📂Partial Differential Equations

Smoothness of Boundaries

Definition¹

Let’s call $U \subset \mathbb{R}^{n}$ a bounded open set. Let $\partial U$ be the boundary of $U$. If there exists a $C^{k}$ function $\gamma = \mathbb{R}^{n-1} \to \mathbb{R}$ satisfying the following for each point $x = (x_{1}, \dots, x_{n}) \in \partial U$ on the boundary, then we say ’the boundary $\partial U$ is $C^{k}$'.

$$ \gamma (x_{1}, x_{2}, \dots, x_{n-1}) = x_{n} $$

Explanation

To rephrase the condition in the definition, there exists a $C^{k}$ function $\gamma$ that makes the below equation hold. Regarding $x \in \partial U$,

$$ U \cap B(x,r) = \left\{ y \in B(x,r) \vert y_{n} \gt \gamma (y_{1},\dots,y_{n-1}) \right\} $$

The reason $\gamma$ is not defined for the entirety of $\partial U$ is because, as shown below, it can have more than one function value at a single point. This is represented in two and three dimensions as follows.