Structure of positive solutions to second order elliptic and parabolic partial differential equations

M.Murata和K.Ishige研究了二阶抛物型方程Cauchy问题非负解的唯一性，并利用方程的内在度量给出了方程系数在无穷远处的最优增长率.他们进一步深入研究了唯一性问题，建立了一个尖锐而普遍的唯一性定理，统一了以往关于唯一性的所有结果。这一结果由A.安科纳、L.萨洛夫-科斯特、K. Sturm，E. B. Davies，A.Grigor'yan等，M.Murata在二阶椭圆型方程正解的Martin理论中引入了半小扰动的概念，稳定地建立了半小扰动下正解的结构，并给出了扰动是半小扰动的充分条件。这一结果与二阶抛物型方程Cauchy问题非负解的非唯一性、概率论中扩散过程的寿命估计以及椭圆型方程正解的结构有关。平肖弗（Y.Pinchover）和平斯基（R. O. Pinsky）等专家对它作了很高的估计，最近相川（H.Aikawa）和平肖弗（Y. Pinchover）也给出了与之相关的结果。村田还出版了一份调查的结构正解固定薛定谔方程。M.村田和K.仓田出版了一本书的理论椭圆和抛物偏微分方程是基本的上述和其他调查偏微分方程。K.Kurata也给出了几个关于椭圆型方程的结果。

M.Murata and K.Ishige studied uniqueness of nonnegative solutions of the Cauchy problem to second order parabolic equations, and gave optimal growth rates at infinity of the coefficients of equations via intrinsic metrics for equations. They further investigated the uniqueness problem deeply, and established a sharp and general uniqueness theorem which unifies all previous results on the uniqueness. This result was reviewed by such experts as A.Ancona, L.Salloff-Coste, K.Th. Sturm, E.B.Davies, A.Grigor'yan, and has been submitted.M.Murata introduced a notion of semismall perturbation in the Martin theory for positive solutions of second order elliptic equations, established stably of the structure of positive solutions under semismall perturbations, and gave sufficient conditions for perturbations to be semismall. This result is related to non-uniqueness of nonnegative solutions of the Cauchy problem to second order parabolic equations, lifetime estimates of diffusion processes in probability theory, and the structure of positive solutions to elliptic equations. It is highly estimated by such experts as Y.Pinchover and R.O.Pinsky, and results related to it have been given recently by H.Aikawa and Y.Pinchover. M.Murata also published a survey on the structure of positive solutions to stationary Schrdinger equations.M.Murata and K.Kurata published a book on the theory of elliptic and parabolic partial differential equations which is fundamental for the above and other investigations on partial differential equations . K.Kurata also gave several results on elliptic equations.