Ishii's Lemma: Test Data and Matrix Bounds at a Maximum of a Quadratically Penalised Difference

Throughout we work in the setting of Second-Order Equations on Euclidean Open Sets, whose notation, including that of Euclidean Space and Real Matrices: Standing Notation on which it rests, is in force in the dimensions $n$ and $2n=n+n$, for a natural number $n$ with $1\le n$. In addition $\iota:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{2n}$ is the concatenation map, a bijection; for $X,Y\in\mathcal{S}(n)$, $X\oplus Y\in\mathcal{S}(2n)$ is the block diagonal matrix they determine and $-Y=(-1)Y$; and $J\in\mathcal{S}(2n)$ is the doubling matrix. That a quadruple is approximable by test data from above or from below for a function is as defined there. We abbreviate $\lVert z\rVert^{2}=\lVert z\rVert\lVert z\rVert$; $\tfrac{s}{2}$ 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; and $3=2+1$, $6=3+3$.

Let $\Omega\subseteq\mathbb{R}^{n}$ be open, let $u:\Omega\to\mathbb{R}$ be upper semicontinuous on $\Omega$, let $v:\Omega\to\mathbb{R}$ be lower semicontinuous on $\Omega$, and let $\alpha\in\mathbb{R}$ be positive. Let $\hat{x},\hat{y}\in\Omega$ and suppose there is a positive $\delta\in\mathbb{R}$ such that

$$u(x)-v(y)-\tfrac{\alpha}{2}\lVert x-y\rVert^{2}\ \le\ u(\hat{x})-v(\hat{y})-\tfrac{\alpha}{2}\lVert\hat{x}-\hat{y}\rVert^{2}$$

for all $x,y\in\Omega$ with $d_{E}(x,\hat{x})<\delta$ and $d_{E}(y,\hat{y})<\delta$. Put $p=\alpha(\hat{x}-\hat{y})\in\mathbb{R}^{n}$.

Then there exist $X,Y\in\mathcal{S}(n)$ for which the following hold.

1. (Test data)(i) The quadruple $\bigl(\hat{x},u(\hat{x}),p,X\bigr)$ is approximable by test data from above for $u$, and the quadruple $\bigl(\hat{y},v(\hat{y}),p,Y\bigr)$ is approximable by test data from below for $v$; in both cases the open set is $\Omega$.

2. (Matrix bound)(ii)

$$-3\alpha I_{2n}\ \preceq\ X\oplus(-Y)\ \preceq\ 3\alpha J .$$

3. (Quadratic form bound)(iii) For all $\xi,\eta\in\mathbb{R}^{n}$,

$$-3\alpha\bigl(\lVert\xi\rVert^{2}+\lVert\eta\rVert^{2}\bigr)\ \le\ \xi\cdot(X\xi)-\eta\cdot(Y\eta)\ \le\ 3\alpha\lVert\xi-\eta\rVert^{2}.$$

4. (Ordering)(iv) $X\preceq Y$.

5. (Norm bound)(v) $\lVert X\rVert\le 6\alpha$ and $\lVert Y\rVert\le 6\alpha$.