III. Approximations by spherical harmonics.
III. Approximations by spherical harmonics.
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DOI：
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发表时间：
2016
影响因子：
1.1
通讯作者：
P. Hartman;A. Wintner
中科院分区：
文献类型：
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作者：
P. Hartman;A. Wintner
本文对自变量个数大于2的椭圆型偏微分方程解，得到与文[4]、[51]中的定理类似的结果。为方便起见，假定自变量个数为3。所考虑的方程是Au+0型方程，其中Au是i的欧几里得拉普拉斯算子，方程的其余部分不出现i的二阶偏导数。此时不考虑用u的二阶导数的更一般的线性组合来代替Au。(在平面上，在适当的光滑性假设下，这种更一般的情况可以通过共形映射化为特例；在空间上，可以使用Korn和Lichstein的摄动方法。)为简单起见，将假设要考虑的偏微分方程式是线性的(这些方法适用于以下类型的非线性方程式
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