive constant initial data over a bounded convex polygonal domain Ω in the plane. When Ω has m (m≦5) sides and every side of ∂ Ω touches the inscribed circle, we obtain a new necessary condition for Ω having a stationary hot spot. (submitted for the publication ) 5. In the initial-boundary value problem for the linear diffusion equation with zero initial data and with positive constant boundary value, a result of Varadhan (1967) is such that the initial behavior of the solution is described through the distance function to the boundary. We extend this result to some nonlinear diffusion equations, which are uniformly parabolic, with the aid of the theory of viscosity solutions. Moreover, we give a characterization of the sphere through the solution having a stationary level surface in case of nonlinear diffusion equations. (in preparation )
Study of finite element methods for nonlinear problems and its error analysis

TSUCHIYA Takuya, SUZUKI Takashi, SAKAGUCHI Shigeru

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: Ehime University

2005 - 2006

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1. We have studied on the free boundary problem called the filtration problem or the dam problem. In any cases, some level-set approaches are taken to analyze the dam problem theoretically. However, to compute the numerical solutions, iterative schemes are used in engineering usually. Since rigorous analysis of iterative schemes is very difficult, there are very few mathematical results on iterative schemes for the dam problem. In our study, we present an mathematical framework for convergence analysis of numerical iterative methods for the dam problem (see [1,3]). 2. To design "good" iterative scheme for free boundary problems, it is important to understand how quantities related to the problem will vary when the boundary of the domain is perturbed. Such a variation is called the Hadamard variation. In our study, we have succeeded to compute the first variation of the velocity potential with respect to boundary perturbation. T. Suzuki, T. Tsuchiya, "Weak formulation of Hadamard variation and its application to the filtration problem", preprint. 3. We analyze the piecewise quadratic finite element method applied to 2-point boundary value problems. We use "Yamamoto's principle" for it. Since Yamamoto's principle is a powerful tool, we can deal with cases which the standard theory cannot handle. We conform that all standard results are still valid even if coefficient functions are only piecewise smooth (see [3]).
Boundary value problems for higher-order nonlinear ordinary differential equations

NAITO Manabu, SAKAGUCHI Shigeru, HASHIMOTO Takahiro, USAMI Hiroyuki

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B)

Institution: EHIME UNIVERSITY

2003 - 2005

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The aim of this research is to study the existence, the uniqueness, the number of zeros and the distribution of zeros of solutions of boundary value problems for higher-order ordinary differential equations, and to obtain the detailed information for the set of solutions of higher-order elliptic differential equations on the base of the results for ordinary differential equations. The new results are as follows : 1.For a fourth-order or even-order quasilinear ordinary differential equation, necessary and sufficient conditions for the existence of a positive solution are obtained. For a 2-system of the second-order ordinary differential equations and a 2-system of the second-order elliptic equations, the existence of a positive solution is discussed. 2.A degenerate elliptic equation with arbitrary nonlinearity is considered on exterior domain, and necessary and sufficient conditions for all solutions to be oscillatory are established. A degenerate elliptic equation is considered on strip-like domain, and the nonexistence of a positive solution is discussed. 3.For an initial value problem for the heat equation, stationary isothermic surfaces and uniformly dense domains are discussed, and an interaction between degenerate diffusion and shape of domain is discussed. 4.For a singular eigenvalue problem to a higher-order ordinary differential equation on an infinite interval, it is shown that there is a countable sequence of eigenfunctions having exactly n zeros. 5.For a fourth-order nonlinear elliptic differential equation including the poly-harmonic operator, the existence results and nonexistence results are obtained.
A new development of variational problems under constraints with mixed boundary conditions Competitive

SAKAGUCHI Shigeru

Offer Organization: Japan Society for the Promotion of Science

System: Grant-in-Aid for Exploratory Research

2002/04 - 2004/03
Behavior of spatial critical points and zeros of solutions of partial differential equations Competitive

SAKAGUCHI Shigeru

Offer Organization: Japan Society for the Promotion of Science

System: Grant-in-Aid for Scientific Research (B)

Category: Grant-in-Aid for Scientific Research (B)

