Theorem on Sums for Two Upper Semicontinuous Functions

Throughout we work in the setting of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, whose notation, including that of Euclidean Space and Real Matrices: Standing Notation on which it rests, is in force in the dimensions $n_{1}$, $n_{2}$ and $N=n_{1}+n_{2}$, for natural numbers $n_{1}$ and $n_{2}$ with $1\le n_{1}$ and $1\le n_{2}$. In particular $\iota:\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}\to\mathbb{R}^{N}$ is the concatenation map, a bijection. We use in addition the square $A^{2}=AA$ of a square real matrix and the notation $-aI_{N}$ for $(-a)I_{N}$, an element of $\mathcal{S}(N)$ for every real $a$; and for $X_{1}\in\mathcal{S}(n_{1})$ and $X_{2}\in\mathcal{S}(n_{2})$, $X_{1}\oplus X_{2}\in\mathcal{S}(N)$ is the block diagonal matrix of Block Diagonal Symmetric Matrices and the Blocks of a Symmetric Matrix (i). That a quadruple is approximable by test data from above for a function on an open set is as defined there.

Let $\Omega_{1}\subseteq\mathbb{R}^{n_{1}}$ and $\Omega_{2}\subseteq\mathbb{R}^{n_{2}}$ be open and, for $i\in\{1,2\}$, let $u_{i}:\Omega_{i}\to\mathbb{R}$ be upper semicontinuous on $\Omega_{i}$. Put

$$\Omega=\{\,\iota(\xi,\eta)\ :\ \xi\in\Omega_{1},\ \eta\in\Omega_{2}\,\},$$

which is open in $\mathbb{R}^{N}$ by Twice Differentiability of a Sum in Separated Variables (i), and let $w:\Omega\to\mathbb{R}$ be the function determined by

$$w\bigl(\iota(\xi,\eta)\bigr)=u_{1}(\xi)+u_{2}(\eta)\qquad(\xi\in\Omega_{1},\ \eta\in\Omega_{2}),$$

which is well defined because $\iota$ is injective.

Let $V\subseteq\mathbb{R}^{N}$ be open with $\Omega\subseteq V$, let $\varphi:V\to\mathbb{R}$ be of class $C^{2}$ on $V$, and let $\hat{x}\in\Omega$ be a point at which $w-\varphi$ has a local maximum relative to $\Omega$. Let $(\hat{x}_{1},\hat{x}_{2})\in\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}$ be the unique pair with $\iota(\hat{x}_{1},\hat{x}_{2})=\hat{x}$, so that $\hat{x}_{i}\in\Omega_{i}$ for $i\in\{1,2\}$, let $(p_{1},p_{2})$ be the unique pair with $\iota(p_{1},p_{2})=D\varphi(\hat{x})$, and put $A=D^{2}\varphi(\hat{x})\in\mathcal{S}(N)$.

Let $\varepsilon\in\mathbb{R}$ be positive, so that $\varepsilon^{-1}$ exists and is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field; the matrix $A+\varepsilon A^{2}$ lies in $\mathcal{S}(N)$ by A Weighted Young Inequality and the Splitting of a Quadratic Form (ii).

Then there exist $X_{1}\in\mathcal{S}(n_{1})$ and $X_{2}\in\mathcal{S}(n_{2})$ for which both of the following hold.

1. (Test data for each summand)(i) For each $i\in\{1,2\}$ the quadruple $\bigl(\hat{x}_{i},u_{i}(\hat{x}_{i}),p_{i},X_{i}\bigr)$ is approximable by test data from above for $u_{i}$, the domain being $\Omega_{i}$.

2. (Two-sided matrix bound)(ii)

$$-\bigl(\varepsilon^{-1}+\lVert A\rVert\bigr)I_{N}\;\preceq\;X_{1}\oplus X_{2}\;\preceq\;A+\varepsilon A^{2}.$$