1999 年度実績報告書

偏微分方程式,変分問題,逆問題の解の総合的研究

研究課題

研究課題/領域番号	11640175
研究機関	東京都立大学
研究代表者	倉田和浩東京都立大学, 理学研究科, 助教授 (10186489)
研究分担者	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) 石井仁司東京都立大学, 理学研究科, 教授 (70102887) 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, 理学研究科, 教授 (80201565)
キーワード	楕円型作用素 / 磁場シュレディンガー / 固有値最小化問題 / ギンツブルグ-ランダウ方程式 / 非線形楕円型方程式 / 曲面の曲率流 / Kirchhoff方程式 / ヘレショウ流れ
研究概要	1.倉田は次の研究を行なった。 (1)2階の(退化)楕円型偏微分作用素で、方向によって増大度がことなるような一般の非負のポテンシャルをもつ場合にその基本解の減衰度の評価とその固有値の漸近挙動などへの応用、また磁場の効果を考慮したシュレヂンガー作用素の熱核の評価の研究。 (2)非相対論的なChern-Simons-Higgsゲージ理論における定在波解の存在を適当な外電磁場のもとに証明した。また、いくつかの楕円型作用素のDirichlet第一固有値に関する最小化問題の研究で、対称性の破れなどの現象を研究。 2.分担者の神保は、Neumann境界条件のもとでのギンツブルグ-ランダウ方程式に対して、渦点の特異極限でのダイナミックスの公式を見つけ、田中は、ある非線形楕円型偏微分方程式に対して、非線形項の係数の非斉次性の解の多重性への影響を研究し、4個の正値解の存在を証明した。 3.石井は、Hamilton-Jacobi方程式に対する均質化理論と曲面の曲率流に関する研究を行なった。望月は、弾性弦の振動を記述するKirchhoff方程式について無限遠の近傍に摩擦が働く状況のもとで、小さな解の時間大域的存在とエネルギー減衰を示した。酒井は、ヘレショウ流れの初期領域の境界上に角がある場合の研究を継続し、角の内角が全角(=360度)より小さいときは直角の場合も含めて角のまわりの流れのほぼ完全な記述を得た。

研究成果

(17件)

すべてその他

すべて文献書誌 (17件)

[文献書誌] Kazuhiro Kurata (with S.Sugano): "'A Remark on Estimares for Uniformly Elliptic Operators on Weighted L^p spaces and Morrey spaces",Math.Nachr.. 209,No.4. 137-150 (2000)
[文献書誌] Kazuhiro Kurata (with S.Sugano): "Fundamental solution,eigenvalue asymptotics and eigenfunction of degenerate elliptic operators with positive potentials"to appear in Studia Math..
[文献書誌] Kazuhiro Kurata,: "An estimate on the Heat kernel of magnetic Schrodinger operators and uniformlly elliptic operators with non-negative potentials,"to appear in J.London Math.Soc..
[文献書誌] Kazuhiro Kurata,: "Standing wave solutions for the Planer Chern-Simons gauged nonlinear Schrodinger equation with an external electromagnetic field,"to appear in J.Korean Math.Soc..
[文献書誌] Masaki Imai (with K.Kurata and I.Ohnishi): "A Mimimizing Problem for the First Dirichlet Eigenvalue and Numerical Results,"「ポテンシャル論とその周辺」(数理研講究録). 1116. 121-138 (1999)
[文献書誌] H.Ishii (with D.Chopp,L.C.Evans): "Waiting time effects for Gauss curvature flows"Indiana Univ.Math.J.. 48. 311-334 (1999)
[文献書誌] K.Tanaka (with S.Adachi): "Four positive solutions for the semilinear elliptic equation:-Δu+u=a(x)u^p+f(x)inR^N"to appear in Calculus of variations and Partial Differential Equations..
[文献書誌] K.Tanaka (with S.Adachi): "Trudinger type inequalities in R^N and their best exponents,"to appear in Proc.Amer.Math.Soc..
[文献書誌] K.Tanaka (with P.Felmer): "Hyperbolic-like solutions for singular Hamiltonian systems,"to appear in NoDEA Nonlinear Differential Equations Appl..
[文献書誌] 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..
[文献書誌] K.Tanaka (with Y.Kabeya): "Uniqueness of positive radial solutions of semilinear elliptic equaitons in R^N and Sere's non-degeneracy condition,"Comm.Partial Differential Equations. 24. 563-598 (1999)
[文献書誌] H.Ishii (with G.E.Pires,P.E.Souganidis): "Threshold dynamics type approximation schemes for propagating fronts"J.Math.Soc.Japan. 51. 267-308 (1999)
[文献書誌] K.Mochizuki (with K.Mukai and Q.Huang): "Large time behavior and life span for quasilinear parabolic equation with slowly decaying initial values,"Nonlinear Analysis,. 39. 33-45 (2000)
[文献書誌] K.Mochizuki: "Global existence and energy decay of small solutions to the Kirchhoff equation with linear dissipation localized near infinity,"J.Math.Kyoto Univ.,. 39. 347-363 (1999)
[文献書誌] B.Gustafsson (with M.Sakai,): "On potential theoretic skeletons of polyhedra,"Geometricae Dedicata. 76. 1-30 (1999)
[文献書誌] Makoto Sakai,: "Linear combinations of harmonic measures and quadrature domains of signed measures with small supports,"Proc.Edinburgh Math.Soc.. 42. 433-444 (1999)
[文献書誌] S.Jimbo (with Y.Morita,): "Stable vortex solutions to the Ginzburg-Landau equation with a variable coefficient in a disk,"J.Diff.Eq.. 155. 153-176 (1999)

