Fundamental solutions of parabolic differential equations and boundary value problems
Fundamental solutions of parabolic differential equations and boundary value problems
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抛物型微分方程和边值问题的基本解
DOI：
10.4099/jjm1924.27.0_55
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发表时间：
1957
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影响因子：
--
通讯作者：
Seizô Itô
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文献类型：
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作者：
Seizô Itô
利用抛物型微分方程的基本解定义的有界算子的参数半群，证明了椭圆型微分方程混合型边界条件下绿色函数的存在性.第三章利用前两章的结果，给出了带边界条件的椭圆型微分方程的特征函数展开式。抛物型微分方程的基本解已被许多作者用稍有不同的方法研究过，特别是W。Feller [11，12]0），F. G. Dressel [9]，K. [57][58][59] (See在[33]中我们证明了有界区域上具有混合型边界条件的抛物型微分方程基本解的存在性，并给出了在给定初始条件和边界条件下抛物型方程解的显式表达式。在本文的第一章中，我们将把文[33]中的结果推广到微分方程所考虑的区域不一定有界的情形。为了构造满足混合型边界条件的基本解，考虑依赖于给定微分方程系数的局部坐标似乎是有用的（见·3），即使该方程是在欧几里德m-空间的子域中考虑的。因此，我们考虑微分方程在一个子域的流形在本文中，我们在以前的文件[32，33]。Banach空间中有界算子的单参数半群理论是由E。Hille [23]和K.”Yosida [51]，它与抛物型微分方程理论密切相关，例如，可以在[12]，[13]和[52]中看到。在第二章中，我们将证明由基定义的半群的强可微性的几个定理。
parameter semi-group of bounded operators defined by the fundamental solution of a parabolic differential equation, and show the existence of Green functions of elliptic differential equations with a mixed-type boundary condition. In the third chapter, making use of results of above two chapters, we shall give the formula of eigenfunction expansions for elliptic differential equations with boundary conditions. Fundamental solutions of parabolic differential equations have been studied by many authors by means of slightly different approach, especially by W. Feller [11, 12]0), F. G. Dressel [9], K. Yosida [57, 58] and the present author [32, 33]. (See also other papers referred at the end of this papers) we have shown in [33] the existence of the fundamental solution of parabolic differen tial equations with mixed-type boundary condition in a bounded domain, and an explicit formula to express the solution of the parabolic equation with given initial condition and boundary condition. In Chapter I of the present paper, we shall extend those results in [33] to the case where the domain in which the differential equation is considered is not necessarily bounded. In order to con struct a fundamental solution satisfying a mixed-type boundary condition, it seems to be useful to consider local coordinates depending on coefficients of given differential equation (see • ̃3) even if the equation is considered in a subdomain of the euclidean m-space. So we consider differential equations in a subdomain of a manifold in this paper as we did in previous papers [32, 33]. The theory of one-parameter semi-groups of bounded operators in Banach spaces has been established by E. Hille [23] and K. "Yosida [51], and it is closely related with the theory of parabolic differential equations as may be seen, for example, in [12], [13] and [52]. In Chapter II, we shall prove some theorems concerning the strong differentiability of the semi-group defined by the funda