A synthetic study of positive solutions to elliptic and parabolic partial differential equations

M.Murata和K.Ishige研究了二阶抛物型方程在riemann流形和R^n定域上Cauchy问题非负解的唯一性，并通过方程和流形在无穷远处的渐近性质给出了方程和流形唯一性的一个尖锐而一般的充分条件。师范学报，30(2001),171-223。这是一个简单而一般的结果，它统一了以往所有关于唯一性的结果。通过引入热逸的概念，M.Murata也给出了非唯一性的一个尖锐而一般的充分条件（数学与数学，327(2003),203-226）。村田研究了二阶椭圆型偏积方程正解的结构，并利用微扰理论、抛物型方程基本解的估计和抛物型方程非负解的唯一性定理确定了马丁边界和马丁核(j .数学学报，1994 （2002),53-141;j .数学社会学报，57(2005),1-27）。Murata和T.tsuchida给出了R^n上具有周期系数的椭圆方程的Green函数在无穷远处的渐近性，并明确地确定了它的Martin边界（J.Diff.Eq.,195(2003),82-118）。这解决了S.Agmon自1984年以来的一个开放性问题。

M.Murata and K.Ishige studied uniqueness of nonnegative solutions of the Cauchy problem to second order parabolic equations on Riemannian manifolds and domains of R^n, and gave a sharp and general sufficient condition for the uniqueness via asymptotic properties at infinity of the equation and manifolds (Ann.Scuola Normale Sup. Pisa,30(2001),171-223). This is a simple and general result which unifies all previous results on the uniqueness. By introducing tne notion of heat escape, M.Murata also gave a sharp and general sufficient condition for the non-uniqueness (Math.Ann.,327(2003),203-226).M.Murata investigated the structure of positive solutions of second order elliptic equations in skew product form, and determined the Martin boundary and Martin kernel by exploiting and developing perturbation theory, estimates of fundamental solutions for parabolic equations, and uniqueness theorems for nonnegative solutions of parabolic equations (J.Func.Anal.,194(2002),53-141;J.Math.Soc.Japan,57(2005),1-27).M.Murata and T.tsuchida gave the asymptotics at infinity of the Green function for an elliptic equation with periodic coefficients on R^n, and explicitly determined the Martin boundary for it (J.Diff.Eq.,195(2003),82-118). This solves an open problem by S.Agmon since 1984.