Penalty-Subordinate Growth of a Function 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, let $D\subseteq\mathbb{R}^{n}$ be open, let $P$ be a penalty on $D$ and let $f:D\to\mathbb{R}$.

1. (Growth from above)¶ The function $f$ has $P$-subordinate growth from above if for every positive $\delta\in\mathbb{R}$ there is $C\in\mathbb{R}$ such that

2. (Growth from below)¶ The function $f$ has $P$-subordinate growth from below if for every positive $\delta\in\mathbb{R}$ there is $C\in\mathbb{R}$ such that