The classification of possible singular supports

Question

I need to find the solutions of $D_{x_1}u=0$ on $\mathbb{R}^{n}$ and to classify the possible singular supports.

Any one have an idea how to solve this kind of question?

Thanks!

Answer 1

The equation says that $u$ kills all test functions $\varphi$ such that $\varphi=\psi_{x_1}$ for some other test function $\psi$. So, the first step is to observe that

• $\varphi=\psi_{x_1}$ for some $\psi$ if and only if the integral of $\varphi$ over every line in $x_1$ direction is zero.

To move this further, introduce the operator $I:\mathcal D(\mathbb R^n)\to\mathcal D(\mathbb R^{n-1})$ which sends $\varphi$ to $\int \varphi\,dx_1$. We just saw that $u$ vanishes on the kernel of $I$. Therefore, $u$ factors through $I$: namely, $u=v\circ I$ for some $v\in\mathcal D'(\mathbb R^{n-1})$. This is a characterization of the solutions of $u_{x_1}=0$.

The singular support of $v$ above can be any closed set $A\subset \mathbb R^{n-1}$. The singular support of $u$ is $\mathbb R\times A$.