Quadratic Test Functions, Limits, Translation and Locality for Approximability by Test Data

In the setting of Second-Order Equations on Euclidean Open Sets we use the real numbers and absolute value $|\cdot|$, the Euclidean norm $\lVert\,\cdot\,\rVert$, dot product, sum and difference of points, Euclidean distance $d_{E}$ and notion of openness, the set $\mathcal{S}(q)$ of symmetric real matrices, the identity matrix $I_{q}$, the matrix-vector product, the norm $\lVert\,\cdot\,\rVert$ and the distance $d_{\mathcal{S}(q)}$, the notion of a function of class $C^{2}$ together with its gradient $D\varphi$ and Hessian $D^{2}\varphi$, and the notions of local maximum and local minimum relative to a subset, all with a natural number $n$ satisfying $1\le n$. We abbreviate $\lVert z\rVert^{2}=\lVert z\rVert\cdot\lVert z\rVert$, and $\tfrac{s}{2}$ denotes the product of $s\in\mathbb{R}$ with the multiplicative inverse of $2=1+1$, which exists by claim 8 of Elementary Order Arithmetic in an Ordered Field.

Convergence of a sequence in a metric space is as defined there; convergence in $\mathbb{R}^{n}$ refers to $d_{E}$, convergence in $\mathcal{S}(n)$ to $d_{\mathcal{S}(n)}$, and convergence in $\mathbb{R}$ to the metric of The Absolute Value Metric on the Real Numbers. Twice differentiability at a point with a given first-order coefficient and Hessian is as defined there. That a quadruple is approximable by test data from above or from below for a function is as defined there.

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 $X\in\mathcal{S}(n)$. Then the following hold.

1. (Quadratic test functions suffice) (i) The quadruple $\bigl(x_{0},u(x_{0}),p,X\bigr)$ is approximable by test data from above for $u$ if and only if for every positive $\varepsilon\in\mathbb{R}$ there exist $y\in U$, $r\in\mathbb{R}^{n}$, $Y\in\mathcal{S}(n)$ and a positive $\delta\in\mathbb{R}$ such that

$$d_{E}(y,x_{0})<\varepsilon,\qquad |u(y)-u(x_{0})|<\varepsilon,\qquad \lVert r-p\rVert<\varepsilon,\qquad d_{\mathcal{S}(n)}(Y,X)<\varepsilon,$$

and

$$u(z)\;\le\;u(y)+r\cdot(z-y)+\tfrac{1}{2}\,(z-y)\cdot\bigl(Y(z-y)\bigr)\qquad\text{for every }z\in U\text{ with }d_{E}(z,y)<\delta .$$

The corresponding statement holds for approximability from below, with the last displayed inequality replaced by

$$u(z)\;\ge\;u(y)+r\cdot(z-y)+\tfrac{1}{2}\,(z-y)\cdot\bigl(Y(z-y)\bigr).$$

2. (Points of twice differentiability) (ii) If $u$ is twice differentiable at $x_{0}$ with first-order coefficient $p$ and Hessian $X$, then $\bigl(x_{0},u(x_{0}),p,X\bigr)$ is approximable by test data both from above and from below for $u$.

3. (Limits of test data) (iii) Let $(y_{k})_{k\in\mathbb{N}}$ be a sequence in $U$, let $(p_{k})_{k\in\mathbb{N}}$ be a sequence in $\mathbb{R}^{n}$ and let $(X_{k})_{k\in\mathbb{N}}$ be a sequence in $\mathcal{S}(n)$ such that for every $k\in\mathbb{N}$ the quadruple $\bigl(y_{k},u(y_{k}),p_{k},X_{k}\bigr)$ is approximable by test data from above for $u$. Suppose $(y_{k})$ converges to $x_{0}$ in $\mathbb{R}^{n}$, $\bigl(u(y_{k})\bigr)$ converges to $u(x_{0})$ in $\mathbb{R}$, $(p_{k})$ converges to $p$ in $\mathbb{R}^{n}$ and $(X_{k})$ converges to $X$ in $\mathcal{S}(n)$. Then $\bigl(x_{0},u(x_{0}),p,X\bigr)$ is approximable by test data from above for $u$. The corresponding statement holds for approximability from below.

4. (Translation and affine perturbation) (iv) Let $b,q\in\mathbb{R}^{n}$ and $c\in\mathbb{R}$, put $U-b=\{z\in\mathbb{R}^{n}:z+b\in U\}$, which is open by Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity (ii), and let $\tilde{u}:U-b\to\mathbb{R}$ be given by $\tilde{u}(z)=u(z+b)+q\cdot z+c$. Then $\bigl(x_{0},u(x_{0}),p,X\bigr)$ is approximable by test data from above for $u$ if and only if $\bigl(x_{0}-b,\tilde{u}(x_{0}-b),p+q,X\bigr)$ is approximable by test data from above for $\tilde{u}$. The corresponding statement holds for approximability from below.

5. (Locality) (v) Let $U'\subseteq\mathbb{R}^{n}$ be open, let $u':U'\to\mathbb{R}$, and let $W\subseteq\mathbb{R}^{n}$ be open with $x_{0}\in W$, $W\subseteq U$, $W\subseteq U'$ and $u(z)=u'(z)$ for every $z\in W$. If $\bigl(x_{0},u(x_{0}),p,X\bigr)$ is approximable by test data from above for $u$, then $\bigl(x_{0},u'(x_{0}),p,X\bigr)$ is approximable by test data from above for $u'$. The corresponding statement holds for approximability from below.