Test Function with Vanishing Partial Derivative
Theorem
Let $\phi \in \map \DD {\R^2}$ be a test function such that:
- $\tuple {x, y} \stackrel \phi {\longrightarrow} \map \phi {x, y}$
Suppose $\phi$ is a solution to the following partial differential equation:
- $\ds \dfrac {\partial \phi}{\partial x} = 0$
Then $\phi$ is identically $0$.
Proof
$\ds \dfrac {\partial \phi}{\partial x} = 0$ implies that:
- $\forall x \in \R : \map \phi {x, y} = \map C y$
where $C : \R \to \C$ is a complex-valued function.
By definition, $\phi$ is a test function.
Hence, $\phi$ must have a compact support $\Omega \subset \R^2$.
Let $\map {B^-_\epsilon} 0 \subset \R^2$ be a closed ball in Euclidean space such that:
- $\Omega \subseteq \map {B^-_\epsilon} 0$
By Closed Ball in Euclidean Space is Compact, $\map {B^-_\epsilon} 0$ is a compact.
Then $\map {B^-_\epsilon} 0$ also qualifies as a compact support of $\phi$.
By definition of a test function:
- $\forall y \in \R : \size y > \epsilon \implies \map \phi {x, y} = 0$
But for each $y \in \R$ we have that $\map \phi {x, y} = \map C y$ is a constant.
By smoothness of test functions, this constant has to be the same for all $y \in \R$.
Hence, $\map \phi {x, y} = 0$.
$\blacksquare$
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 6.1$: A glimpse of distribution theory. Test functions, distributions, and examples