Study on Nonlinear Evolution Equations and Nonlinear Elliptic Equations

(1)提出了“L^∞-能量法”。这保证了拟线性抛物方程解的高可微性。通过这种方法，存在W^<1。讨论了一类一般双非线性抛物方程的C ^∞-解和“多孔介质方程允许C^∞-解？“得到了肯定的解决。最近的研究表明，这为解决各种问题提供了一个非常强大的工具。(2)将“次微分的非单调扰动理论“推广到Banach空间。利用这一理论，我们可以用比Galerkin方法更自然的方法来处理退化抛物方程解的存在性和正则性，并解决了通常方法未解决的公开问题。(3)A给出了具有部分对称性的集中紧性（CC）理论。通常的CC理论被认为是有用的，以分析缺乏紧性的问题。另一方面，高对称性，如径向对称性往往恢复紧凑性。研究了不足以恢复紧致性的部分对称性如何反映到CC理论中。利用这一理论，证明了无限圆柱区域上拟线性椭圆型方程非平凡解的存在性。（4）R.Palais的经典的“对称临界性原理（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在整个空间中给出真实的临界点，但仅限于具有变分结构的系统。PSC被扩展到一个更一般的理论，它涵盖了椭圆系统没有充分的对称性或发展方程，包括时间演化项。(5)A建立了新的度理论。它可以处理包括次微分算子在内的多值算子，也可以处理具有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，使之适用于磁微极流体方程的初边值问题和时间周期问题。

(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.