Institution: Ehime University

2000/04 - 2003/03

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1. Let Ω be a domain in the N-dimensional Euclidean space, and consider the initial-Dirichlet problem for initial data being a positive constant. Suppose that D is a domain satisfying the interior cone condition and D^^-⊂Ω. We considered the question how the boundary ∂D is a stationary isothermic surface of the solution, and obtained the following two theorems : (i) Let Ω be either a bounded domain or an exterior domain satisfying the exterior sphere condition. If ∂D is a stationary isothermic surface, then ∂Ω must be a sphere. (ii) Let Ω be an unbounded domain satisfying the uniform exterior sphere condition, and suppose that ∂Ω contains a nonempty open subset where the principal curvatures of ∂Ω with respect to the exterior normal direction to ∂Ω are nonnegative. Furthermore, assume that, for any r > 0, ∂Ω contains the graph over a (N -1)-dimensional ball with radius r > 0. If ∂D is a stationary isothermic surface, then ∂Ω must be either a hyperplane or two parallel hyperplanes. 2. There is a conjecture of Chamberland and Siegel (1997) concerning the hot spots of solutions of the heat equation. Let Ω be a bounded domain in the Euclidean space containing the origin, and consider the initial-Dirichlet problem for initial data being a positive constant. The conjecture stated that if the origin is a stationary hot spot, then Ω is invariant under the action of an essential subgroup G of orthogonal transformations. Concerning this conjecture, we obtained the following four theorems when the space dimension is two : (i) Let Ω be a triangle. If the origin is a stationary hot spot, then Ω must be an equilateral triangle centered at the origin. (ii) Let Ω be a convex quadrangle, then Ω must be a parallelogram centered at the origin. (iii) If the origin is a stationary hot spot, then Ω is not a non-convex quadrangle. (iv) Let Ω be a convex m-polygon ( m = 5 or 6 ). Suppose that the inscribed circle centered at the origin touches every side of Ω, and suppose that the origin is a stationary hot spot. Then, if m = 5, Ω must be a regular pentagon centered at the origin, and if m = 6, Ω must be invariant under the rotation of one of three angles, π/3, 2π/3, and π.
Mathematical Theory of Error Analysis of Finite Element Methods

TSUCHIYA Takuya, SAKAGUTI Sigeru, FANG Qing

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: Ehime University

2002 - 2003

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In this research project, we have tried to develop a mathematical theory of error analysis of finite element methods, and have obtained the following results. (1)We have consider on error analysis of Galerkin methods applied to (non-) linear equations with a (linear or nonlinear) compact term defined in Hilbert space setting. We have found a simple way of developing mathematical theory of error analysis of Galerkin methods applied to such equations. (2)We have proved rigorously the convergence of trial free boundary methods applied to the Dam Problem, which is a typical elliptic free boundary problem. (3)We have consider on error analysis of the piecewise quadratic finite element method applied to 2-point boundary value problems defined on 1-dimensional bounded interval. We have found that all known error bounds are still valid even if coefficient function of the principle term is not-continuous or entirely positive.
Study on Nonlinear Evolution Equations and Nonlinear Elliptic Equations

OTANI Mitsuharu, ISHII Hitoshi, TANAKA Kazunaga, YAMADA Yoshio, SAKAGUCHI Shigeru, SUZUKI Takashi

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B)

Institution: Waseda University

2000 - 2003

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(1)"L^∞-energy method" is invented. This assures the high differentiablity of solutions of quasilinear parabolic equations. By this method, the existence of W^<1. ∞>-solutions for a general doubly nonlinear parabolic equations and the open problem : "porous medium equations admit C^∞-solutions?" is solved affirmatively. Recent studies suggest that this gives a quite powerful tool for various problems. (2)"The theory of nonmonotone perturbations for subdifferentials " is extended to Banach space setting. By this theory, we can treat the existence and regularity of solutions for degenerate parabolic equations in a more natural way than Galerkin' s method and open problems, left unsolved in the usual way, were solved. (3)A Concentration Compactness (CC) theory with partial symmetry is given. The usual CC theory is known to be useful to analyze the problem with lack of compactness. On the other hand, the high symmetry such as the radial symmetry often recovers the compactness. It is studied how the partial symmetry not enough to recover compactness is reflected to CC theory. By this theory, the existence of nontrivial solutions is proved for some quasilinear elliptic equations in infinite cylindrical domains. (4)The classical "Principle of Symmetric Criticality (PSC)" by R.Palais assures that under suitable conditions, critical points in the subspace with the symmetry give real critical points in the whole space, but is restricted to the system with variational structures. PSC is extended to a more general theory which covers the elliptic systems without full symmetry or evolution equations including time evolution terms. (5)A new degree theory is established. It can teat mutivuled operators including subdifferential operators and cover nonlinear PDE with various multivaluedness nature. (6)The theory of nonmonotone perturbations for subdifferentials is ameliorated to cover the initial-boundary value problems and time periodic problems for magneto-micropolar fluid equations.
Study on the number of zeros of solutions to higher-order nonlinear differential equations

