Fundamental solutions and Green’s functions in higher dimensions

Abstract

Distributions of several variables $x_1 \ldots, x_n$ and operations with them are defined similarly to the case n = 1 (see Sections 19, 20).

• a. For example, the Dirac δ-function in $\mathbb{R}^n$ is defined by

  $$\langle \partial_{(n)},\varphi\rangle \equiv \varphi(0), \qquad \forall \varphi \in C^{\infty}_0 (\mathbb{R}^n);$$

  (28.1)

• b. For each distribution $u(x) \in \mathcal{D}^\prime (\mathbb{R}^n),$

  $$\left\langle \frac{\partial u}{\partial x_2},\varphi \right\rangle \equiv - \left\langle u, \frac{\partial \varphi}{\partial x_2} \right\rangle, \qquad \forall\varphi \in C^{\infty}_0 (\mathbb{R}^n).$$

  (28.2)