矩形区域上Poincaré不等式最佳常数

Poincaré不等式在调和分析、微分方程理论及其数值方法等领域的研究中具有极其重要的作用。但是，Poincaré不等式中最佳常数的确定问题至今仍然未被系统地研究过。运用Hilbert空间广义Fourier正交级数理论，对于一维区间和二维矩形区域上带有Dirichlet边界条件的函数，获得Poincaré不等式中的最佳常数。本方法可推广到高维空间中矩形区域上的问题。

Optimal Constant of Poincaré Inequality on Rectangle Regions

Poincaré inequalities play an important role in the research of many fields such as in harmonic analysis and in theory and numerical method of differential equations. However, to determine the optimum constants of Poincaré inequalities is very difficult in application. Up to now,this problem has been unsolved. In the present paper,based on the theory of the generalized Fourier orthogonal series of Hilbert space,we give some answers for the functions on a interval of R or rectangle region of R^2 with Dirichlet boundary conditions. The method used in this paper can be extended to higherdimensional rectangles to obtain the optimal constants of Poincar inequalities.