Quadruple Approximable by Test-Function Data

In the setting of Second-Order Equations on Euclidean Open Sets, let $n\ge1$ be a natural number, let $U\subseteq\mathbb{R}^{n}$ be open, let $u:U\to\mathbb{R}$, let $x_{0}\in U$, let $p\in\mathbb{R}^{n}$ and let $X\in\mathcal{S}(n)$.

For a function $\varphi:U\to\mathbb{R}$ of class $C^{2}$ on $U$ and $y\in U$, the gradient $D\varphi(y)$ lies in $\mathbb{R}^{n}$ and the Hessian $D^{2}\varphi(y)$ lies in $\mathcal{S}(n)$ by that clause.

1. (From above)(i) The quadruple $\bigl(x_{0},u(x_{0}),p,X\bigr)$ is approximable by test data from above for $u$ if for every positive $\varepsilon\in\mathbb{R}$ there exist $y\in U$ and a function $\varphi:U\to\mathbb{R}$ of class $C^{2}$ on $U$ such that $u-\varphi$ has a local maximum at $y$ relative to $U$ and

$$d_{E}(y,x_{0})<\varepsilon,\qquad |u(y)-u(x_{0})|<\varepsilon,\qquad \lVert D\varphi(y)-p\rVert<\varepsilon,\qquad d_{\mathcal{S}(n)}\bigl(D^{2}\varphi(y),X\bigr)<\varepsilon .$$

2. (From below)(ii) The quadruple $\bigl(x_{0},u(x_{0}),p,X\bigr)$ is approximable by test data from below for $u$ if for every positive $\varepsilon\in\mathbb{R}$ there exist $y\in U$ and a function $\varphi:U\to\mathbb{R}$ of class $C^{2}$ on $U$ such that $u-\varphi$ has a local minimum at $y$ relative to $U$ and the four displayed inequalities of clause 1 hold.