1998 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 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	超伝導 / Ginzburg-Landau方程式 / 半古典極限 / Schrodinger作用素 / スペクトエル / Morrey空間 / 楕円型方程式 / ハミルトン・ヤコビ方程式
Research Abstract	1. 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、磁場の効果の入った非線形Schrodinger方程式のenergy最小のstanding waveの存在を示し、その解の半古典極限でのprofileを調べた。 2. 分担者の西垣氏らと、非負のポテンシャルを持つSchrodinger作用素の基本解を積分核にもつ積分作用素の一般Morrey空間上での有界性評価を与え、さらに小さい摂動をくわえた作用素のmapping propertyを調べた。 3. 2階の線形-様楕円型作用素に加藤class のpotentialを加えた作用素のessential spectrumより下にあるspectrumの有限性を調べた。2.また、正の多項式的なポテンシャルに負の小さい挾摂動ポテンシャルを加えたSchrodinger作用素を考え、対応するRiesz型積分作用素の一般化Morrey空間上の有界性を与え、その応用を研究した。 4. 分担者の神保は、Ginzburg-Landau方程式の安定な非定数解の存在およびその零点の位置を論じ、田中は非線型楕円型方程式の正値解の一意性、非退化性を論じ、multi-bump solutionの構成を行なった。また、村田は2階の楕円型方程式に対するMartin理論における半小摂動の概念を導入し、半小摂動の下での正値解の構造安定性及び半小摂動になるための判定条件を与えた。望月は、反応拡散系の簡単なモデル方程式系の大域解の存在、非存在に関し、べき状の非線形項について存在ー非存在の臨界指数の決定、臨界爆発、存在時間の評価、又大域解については、時間の経過に伴う漸近挙動を調べた。石井は、ハミルトン・ヤコビ方程式に対する周期的領域上でのノイマン型とディリクレ型の境界値問題の均質化を扱い、ハミルトニアンと領域が周期的に振動するときの極限方程式を決定した。

Research Products
(20 results)

All Other

All Publications (20 results)

[Publications] K.Mochizuki(with Q.Huang): "Existence and behavior of solutions for a weakly coupled system of reaction-diffusion equations" Meth.Appl.Anal.5. 109-124 (1998)
[Publications] H.Ishii: "Homogenization of the Cauchy problem for Hamilton-Jacobi equations," Stochastic Analysis, Control, Optimization and Applications : a volume in honor of W.H.Fleming (ed.W.M.McEneaney, G.G.Yin, and Q.Zhang),. 305-324 (1999)
[Publications] H.Ishii(with K.Horie): "Homogenization of Hamilton-Jacobi equations on domains with small scale peri-odic structure," Indiana Univ.Math.J.47. 1011-1058 (1998)
[Publications] H.Ishii(with G.E.Pires and P.E.Souganidis): "Threshold dynamics type approximations schemes for propagating fronts," J.Math.Soc.Japan, in press.
[Publications] H.Ishii(with M.Arisawa): "Some properties of ergodic attractors for controlled dynamical systems." Discrete Contin.Dynam.Systems. 4. 43-54 (1998)
[Publications] Makoto Sakai,: "Sharp estimates of the distance from a fixed point to the frontier of a Hele-Shaw flow," Potential Analysis. 8. 227-302 (1998)
[Publications] M.Sakai(with B.Gustafsson): "On potential theoretic skeletons of polyhedra," to appear in Geometricae Dedi-cata.
[Publications] Makoto Sakai,: "Linear combinations of harmonic measures and quadrature domains of signed measures with small supports," to appear in Proc.Edinburgh Math.Soc.
[Publications] S.Takakuwa: "A compactness theorem for harmonic maps," Differential and Integral Equation,. 11. 169-178 (1998)
[Publications] K.Hidano,: "Nonlinear small data scattering for the wave equation in R^<4+1>" J.Math.Soc.Japan. 50. (1998)
[Publications] M.Murata(with Kazuhiro Ishige): "An intrinsic metric approach to uniqueness of the positive Cauchy problem for parabolic equations," Meth.Z.,. 227. 313-335 (1998)
[Publications] M.Murata,: "Structure of positive solutions to Schrodinger equations," Sugaku Expositions,. 11. 101-121 (1998)
[Publications] S.Jimbo(with X.Y.Chen, Y.Morita): "Stabilization of vortices in the Ginzburg-Landau equation with variable diffusion coefficients," SIAM J.Math.Anal.29. 903-912 (1998)
[Publications] S.Jimbo(with J.Zhai): "Ginzburg-Landau equation with magnetic effect : Non-simply-connected do-mains," J.Math.Soc.Japan.50. 663-684 (1998)
[Publications] K.Tanaka(with S.Adachi): "Trudinger type inequalities in R^N and their best exponents," to appear in Proc.Amer.Math.Soc.
[Publications] K.Tanaka(with P.Felmer): "Hyperbolic-like solutions for singular Hamiltonian systems," to appear NoDEA Nonlinear Differential Equations Appl.
[Publications] K.Tanaka: "Periodic solutions for singular Hamiltonian systems and closed geodesics on non-compact Riemannian manifolds," to appear in Ann.Inst.H.Poincare Anal.Non Lineaire.
[Publications] K.Tanaka(with Y.Kabeya): "Uniqueness of positive redial solutions of semilinear elliptic equaitons in R^N and Sere's non-degeneracy condition," to appear in Comm.Partial Differential Equations.
[Publications] K.Tanaka(with A.Ambrosetti): "On Keplerian N-body type problems," in "Nonlinear Analysis and continuum mechanics : Paper for the 65-th birthday of James Serrin" (G.Buttazzo, G.P.Galdi, E.Lanconelli, P.Pucci ed.) Springer. 15-25 (1998)
[Publications] 西垣誠一: "2階楕円型方程式に対する境界値問題について(III)" 沼津工業高等専門学校研究報告. 32. (1998)

