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

M.Murata和K.Ishige研究了R^n上黎曼流形和区域上二阶抛物型方程柯西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(出现在ANN中)给出了一个简洁而一般的唯一性充分条件.斯库拉·诺曼·苏普。比萨。)。这是一个简单而普遍的结果，统一了前人关于唯一性的所有结果。M.Murata研究了二阶椭圆型方程斜积形式正解的结构，并利用和发展了扰动理论和抛物型方程基本解的估计，确定了Martin边界和Martin核。这一结果被Y.Pinchover，A.Grigor‘yan，V.Maz’ya，S.Gardiner等人所回顾，并被提交。K.Ishige研究了R^n区域上二阶抛物型方程Dirichlet和Neumann边值问题非负解的唯一性，并通过方程和区域的无穷远处的渐近性质给出了其唯一性的精确充分条件。对于Dirichlet问题，他建立了最优唯一性定理(《微分方程式杂志》，158(1999)，251-290)。对于Neumann问题，他揭示了与Dirichlet问题的很大不同，并建立了属于比Dirichlet问题更广的一类区域的唯一性定理。

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 (to appear in Ann. Scuola Normale Sup. Pisa.). This is a simple and general result which unifies all previous results on the uniqueness.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 developping perturbation theory and estimates of fundamental solutions for parabolic equations. This result was reviewed by such experts as Y.Pinchover, A.Grigor'yan, V.Maz'ya, S.Gardiner, and has been submitted.K.Ishige studied uniqueness of nonnegative solutions of the Dirichlet and Neumann boundary value problems to second order parabolic equations on domains of R^n, and gave sharp sufficient conditions for the uniqueness via asymptotic properties at infinity of the equation and domains. For the Dirichlet problem, he established an optimal uniqueness theorem (Journal of Differential Equations, 158(1999), 251 -290). As for the Neumann problem, he revealed a large difference from the Dirichlet problem and established a uniqueness theorem for domains belonging to a class wider than that for the Dirichlet probelm.