1997 Fiscal Year Annual Research Report

調和解析,変分問題,偏微分方程式の解の研究とその応用

Research Project

Project/Area Number	09640208
Research Category	Grant-in-Aid for Scientific Research (C)
Research Institution	Tokyo Metropolitan University
Principal Investigator	倉田和浩東京都立大学, 理学研究科, 助教授 (10186489)
Co-Investigator(Kenkyū-buntansha)	田中和永早稲田大学, 理工学部, 助教授 (20188288) 神保秀一北海道大学, 大学院・理学研究科, 助教授 (80201565) 村田実東京工業大学, 理学部, 教授 (50087079) 酒井良東京都立大学, 理学研究科, 教授 (70016129) 望月清東京都立大学, 理学研究科, 教授 (80026773)
Keywords	一意接続性定理 / Schrodinger作用素 / Ginzburg-Landau方程式 / Chern-Simons-Higgs理論 / スカラー曲率方程式 / Fefferman-Phong不等式 / Morrey空間 / 対称性
Research Abstract	1.特異的な磁場を持つSchrodinger作用素の解の振動関数の単調性を証明し、一意接続性定理および解の零点集合の評価を与えた。また、弱解の有界性、連続性も係数がKato classに属する場合に示した。 2.超伝導のモデルであるGinzburg-Landau方程式の解が局在しない、という一種のLiouville型定理を与えた。また、ゲージ理論の1つであるChern-Simons-Higgs理論に現れる偏微分方程式を解析し、特に背景となる空間の計量が一般の場合に、non-topologicalな解の存在および解の球対称性を示した。 3.Moving Sphere methodを応用して、幾何学におけるスカラー曲率方程式の解に対してKelvin変換に関する対称性を示した。この方法をChern-Simons-Higgs方程式にも応用した。 4.重み付きFeffermann-Phongの不等式を示し、2階の楕円型偏微分作用素で非負のポテンシャルを持つものに対して、その基本解の評価を与え、Morrey空間等さまざまな空間上での不等式を与えた。また、負のポテンシャルを持つ場合でも、一般化Morrey空間上のRiesz型積分作用素の有界性を与え、摂動論を展開した。また対応する熱核の評価も与えた。 5.分担者の神保は、Ginzburg-Landau方程式の安定な解の存在を論じ、田中はロトカーボルテラ型の非線型楕円型方程式の正値解の個数を論じた。また、村田は2階の放物型方程式の初期値問題の非負解の一意性を論じ、望月は臨界指数をもつ準放物型方程式の解の爆発を論じ、酒井は一般化された平均値の性質を調和関数が満たす領域について論じた。

Research Products
(17 results)

All Other

All Publications (17 results)

[Publications] Kazuhiro Kurata: "Local boundedness and continuity for weak solutions of(-(∇-ib)^2u+Vu=0" Math.Z.224. 641-653 (1997)
[Publications] Kazuhiro Kurata: "A unique continuation theorem for the Schrodinger equation with singular magnetic field" Proc.A.M.S.125. 853-860 (1997)
[Publications] Kazuhiro Kurata: "On some symmetry property of solutions to the scalar curvature equation" in Functional Analysis and Global Analysis,Eds.,by T.Sunada and P.W.Sy. 179-184 (1997)
[Publications] Kazuhiro Kurata: "A Liouville type theorem for the Ginzburg-Landau equations with general potentials." to appear in Discrete and Continuous Dynamical Systems.
[Publications] Kiyoshi Mochizuki: "A note on the global solutions of a degenerate parabolic system" Tokyo J.Math.20. 63-66 (1997)
[Publications] Kiyoshi Mochizuki: "Critical exponent and critical blow-up for quasilinear parabolic equations" Israel J.Math.98. 141-156 (1997)
[Publications] B.Gustafsson: "On domains in which harmonic functions satisfy generalized mean value properties." Potential Analysis. 7. 467-484 (1997)
[Publications] Makoto Sakai: "Sharp estimates of the distance from a fixed point to the frontier of a Hele-Shaw flow" to appear in Potential Analysis.
[Publications] Hitoshi Ishii: "Some properties of ergodic attractors for controlled dynamical systems." Discrete Conti.Dynam.Systems. 4. 43-54 (1998)
[Publications] Minoru Murata: "Semismall perturbations in the Martin theory for elliptic equations." Israel J.Math.102. 29-60 (1997)
[Publications] Minoru Murata: "An intrinsic metric approach to uniqueness of the positive Cauchy problem for parabolic equations." Math.Z.(to appear).
[Publications] Shuichi Jimbo: "On the existence of nontrivial stable solutions to the Ginzburg-Landau equation." Nonlinear Anal.TMA. 30. 811-817 (1997)
[Publications] A.Ambrosetti: "On Keplerian N-body type problems" in"Nonlinear Analysis and continuum mechanics:"Buttazgo eds.15-25 (1997)
[Publications] Kazunaga Tanaka: "Multiple positive solutions for some nonlinear elliptic systems" Top.Methods in Nonlinear Ana.10(to appear).
[Publications] Kunio Hidano: "Small data scattering for wave equations with supercritical nonlinearity in low space dimensions" Math.Z.(to appear).
[Publications] Shoichiro Takakuwa: "A compactness theorem for harmonic maps" Diff.and Integ.Eq.11. 169-178 (1998)
[Publications] 村田実: "「偏微分方程式1」岩波講座現代数学の基礎" 岩波書店, 258 (1997)