NAITO Manabu, USAMI Hiroyuki, HASHIMOTO Takahiro, SAKAGUCHI Sigeru

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: EHIME UNIVERSITY

2001 - 2002

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The aim of this research is to study the number of zeros and the distribution of zeros of solutions of higher-order ordinary differential equations with Emden-Fowler type nonlinearity, and to discuss the oscillatory properties of solutions of higher-order elliptic differential equations on the base of the results for ordinary differential equations. The new results and knowledge obtained in the two years are as follows : 1. For a singular eigenvalue problem to the linear and sublinear higher-order ordinary differential equations on an infinite interval [a, +∞), it is shown that there is a countable sequence of eigenvalues and that the n-th eigenfunction has exactly n zeros. 2. For a regular eigenvalue problem to the sublinear higher-order ordinary differential equations on a finite interval, it is shown that there is a countable sequence of eigenvalues and that the n-th eigenfunction has exactly n zeros. 3. For the second-order half-linear ordinary differential equations, it is shown that the number of zeros of specific nonoscillatory solutions changes one by one as a parameter varies. 4. For the second-order half-linear ordinary differential equations, a generalization and an analogue of the Sturm-Liouville linear regular eigenvalue problem are obtained. 5. For the four-dimensional Emden-Fowler differential systems and the fourth-order quasilinear differential equations of Emden-Fowler type, a necessary and sufficinet condition for the existence of nonoscillatory solutions with specific asymptotic properties as t →∞ is established, and a sufficient condition for oscillation of all solutions is also obtained. 6. For the two-dimensional semilinear elliptic differential systems of the Laplace type, an analogue of Liouville's theorem is established. 7. For the fourth-order nonlinear elliptic differential equations including the poly-harmonic operator, it is shown that a duality between the existence and nonexistence in an interior/exterior/entire the problems still holds.
Asymptotic behaviors of spatial critical points and zeros of solutions of parabolic equations Competitive

SAKAGUCHI Shigeru

Offer Organization: Japan Society for the Promotion of Science

System: Grant-in-Aid for Scientific Research (C)

Category: Grant-in-Aid for Scientific Research (C)

Institution: Ehime University

1998/04 - 2000/03

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(1) Level surfaces invariant with time of solutions of diffusion equations We consider solutions of the initial-Neumann problem for the heat equation on bounded Lipschitz domains in Euclidean space, and with the help of the classification theorem of isoparametric hypersurfaces in Euclidean space of Levi-Civita (1937) and Segre (1938), we classify the solutions whose isothermal surfaces are invariant with time. Furthermore, we can deal with nonlinear diffusion equations such as the porous medium equation, and we get similar classification theorems. (2) Asymptotic behaviors of the interfaces with sign changes of solutions of the one-dimensional porous medium equation We consider the Cauchy and the initial-Dirichlet problems for the one-dimensional evolution p-Laplacian equation with p>1 for nonzero, bounded, and nonnegative initial data having compact support. It was shown that after a finite time the set of spatial critical points of the solution u in {u > 0} consists of one point, say x = x(t) for time t. In this research, we show that after a finite time x(t) is CィイD11ィエD1 in t. Furthermore, we can deal with generalized porous medium equations with sign changes, and we get CィイD11ィエD1 regularity of the interfaces with sign changes. Also, in the initial-Dirichlet problem for the one-dimensional evolution p-Laplacian equation, we show that there exists a positive constant β=β(ρ) such that x(t)tィイD1-βィエD1 tends to some positive constant as t → ∞. (3) Stationary critical points of the heat flow and the symmetries of the domains We consider the initial-Dirichlet problem for the heat equation on bounded and simply connected domains in the plane. By a new method with the help of the Riemann Mapping theorem in complex analysis, we give a characterization of domains invariant under the rotation of angle 2π/3 by making use of the stationary critical points of the heat flow. (Previously, only the characterizations of balls and centrosymmetric domains were obtained.) Furthermore, we consider stationary critical points of the heat flow in sphere SィイD1NィエD1 and in hyperbolic space HィイD1NィエD1, and prove several results corresponding to those in Euclidean space which have been proved in Magnanini and Sakaguchi (1997, 1999). Precisely. We get the characterizations of geodesic balls and centrosymmetric domains by making use of the stationary critical points of the heat flow.
Oscillatory properties of solutions of higher order differential equations