1999 年度 実績報告書

偏微分方程式,変分問題,逆問題の解の総合的研究

研究代表者

倉田 和浩 東京都立大学, 理学研究科, 助教授 (10186489)

研究成果

[文献書誌] Kazuhiro Kurata (with S.Sugano): "'A Remark on Estimares for Uniformly Elliptic Operators on Weighted L^p spaces and Morrey spaces",Math.Nachr.. 209,No.4. 137-150 (2000)

[文献書誌] Kazuhiro Kurata (with S.Sugano): "Fundamental solution,eigenvalue asymptotics and eigenfunction of degenerate elliptic operators with positive potentials"to appear in Studia Math..

[文献書誌] Kazuhiro Kurata,: "An estimate on the Heat kernel of magnetic Schrodinger operators and uniformlly elliptic operators with non-negative potentials,"to appear in J.London Math.Soc..

[文献書誌] Kazuhiro Kurata,: "Standing wave solutions for the Planer Chern-Simons gauged nonlinear Schrodinger equation with an external electromagnetic field,"to appear in J.Korean Math.Soc..

[文献書誌] Masaki Imai (with K.Kurata and I.Ohnishi): "A Mimimizing Problem for the First Dirichlet Eigenvalue and Numerical Results,"「ポテンシャル論とその周辺」(数理研講究録). 1116. 121-138 (1999)

[文献書誌] H.Ishii (with D.Chopp,L.C.Evans): "Waiting time effects for Gauss curvature flows"Indiana Univ.Math.J.. 48. 311-334 (1999)

[文献書誌] K.Tanaka (with S.Adachi): "Four positive solutions for the semilinear elliptic equation:-Δu+u=a(x)u^p+f(x)inR^N"to appear in Calculus of variations and Partial Differential Equations..

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

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

[文献書誌] 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..

[文献書誌] K.Tanaka (with Y.Kabeya): "Uniqueness of positive radial solutions of semilinear elliptic equaitons in R^N and Sere's non-degeneracy condition,"Comm.Partial Differential Equations. 24. 563-598 (1999)

[文献書誌] H.Ishii (with G.E.Pires,P.E.Souganidis): "Threshold dynamics type approximation schemes for propagating fronts"J.Math.Soc.Japan. 51. 267-308 (1999)

[文献書誌] K.Mochizuki (with K.Mukai and Q.Huang): "Large time behavior and life span for quasilinear parabolic equation with slowly decaying initial values,"Nonlinear Analysis,. 39. 33-45 (2000)

[文献書誌] K.Mochizuki: "Global existence and energy decay of small solutions to the Kirchhoff equation with linear dissipation localized near infinity,"J.Math.Kyoto Univ.,. 39. 347-363 (1999)

[文献書誌] B.Gustafsson (with M.Sakai,): "On potential theoretic skeletons of polyhedra,"Geometricae Dedicata. 76. 1-30 (1999)

[文献書誌] Makoto Sakai,: "Linear combinations of harmonic measures and quadrature domains of signed measures with small supports,"Proc.Edinburgh Math.Soc.. 42. 433-444 (1999)

[文献書誌] S.Jimbo (with Y.Morita,): "Stable vortex solutions to the Ginzburg-Landau equation with a variable coefficient in a disk,"J.Diff.Eq.. 155. 153-176 (1999)

1999 年度実績報告書

倉田和浩東京都立大学, 理学研究科, 助教授 (10186489)