CO-operative Research on Functional Analysis, Real Analisis, and Partial Diffential Equations

这项研究项目旨在本着合作精神，从独特的角度调查与标题中提到的三个学科有关的问题。三年来取得的成果如下：1.泛函和实数分析方法在偏微分方程组中的应用。利用这些工具，利用函数空间上的各种结果，研究了非线性偏微分方程解的性质。2.调和分析与复变分析。得到了关于强伪凸域上退化椭圆型偏微分方程解的结果。这些结果与几何和概率论有一定的联系。3.小波分析。将小波理论和威尔逊基的相关理论应用于薛定谔算子本征值的渐近分布和拟微分算子的有界性问题。4.渐近分析，数学物理。(1)渐近分析在量子理论的数学物理中的应用。(2)得到了关于散射理论的各种结果，特别是波算子的L有界性。5.正解的结构。更详细地研究了非相对紧致区域上二阶抛物型和椭圆型偏微分方程正解的具体结构。这里的研究还涉及到位势理论、几何和概率论。这些结果清楚地表明，后向分析、傅立叶分析、线性和非线性偏微分方程与数学物理之间存在着密切的联系。每一门学科都有自己的发展历史。我们认为，在把各个领域的研究不断推向深入的同时，从整体上看合作研究项目是非常必要的，应该进一步推动。

This research project aimed at investigating problems related to three disciplines mentioned in the title in the spirit of co-operative work and from a unique viewpoint. Results obtained during three years will be itemized below.1.Methods of Functional and Real Analysis applied to Partial Differential Equations(PDE). Properties of solutions of nonlinear PDE are investigated by these tools and using various results on function spaces. 2.Harmonic Analysis and Complex Analysis. Results are obtained on degenerate elliptic PDE on strongly pseudo-convex domains. These results have some connection with geometry and the theory of probability.3.Wavelet Analysis. Wavelets theory and related theory of Wilson basis are applied to the prob-lem of asymptotic distribution of eigenvalues of Schrodinger operators and the boundedness of pseudo-differential operators.4.Asymptotic Analysis, Mathematical Physics.(1)Application of Asymptotic Analysis to Mathematical Physics of quantum theory.(2)Various results on scattering theroy are obtaind, notably L^P boundedness of wave operators.5.Structure of Positive Solutions.Broard investigations are made on the concrete structure of positive solutions of 2nd order parabolic and elliptic PDEs on non relatively compact domains. Here, the investigation is also related to potential theory, geometry, and the theory of probability.These results manifest clearly strong ties which exist between Rear Analysis, Fourier Analysis, Linear and nonliner PDEs, and Mathematical Physics.Each of these disciplines has certainly its own history of development. We feel that, while pushing researches in each field deeper and deeper, project of co-operative researches from overall viewpoints are much necessary and should be pushed further.