Functions Differentiable on the Boundaries of Regions
Functions Differentiable on the Boundaries of Regions
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DOI：
10.1007/978-1-4612-2972-8_18
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发表时间：
1934-07
影响因子：
4.9
通讯作者：
H. Whitney
中科院分区：
文献类型：
--
作者：
H. Whitney
1.引言。设函数I(XI，…，x，…)定义在n维空间E的有界区域2R上，设I在R中有连续的第m阶偏导数，即I在R中“属于em类”，如果B是R的边界，我们如何判定I在R+B中是否属于em类？如果I的导数在B上取边值，则将B上的导数定义为它们在R中取值的极限是很自然的，但很容易构造一个区域R和一个函数I，使得1(0和lt；k~m)的第k个偏导数在R+B中连续，而在B的某一边界点P，I是P或P连续的；3在这种情况下，说R+B中的em类FI似乎是不合理的。如果可以将I的定义推广到包含R+B的区域，使得它在那里有连续的第m个偏导数，那么我们可以肯定地说，I在R+B中是em类；这就是我们要使用的定义。在本文中，我们证明了对于某些区域，对于闭区域中具有em类的函数，m阶偏导数在边界上连续是充分的。
1. Introduction. Let the function I (XI,..., x,,) be defined in the bounded region2 R of n-space E, and suppose I has continuous mth partial derivatives in R, ie I" is of class em" in R. If B is the boundary of R, how shall we decide whether I is of class em in R+ B? If the derivatives of I take on boundary values on B, it would be natural to define the derivatives on B as the limit of their values in R. But it is easy to construct a region R and a function I such that the kth partial derivatives of 1 (0< k~ m) are continuous in R+ B, whereas at a certain boundary point P of B, I is p. ot continuous; 3 it seems unreasonable in this case to say that fis of class em in R+ B. If it is possible to extend the definition of I throughout a region containing R+ B so that it has continuous mth partial derivatives there, we may then surely say that I is of class em in R+ B; this is the definition we shall use. We show in this note that, for certain regions, for a function to be of class em in the closed region, it is sufficient that the mth partial derivatives be continuous on the boundary.