Basic Properties of the Sublevel Sets of a Penalty

In the setting of Second-Order Equations on Euclidean Open Sets, let $n\ge1$ be a natural number, let $D\subseteq\mathbb{R}^{n}$ be open and let $P$ be a penalty on $D$. For $R\in\mathbb{R}$ write $D_{R}=\{x\in D:P(x)<R\}$, and write $\overline{D_{R}}=\operatorname{cl}_{\mathbb{R}^{n}}(D_{R})$ and $\partial D_{R}=\partial_{\mathbb{R}^{n}}D_{R}$.

Then the following hold.

1. (Lower bound)¶ There is $m\in\mathbb{R}$ with $m\le P(x)$ for every $x\in D$.

2. (Sublevel sets)¶ For every $R\in\mathbb{R}$ the set $D_{R}$ is open and bounded, $\overline{D_{R}}$ is compact, and $\overline{D_{R}}\subseteq\{x\in D:P(x)\le R\}$; in particular $\overline{D_{R}}\subseteq D$.

3. (Boundary)¶ For every $R\in\mathbb{R}$ and every $x\in\partial D_{R}$ one has $x\in D$ and $P(x)=R$.