某类四阶非对称微分算子的正则性

On Regularity of Some Fourth-Order Non-Symmetric Differential Operators

本文处理如下四阶非对称微分算子A_λ,A_λ:K(i,j)→A_λK(i,j)的正则性。对于对称微分算子的传统处理,是利用紧算子的重要性质与E.C.Tichmarsh的函数论方法。对于非对称算子,类似的方法会遇到若干麻烦。然而当采用先验估计法,特别利用Opial不等式与内插不等式后,则能比较容易地得到A_λ的范数下界估计及一系列的正则性结果。

This paper deals with the regularity of the following fourth-order non-symmetric differential operator AλAccording to the classical technique for the symmetric differential operators, it can be tackled by using some important properties of compact operators, or E.C.Tichmarsh's functional method.When the operator is nonsymmetric, it is somewhat difficult to use the preceeding method. However, if some estimation techniques, especially the interpolating inequality and Opial inequality, are used, some lower bounds of Aλ's norm and a lot of results of regularity and common regularity can be obtained with ease