1999 Fiscal Year Annual Research Report

偏微分方程式,変分問題,逆問題の解の総合的研究

Research Project

Project/Area Number	11640175
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)	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)
Keywords	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
Research Abstract	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。

Research Products

(17 results)

All Other

All Publications (17 results)

[Publications] 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)
[Publications] Kazuhiro Kurata (with S.Sugano): "Fundamental solution,eigenvalue asymptotics and eigenfunction of degenerate elliptic operators with positive potentials"to appear in Studia Math..
[Publications] 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..
[Publications] 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..
[Publications] 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)
[Publications] H.Ishii (with D.Chopp,L.C.Evans): "Waiting time effects for Gauss curvature flows"Indiana Univ.Math.J.. 48. 311-334 (1999)
[Publications] 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..
[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 in 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 radial solutions of semilinear elliptic equaitons in R^N and Sere's non-degeneracy condition,"Comm.Partial Differential Equations. 24. 563-598 (1999)
[Publications] 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)
[Publications] 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)
[Publications] 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)
[Publications] B.Gustafsson (with M.Sakai,): "On potential theoretic skeletons of polyhedra,"Geometricae Dedicata. 76. 1-30 (1999)
[Publications] Makoto Sakai,: "Linear combinations of harmonic measures and quadrature domains of signed measures with small supports,"Proc.Edinburgh Math.Soc.. 42. 433-444 (1999)
[Publications] 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 Fiscal Year Annual Research Report

偏微分方程式,変分問題,逆問題の解の総合的研究

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 (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)

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

[Publications] 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..

[Publications] 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..

[Publications] 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)

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

[Publications] 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..

[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 in 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 radial solutions of semilinear elliptic equaitons in R^N and Sere's non-degeneracy condition,"Comm.Partial Differential Equations. 24. 563-598 (1999)

[Publications] 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)

[Publications] 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)

[Publications] 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)

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

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

[Publications] 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)

倉田和浩東京都立大学, 理学研究科, 助教授 (10186489)