NAITO Manabu, USAMI Hiroyuki, HASHIMOTO Takahiro, SAKAGUCHI Shigeru

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: EHIME UNIVERSITY

1999 - 2000

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The aim of this research is to investigate the oscillatory properties of solutions of higher-order (including second-order) ordinary differential equations of the Emden-Fowler type, and to investigate the oscillatory properties of solutions of elliptic differential equations on the base of the results for ordinary differential equations. The new results and knowledge obtained in the two years are as follows : 1. For the second-order half-linear ordinary differential equations, a generalization and an analogue of the Sturm-Liouville linear regular eigenvalue problem are obtained. 2. For the four-dimensional Emden-Fowler differential systems, a complete characterization for the existence of nonoscillatory solutions with specific asymptotic properties as t→∞ is established, and a characterization for the nonexistence of nonoscillatory solutions is also obtained. 3. For higher-order ordinary differential equations with general nonlinearities, a characterization for the existence of nonoscillatory solutions of the Kiguradze classes is established. 4. For a singular eigenvalue value problem to higher-order linear ordinary differential equations, it is shown that there is a countable sequence of eigenvalues and that the n-th eigenfunction has exactly n zeros in an infinite interval under consideration. 5. For the second-order quasilinear ordinary differential equations, the asymptotic forms of positive solutions are completely determined. 6. For the second-order quasilinear elliptic differential equations, a sufficient condition for the oscillation of all solutions is established. 7. For the two-dimensional semilinear elliptic differential systems of the Laplace type, an analogue of the Liouville theorem is established.
Nonlinear Evolution Equations and Elliptic Equations

OTANI Mitsuharu, ISHII Hitoshi, TANAKA Kazunaga, YAMADA Yoshio, SAKAGUCHI Shigeru, SUZUKI Takashi

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B)

Institution: Waseda University

1997 - 1999

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Elliptic Equations (1) Concerning the equation (E) - △u = |u|ィイD1q-2ィエD1u x ∈Ω, u(x) = 0 x ∈∂Ω we obtained the following results. Let Ω = RィイD1NィエD1＼BィイD2R1ィエD2, BィイD2RィエD2 = {x ∈ IRィイD1NィエD1 ; |x|【less than or equal】 R }, 2ィイD1*ィエD1<q< +∞ (2ィイD1*ィエD1 is the critical exponent for Sobolev's embedding HィイD31(/)0ィエD3 (Ω) ⊂ LィイD1qィエD1 (Ω) ), then (E) admits a radially symmetric solution in HィイD11ィエD1 (Ω) ∩ LィイD1qィエD1 (Ω). This fact has been conjectured from the duality between bounded domains and exterior domains. (II) Consider the equation : (E)ィイD2λィエD2 -△u = λu + |u|ィイD1q-2ィエD1u x ∈Ω, u(x) = 0 x ∈∂Ω (1) Let Ω = ΩィイD2dィエD2 × λRィイD1N-dィエD1, (ΩィイD2dィエD2 is a bounded domain in IRィイD1dィエD1), q = 2ィイD1*ィエD1, d【greater than or equal】 1, N 【greater than or equal】 4, then for all λ ∈ (0, λィイD21ィエD2), λィイD21ィエD2 = infィイD2v∈HィイD31(/)0ィエD3 (Ω)ィエD2‖∇ィイD2uィエD2‖LィイD42ィエD4ィイD12ィエD1/‖u‖LィイD42ィエD4ィイD22ィエD2 > 0, (E)ィイD2λィエD2 has a nontrivial solution, which gives a generalization of the well-known result of Brezis-Nirenberg to unbounded cylinders. (2) Let Ω = ΩィイD2dィエD2 x RィイD1N-dィエD1 and let ΩィイD2dィエD2 be a d-dimensional annulus. ・ If q 【greater than or equal】 NィイD2dィエD2 = 2 (N -d+1)/(N-d+1-2) , then (E)ィイD2λィエD2 admits no nontrivial weak solution. ・ If q < NィイD2dィエD2, then (E)ィイD2λィエD2 admits a nontrivial weak solution. These results reveal the fact that the d-dimensional symmetry reduces the effective dimension by (d-1). (III) Consider (E)ィイD21ィエD2 -Δu + u = a(x) |u|ィイD1q-2ィエD1u + f(x) x ∈ IRィイD1NィエD1, 2 < q < 2ィイD1*ィエD1 o < a(x), |a(x) - 1| 【less than or equal】 CeィイD1λ|x|ィエD1, λ > 0 It is shown that if ‖f‖ィイD2H-1(RィイD1NィエD1)ィエD2 is sufficiently small, then (E)ィイD21ィエD2 has at least two positive solutions. Furthermore, we found that for the case where f = 0 and q < 2ィイD1*ィエD1 is close enugh to 2ィイD1*ィエD1,the multiplicity of positive solutions depends upon the topological property (su as category) of the set {x ∈Ω ; u(x) = maxィイD2x∈ΩィエD2 }.The analysis of this phenomenon will be an interesting subject to study in future. Parabolic Equations (I) It has been well known that weak solutions of porous medium equations enjoy the Holder continuity. However, the existence of smooth (local) solutions has been left as an open problem for long time. Otani-Sugiyama gave an affirmative answer to this open problem, by developing the LィイD1∞ィエD1-energy method, which was introduce by themselves to show the local existence of WィイD11,∞ィエD1-solutions for more general doubly nonlinear parabolic equations. This is the most fascinating result among our results obtained in this reseach project. (II) It was left as an unsolved problem to determine the asymptotic behabiour of solutions of (P) uィイD2tィエD2, -Δu = |u|ィイD12ィイD1*ィエD1-2ィエD1u x∈Ω, u(x) = 0 x∈∂Ω. To this problem, the following partial answer was obtained. 「Let Ω = {x ∈ RィイD1nィエD1 : |x|< 1 } and the solution u (x.t) be positive, radially symmetric and monotone decreasing with respect to r = |x|. Then u blows up in a finite time or becomes a global solution and satisfies the following property : 「There exists a sequence {tィイD2nィエD2 } such that |∇u (x,tィイD2nィエD2)|ィイD12ィエD1 - CoィイD1δィエD1(0) (u - x), |u (x,tィイD2nィエD2)|ィイD12ィエD1 - CoィイD1σィエD1(0) (u - x). 」 This result give some information about the problem above to some extent. However, since strong technical condtions are assumed. We need further in vestigation to solve this problem in a natural setting.
有限要素解に対する後験的誤差評価の研究

