Qualitative studies of solutions to elliptic equations in unbounded domains

无界域中椭圆方程解的定性研究

• 批准号: 09640192
• 负责人: USAMI Hiroyuki
• 金额: $ 1.02万
• 依托单位: Hiroshima University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640192/
• 关键词: elliptic equation; degenerate Laplacian; Liouville-type theorem; Sturm-Liouville problem; oscillation; eigenvalue problem; symmetry of solutions; elliptic system; 非線形楕円型方程式; リューヴィユ型定理; 非線形固有値問題

(1) Oscillation criteria for elliptic equations : Effective oscillation criteria are established for second-order quasilinear elliptic equations whose leading terms are degenerate Laplacians. Our method is based on comparison principles and asymptotic theory for quasilinear ordinary differential equations. For one-dimensional case, useful information about numbers of zeros of solutions is obtained via the generalized Prufer transformation.(2) Liouville-type theorems and nonexistence of positive solutions of BVPs : Lioville-type theorems are established for quasilinear elliptic equations whose leading terms are degenerate Laplacians and (generalized) mean curvature operators. Our results can be regarded as a natural extension of classical Liouville theorem.(3) Symmetry of positive solutions for elliptic problems : We prove by means of the moving plane method or moving sphere method that positive solutions of elliptic equations of certain types are radially symmetric. Useful information about self-similar solutions of parabolic problems can be derived from our results.(4) Two-parameter eigenvalue problems : Two-parameter nonlinear Sturm-Liouville problems are considered. The existence of the variational eigenvalues is established. Asymptotic formulas of eigenvalues and eigenfunctions are obtained.(5) Asymptotic theory for solutions of elliptic systems : Semilinear elliptic systems are considered. We establish nonexistence criteria of positive solutions, Liouville-type theorems, and oscillation criteria.)

(1)椭圆型方程的振动准则：建立了主导项为退化拉普拉斯项的二阶拟线性椭圆型方程的有效振动准则。我们的方法是基于比较原理和拟线性常微分方程组的渐近理论。对于一维情形，通过广义Prufer变换得到了关于解的零点个数的有用信息。(2)建立了主项为退化拉普拉斯算子和(广义)平均曲率算子的拟线性椭圆型方程的Liouville-型定理和BVPS：Lioville-型定理的正解的不存在性。我们的结果可以看作是经典的Liouville定理的自然推广。(3)椭圆问题正解的对称性：我们用移动平面法或移动球法证明了某些类型的椭圆型方程的正解是径向对称的。从我们的结果可以得到关于抛物问题自相似解的有用信息。(4)双参数特征值问题：考虑双参数非线性Sturm-Liouville问题。证明了变分本征值的存在性。得到了特征值和特征函数的渐近公式。(5)椭圆组：半线性椭圆组的解的渐近理论。我们建立了正解的不存在准则、Liouville-型定理和振动准则。

项目成果

S.Matsumoto: "On ν-distal flows on 3-manifolds" Bull.London Math.Soc.29. 609-616 (1997)

S.Matsumoto：“论 3 流形上的 ν 远端流”Bull.London Math.Soc.29 (1997)。

T.Kusahara et al: "A barrier method for quasilinear ordinary differential equations of the curvature type" Czechoslovak Math.J.(to appear).

T.Kusahara 等人：“曲率型拟线性常微分方程的障碍法”Czechoslovak Math.J.（即将出现）。

T.Shibata: "Nonlinear multiparameter eigenvalue problems on general level sets" Nonlinear Anal.29. 823-838 (1997)

T.Shibata：“一般水平集上的非线性多参数特征值问题”非线性分析.29。

A.Elbert: "Singular eigenvalue problems for second order linear ordinary differential equations" Arch.Math.(Brno). 34・1. 59-72 (1998)

A.Elbert：“二阶线性常微分方程的奇异特征值问题”Arch.Math.(Brno) 59-72 (1998)。

A.Elbert et al: "Singular eigenvalue problems for second order linear ordinary differential equations" Arch.Math. (Brno). 34-1. 59-72 (1998)

A.Elbert 等人：“二阶线性常微分方程的奇异特征值问题”Arch.Math。