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Qualitative Theory of Nonlinear Elliptic Differential Equations

非线性椭圆微分方程的定性理论

基本信息

批准号：
09640196
负责人：
FUKAGAI Nobuyoshi
金额：
$ 0.51万
依托单位：
University of Tokushima
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-09640196/
关键词：
nonlinear elliptic differential equations qualitative theory quasilinear weak solutions bifurcation eigenvalue problems 楕円形

项目摘要

We studied the subjects related to qualitative theory of nonlinear elliptic differential equations : (i) bound-ary value problems of quasilinear elliptic equations in a bounded domain ; (ii) oscillatory problem of ordi-nary differential equations which derived from qualitative problem of elliptic equations in an unbounded domain. Our results are the following.[1] The asymptotic behavior of eigenvalues and eigenfunctions of p-Laplace operator is investigated. We obtain the best constant of L*-Poincare inequality, and a limit equation which the limits of eigenvalues and eigenfuncitons satisfy in a weak sense.[2] A Sturm-Liouville equation on (a, *) is examined. Supposing a strongly nonoscillatory condition, we obtain a sequence of positive princial eigenvalues and the corresponding principal eigenfunctions.[3] Concering an initial value problem of parabolic equation, we obtain a new sufficient condition on the initial value which determines the solution to blow up. Furthermore, we investigate the asymtotic behavior of the solution at the blowing lip time.[4] We consider a second order half-linear differential equation on [a, *]. We take up the two classes of nonoscillatory solutions (i.e., princilal solutions and non principal solutions), and show that precise information can be drawn as to how the number of zeros of these solutions changes as lambda varies from zero to infinity.[5] A bifurcation problem or a nonlinear eigenvalue problem for degenerate quasilinear elliptic equations with Dirichelt boundary condition is studied. By virture of our estimates, we can apply the Leray-Shauder degree theory to our problem and obtain the bifurcation of nontrivial weak solutions.[6] Quasiperiodic solutions to Van der Pol type equations driven by two or more distinct freequency input signals are considered. The existence and uniqueness results are obtained from the viewpoint of numerical analysys.

我们研究了与非线性椭圆型微分方程定性理论有关的问题：(I)有界域上拟线性椭圆型方程的边值问题；(Ii)由无界区域上椭圆型方程的定性问题引出的偏微分方程解的振动性问题。我们的结果如下：[1]研究了p-Laplace算子的特征值和特征函数的渐近行为。我们得到了L~*-庞加莱不等式的最佳常数，以及特征值和特征函数的极限在弱意义下所满足的极限方程。在强非振动条件下，我们得到了一个正的主特征值序列和相应的主特征函数。[3]对于抛物型方程的一个初值问题，我们得到了一个新的关于初值的充分条件，它决定了解的爆破。此外，我们还研究了在吹唇时解的渐近性。[4]我们考虑了[a，*]上的一个二阶半线性微分方程解。我们得到了两类非振动解(即基本解和非主解)，并证明了这些解的零点个数随波长从零变化到无穷大而变化的精确信息。[5]研究了具有Dirichelt边界条件的退化拟线性椭圆型方程的分歧问题或非线性特征值问题。借助于我们的估计，我们可以将Leray-Shauder度理论应用到我们的问题中，并得到非平凡弱解的分支。[6]考虑由两个或多个不同的频率输入信号驱动的Van der Pol型方程的拟周期解。从数值分析的角度得到了解的存在唯一性结果。