Penalty on an Open Subset of Euclidean Space

In the setting of Second-Order Equations on Euclidean Open Sets, let $n\ge1$ be a natural number and let $D\subseteq\mathbb{R}^{n}$ be open.

A penalty on $D$ is a function $P:D\to\mathbb{R}$ with the following two properties.

1. (Regularity)¶ $P$ is of class $C^{2}$ on $D$.

2. (Compact sublevel sets)¶ For every $t\in\mathbb{R}$ the set $\{x\in D:P(x)\le t\}$ is compact.