1997 Fiscal Year Annual Research Report

調和解析,変分問題,偏微分方程式の解の研究とその応用

Principal Investigator

倉田 和浩 東京都立大学, 理学研究科, 助教授 (10186489)

Research Products

[Publications] Kazuhiro Kurata: "Local boundedness and continuity for weak solutions of(-(∇-ib)^2u+Vu=0" Math.Z.224. 641-653 (1997)

[Publications] Kazuhiro Kurata: "A unique continuation theorem for the Schrodinger equation with singular magnetic field" Proc.A.M.S.125. 853-860 (1997)

[Publications] Kazuhiro Kurata: "On some symmetry property of solutions to the scalar curvature equation" in Functional Analysis and Global Analysis,Eds.,by T.Sunada and P.W.Sy. 179-184 (1997)

[Publications] Kazuhiro Kurata: "A Liouville type theorem for the Ginzburg-Landau equations with general potentials." to appear in Discrete and Continuous Dynamical Systems.

[Publications] Kiyoshi Mochizuki: "A note on the global solutions of a degenerate parabolic system" Tokyo J.Math.20. 63-66 (1997)

[Publications] Kiyoshi Mochizuki: "Critical exponent and critical blow-up for quasilinear parabolic equations" Israel J.Math.98. 141-156 (1997)

[Publications] B.Gustafsson: "On domains in which harmonic functions satisfy generalized mean value properties." Potential Analysis. 7. 467-484 (1997)

[Publications] Makoto Sakai: "Sharp estimates of the distance from a fixed point to the frontier of a Hele-Shaw flow" to appear in Potential Analysis.

[Publications] Hitoshi Ishii: "Some properties of ergodic attractors for controlled dynamical systems." Discrete Conti.Dynam.Systems. 4. 43-54 (1998)

[Publications] Minoru Murata: "Semismall perturbations in the Martin theory for elliptic equations." Israel J.Math.102. 29-60 (1997)

[Publications] Minoru Murata: "An intrinsic metric approach to uniqueness of the positive Cauchy problem for parabolic equations." Math.Z.(to appear).

[Publications] Shuichi Jimbo: "On the existence of nontrivial stable solutions to the Ginzburg-Landau equation." Nonlinear Anal.TMA. 30. 811-817 (1997)

[Publications] A.Ambrosetti: "On Keplerian N-body type problems" in"Nonlinear Analysis and continuum mechanics:"Buttazgo eds.15-25 (1997)

[Publications] Kazunaga Tanaka: "Multiple positive solutions for some nonlinear elliptic systems" Top.Methods in Nonlinear Ana.10(to appear).

[Publications] Kunio Hidano: "Small data scattering for wave equations with supercritical nonlinearity in low space dimensions" Math.Z.(to appear).

[Publications] Shoichiro Takakuwa: "A compactness theorem for harmonic maps" Diff.and Integ.Eq.11. 169-178 (1998)

[Publications] 村田 実: "「偏微分方程式1」岩波講座現代数学の基礎" 岩波書店, 258 (1997)

倉田和浩東京都立大学, 理学研究科, 助教授 (10186489)

[Publications] 村田実: "「偏微分方程式1」岩波講座現代数学の基礎" 岩波書店, 258 (1997)