Potential theoretic research on elliptic differential equations

利用格林函数和格林算子给出了一类椭圆型微分方程解的精确性质，研究了如下内容：(1)格林算子范数的估计；(2)摄动下的格林函数；(3)正超调和函数的摄动可积性；(4)用余区公式研究摄动。推广了著名的极值长度概念。建立了连接两个集合的极值距离与分隔它们的极值宽度之间的倒数关系。此外，将极值长度推广到向量测度，给出了退化椭圆型方程的极限长度与容量的关系。这是与M.Ohtsuk.教授合作的工作。借助于精细拓扑学证明了不连续超调和函数的广义Cranston-McConnell不等式。这是研究者在J.Analyse Math中的结果的推广。(1996)。M.Mizutani和S.Gardiner研究了精细拓扑、簇集、边值、边界Harnack原理和Martin边界。提交了关于它们的论文。

The research results of the main investigator are mainly stated.Precise properties of solutions to an elliptic differential equation is given by the Green function and the Green operator, We have studied the following :(i) Estimate of the norm of the Green operator.(ii) Green function under perturbation.(iii) Integrability of positive superharmonic functions by perturbation.(iv) Study of perturbation by coarea formula.The well-known notion of extremal length has been generalized. The reciprocal relationship between the extremal distance connecting two sets and the extremal width separating them has been established. Moreover, extremal length is extended to vector measures and the relationship to capacity with respect to a degenerate elliptic equation is given. This is a joint work with Prof. M.Ohtsuka.The generalized Cranston-McConnell inequality for a discontinuous superharmonic function is proved with the aid [of fine topology. This is an extension of the result of the investigator in J.Analyse Math. (1996).Fine topology, cluster sets, boundary values, boundary Harnack principle and the Martin boundary have been studied with M.Mizutani and S.Gardiner. Papers about them are submitted.