常系数线性PDE方程的解在低维中的叠加表示
Representation of solutions of linear PDE's with constant coefficients as superpositions of solutions in lower dimensions
摘要：
本文给出了用高维常系数线性偏微分方程的低维解表示高维常系数线性偏微分方程的统一方法。它是基于这样的观察，即如果一个（N+1）维变量（N * 2）的函数具有收敛的傅里叶积分表示，那么它可以被写为在低于N+1的任何维数上的旋转平面波的叠加。如果该函数是具有常系数的线性偏微分方程的解，并且另外，该方程在空间变换组下是不变的（例如，旋转，或洛伦兹变换），则叠加是根据相同PDE的低维（不仅是二维）解的解。作为例子，拉普拉斯方程在（N+1）维和波动方程在（1+N）维（N * 2）作为低维的解决方案的叠加的解决方案的表示进行了讨论，并提出了简单的后果。
英文摘要：
A unified approach to the representation of solutions of linear PDE's with constant coefficients in high dimensions in terms of solutions of the same PDE's in lower dimensions is presented. It is based on the observation that if a function of an (N+1)-dimensional variable (N * 2) has a convergent Fourier integral representation, then it may be written as a superposition of rotated plane waves in any number of dimensions lower than N+1. If that function is a solution of a linear PDE with constant coefficients, and, in addition, the equation is invariant under a spatial group of transformations (e.g., rotations, or Lorentz transformations), then the superposition is in terms of solutions of lower- (not only two-) dimensiuonal solutions of the same PDE. As examples, representations of solutions of the Laplace equation in (N+1) dimensions and of the wave equation in (1+N) dimensions (N * 2) as superpositions of solutions in lower dimensions are discussed, and simple consequences are presented.
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