土屋卓也, 庭さき隆, 方青, 坂口茂, 内藤学, 山本哲朗

Offer Organization: 日本学術振興会

System: 科学研究費助成事業

Category: 基盤研究(C)

Institution: 愛媛大学

1996 - 1996

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この一年間、主に、元の微分作用素はフレッシェ微分可能だが、有限要素法等で離散化する際、微分不可能な項が出てくるような非線形境界値問題に対しての、有限要素解の誤差評価を行った。例えば、流体の方程式であるナビア-ストークス方程式を離散化する際に、流れの上流の情報を下流の情報より重視するといった、いわゆる上流型有限要素法においてこのような状況が出てくる。得られた結果は以下の通り:真の解がある程度滑らかなら、それに対する上流型有限要素法により定義される解は、真の解に近くに一意に存在し、適当な誤差評価を満たす。この結果をまとめた次の論文を準備中で、今年度中に投稿する予定である。 N.Mastunaga,T.Tsuchiya Non-Differentiable Finite Element Approximations for Parametrized Strongly Nonlinear Boundary Value Problems また、1996年12月に龍谷大学で行われた応用数学合同研究集会で、同じ著者、題目で研究発表を行った。
非線形偏微分方程式論における実解析的方法

鈴木貴, 野倉嗣紀, 方青, 土屋卓也, 山本哲朗, 坂口茂

Offer Organization: 日本学術振興会

System: 科学研究費助成事業

Category: 一般研究(C)

Institution: 愛媛大学

1994 - 1994

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(1)弾性膜の自由境界問題を記述する平均曲率方程式についてnon-Newtonianの場合の非自明解の存在が変分法によりまたpondent dropの大域分岐図の無限図の揺れが常微分作用素の振動定理によって示された。 (2)2次元のSoboleo臨界を表わすTwdonger不等式と関係する半線形楕円型境界値問題の解の漸近挙動に関して、偏微分方程式固有の方法によって従来の結果が精密化され、変分問題に対して適用された。 (3)楕円型境界値問題の不安定な解を数値的に効率良く求める方法としてNehariの変分原理と関係する反復列が大変有効であることを理論的に解明し、数値計算によって実証した。

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