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Study on Nonlinear Evolution Equations and Nonlinear Elliptic Equations

非线性演化方程和非线性椭圆方程的研究

基本信息

批准号：
12440051
负责人：
OTANI Mitsuharu
金额：
$ 9.41万
依托单位：
Waseda University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440051/
关键词：
Nonlinear evolution equation Nonlinear elliptic equation Nonlinear PDE Method of variation Subdifferential operator Nonlinear Evolution Equation Nonlinear Elliptic Equation Variational Method 非線形放物型方程式発展方程式楕円型方程式放物型方程式

项目摘要

(1)"L^∞-energy method" is invented. This assures the high differentiablity of solutions of quasilinear parabolic equations. By this method, the existence of W^<1. ∞>-solutions for a general doubly nonlinear parabolic equations and the open problem : "porous medium equations admit C^∞-solutions?" is solved affirmatively. Recent studies suggest that this gives a quite powerful tool for various problems.(2)"The theory of nonmonotone perturbations for subdifferentials " is extended to Banach space setting. By this theory, we can treat the existence and regularity of solutions for degenerate parabolic equations in a more natural way than Galerkin' s method and open problems, left unsolved in the usual way, were solved.(3)A Concentration Compactness (CC) theory with partial symmetry is given. The usual CC theory is known to be useful to analyze the problem with lack of compactness. On the other hand, the high symmetry such as the radial symmetry often recovers the compactness. It is studied how the partial symmetry not enough to recover compactness is reflected to CC theory. By this theory, the existence of nontrivial solutions is proved for some quasilinear elliptic equations in infinite cylindrical domains.(4)The classical "Principle of Symmetric Criticality (PSC)" by R.Palais assures that under suitable conditions, critical points in the subspace with the symmetry give real critical points in the whole space, but is restricted to the system with variational structures. PSC is extended to a more general theory which covers the elliptic systems without full symmetry or evolution equations including time evolution terms.(5)A new degree theory is established. It can teat mutivuled operators including subdifferential operators and cover nonlinear PDE with various multivaluedness nature.(6)The theory of nonmonotone perturbations for subdifferentials is ameliorated to cover the initial-boundary value problems and time periodic problems for magneto-micropolar fluid equations.

(1)发明了“L^∞能量法”。这保证了拟线性抛物方程解的高可微性。通过此方法，证明了W^<1的存在性。一类一般双非线性抛物方程的∞>-解和“多孔介质方程允许C^∞-解”的开放问题得到了肯定解。最近的研究表明，这为解决各种问题提供了一个相当强大的工具。(2)将“次微分的非单调摄动理论”推广到Banach空间。利用这一理论，我们可以比伽辽金方法更自然地处理退化抛物方程解的存在性和正则性问题，从而解决了用通常方法无法解决的开放问题。(3)给出了具有部分对称性的集中紧性理论。通常的CC理论对于分析紧性不足的问题是有用的。另一方面，高对称性如径向对称性往往能恢复紧致性。研究了不足以恢复紧性的部分对称性如何反映到CC理论中。利用这一理论，证明了一类拟线性椭圆型方程在无限圆柱域上非平凡解的存在性。(4) palais的经典“对称临界原理”（Principle of Symmetric Criticality， PSC）保证了在适当的条件下，具有对称性的子空间中的临界点给出了整个空间中的实临界点，但仅限于具有变分结构的系统。将PSC扩展为一个更一般的理论，它涵盖了不完全对称的椭圆系统或包含时间演化项的演化方程。(5)建立了一种新的学位理论。它可以处理包括子微分算子在内的多算子，涵盖各种多值性质的非线性偏微分方程。(6)改进了次微分的非单调摄动理论，使其涵盖了磁微极流体方程的初边值问题和时间周期问题。