On the Local Behavior of Solutions of Non-Parabolic Partial Differential Equations
On the Local Behavior of Solutions of Non-Parabolic Partial Differential Equations
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DOI：
10.2307/2372496
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发表时间：
1953-07
影响因子：
1.7
通讯作者：
P. Hartman;A. Wintner
中科院分区：
文献类型：
--
作者：
P. Hartman;A. Wintner
本文对于l个自变量超过2的情况，将得到[4]、[5 1 oIn椭圆偏微分方程定理的类比。为了符号简单起见，假设自变量个数为3。所考虑的方程类型为Au+0，其中Au为i的欧几里德拉普拉斯算子，其余部分不出现i的二阶偏导数。方程。此时将不考虑用u的二阶导数的更一般的线性组合i代替Au。 （在平面中，这种更一般的情况可以通过在系数上适当的平滑度假设下进行共形映射来简化为特殊情况；在空间中，可以使用 Korn 和 Lichtenstein 的微扰方法。）为了简单起见，将假定要考虑的偏微分方程是线性的（这些方法适用于下面类型（5）-（6）的非线性方程）。本文的第一部分涉及零附近的解，第二部分涉及孤立奇点附近的解。
In this paper, the analogues of the theorems of [4], [5 1 oIn solutions of elliptic partial differential equations will be obtained for the case where the lnumber of independent variables exceeds 2. For the sake of notational simplicity, it will be assumed that the number of independent variables is 3. The equation to be considered is of the type Au + 0, where Au is the Euclidean Laplacian of i, and no second order partial derivative of i occurs in the rest of the equation. The replacement of Au by a more general linear combinationi of second derivatives of u will not be considered at this time. (In the plane, this more general case can be reduced to the special case by colnformal mappings under suitable smoothness assumptions on the coefficients; in space, perturbatioin imiethods of Korn and Lichtenstein can be used.) For simplicitv, the partial differential equation to be considered will be assumed to be linear (the methods are applicable to non-linear equations of the type (5)-(6) below). The first part of the paper deals with solutions near a zero, the second part with solutions near an isolated singularity.