1998 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] K.Mochizuki(with Q.Huang): "Existence and behavior of solutions for a weakly coupled system of reaction-diffusion equations" Meth.Appl.Anal.5. 109-124 (1998)

[Publications] H.Ishii: "Homogenization of the Cauchy problem for Hamilton-Jacobi equations," Stochastic Analysis, Control, Optimization and Applications : a volume in honor of W.H.Fleming (ed.W.M.McEneaney, G.G.Yin, and Q.Zhang),. 305-324 (1999)

[Publications] H.Ishii(with K.Horie): "Homogenization of Hamilton-Jacobi equations on domains with small scale peri-odic structure," Indiana Univ.Math.J.47. 1011-1058 (1998)

[Publications] H.Ishii(with G.E.Pires and P.E.Souganidis): "Threshold dynamics type approximations schemes for propagating fronts," J.Math.Soc.Japan, in press.

[Publications] H.Ishii(with M.Arisawa): "Some properties of ergodic attractors for controlled dynamical systems." Discrete Contin.Dynam.Systems. 4. 43-54 (1998)

[Publications] Makoto Sakai,: "Sharp estimates of the distance from a fixed point to the frontier of a Hele-Shaw flow," Potential Analysis. 8. 227-302 (1998)

[Publications] M.Sakai(with B.Gustafsson): "On potential theoretic skeletons of polyhedra," to appear in Geometricae Dedi-cata.

[Publications] Makoto Sakai,: "Linear combinations of harmonic measures and quadrature domains of signed measures with small supports," to appear in Proc.Edinburgh Math.Soc.

[Publications] S.Takakuwa: "A compactness theorem for harmonic maps," Differential and Integral Equation,. 11. 169-178 (1998)

[Publications] K.Hidano,: "Nonlinear small data scattering for the wave equation in R^<4+1>" J.Math.Soc.Japan. 50. (1998)

[Publications] M.Murata(with Kazuhiro Ishige): "An intrinsic metric approach to uniqueness of the positive Cauchy problem for parabolic equations," Meth.Z.,. 227. 313-335 (1998)

[Publications] M.Murata,: "Structure of positive solutions to Schrodinger equations," Sugaku Expositions,. 11. 101-121 (1998)

[Publications] S.Jimbo(with X.Y.Chen, Y.Morita): "Stabilization of vortices in the Ginzburg-Landau equation with variable diffusion coefficients," SIAM J.Math.Anal.29. 903-912 (1998)

[Publications] S.Jimbo(with J.Zhai): "Ginzburg-Landau equation with magnetic effect : Non-simply-connected do-mains," J.Math.Soc.Japan.50. 663-684 (1998)

[Publications] K.Tanaka(with S.Adachi): "Trudinger type inequalities in R^N and their best exponents," to appear in Proc.Amer.Math.Soc.

[Publications] K.Tanaka(with P.Felmer): "Hyperbolic-like solutions for singular Hamiltonian systems," to appear NoDEA Nonlinear Differential Equations Appl.

[Publications] K.Tanaka: "Periodic solutions for singular Hamiltonian systems and closed geodesics on non-compact Riemannian manifolds," to appear in Ann.Inst.H.Poincare Anal.Non Lineaire.

[Publications] K.Tanaka(with Y.Kabeya): "Uniqueness of positive redial solutions of semilinear elliptic equaitons in R^N and Sere's non-degeneracy condition," to appear in Comm.Partial Differential Equations.

[Publications] K.Tanaka(with A.Ambrosetti): "On Keplerian N-body type problems," in "Nonlinear Analysis and continuum mechanics : Paper for the 65-th birthday of James Serrin" (G.Buttazzo, G.P.Galdi, E.Lanconelli, P.Pucci ed.) Springer. 15-25 (1998)

[Publications] 西垣 誠一: "2階楕円型方程式に対する境界値問題について(III)" 沼津工業高等専門学校研究報告. 32. (1998)

倉田和浩東京都立大学, 理学研究科, 助教授 (10186489)

[Publications] 西垣誠一: "2階楕円型方程式に対する境界値問題について(III)" 沼津工業高等専門学校研究報告. 32. (1998)