Second-Order Equations on Euclidean Open Sets

This setting fixes the standing notation shared by results on second-order equations on open subsets of Euclidean space. It introduces no new concept and asserts nothing beyond the identifications recorded below, each of which is justified by the reference attached to it. It rests on Euclidean Space and Real Matrices: Standing Notation and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, from which the numerical, Euclidean, matrix and calculus notation is imported clause by clause.

1. (Numbers)(i) The real numbers $\mathbb{R}$, carrying the addition, multiplication and order $\le$ of their ordered field structure together with the associated strict order $a<b$, the difference $a-b$ and the multiplicative inverse $a^{-1}$ of a nonzero $a$, the absolute value $|\cdot|$ on $\mathbb{R}$, the natural numbers $\mathbb{N}$, the initial segments $[q]$, and sequences and their convergence are as fixed there, as are the terms positive and nonnegative.

2. (Euclidean space, its metric and its topology)(ii) Throughout, $q$ denotes a natural number with $q\ge1$, and the notation of this clause and of clauses 3 to 5 is introduced for every such $q$ simultaneously. The Euclidean space $\mathbb{R}^{q}$ as a real vector space, with the sum $x+y$ of points, the scalar multiple $\mu x$ by $\mu\in\mathbb{R}$, the difference $x-y$ and the dot product $x\cdot y$, the Euclidean norm $\lVert\,\cdot\,\rVert$, the Euclidean distance $d_{E}$ satisfying $d_{E}(x,y)=\lVert x-y\rVert$, the metric space $(\mathbb{R}^{q},d_{E})$, the origin $0_{\mathbb{R}^{q}}$, and the notions of openness, closedness, boundedness and compactness are as fixed there. A subset of $\mathbb{R}^{q}$ is open in $(\mathbb{R}^{q},d_{E})$ if and only if it is open in the Euclidean sense, by Euclidean Openness Agrees with Metric Openness on $\mathbb{R}^n$; such a subset is called open without further qualification. The collection $\mathcal{T}_{d_{E}}$ of open subsets of $\mathbb{R}^{q}$ is a topology by The Open Subsets of a Metric Space Form a Topology, and the closure $\operatorname{cl}_{\mathbb{R}^{q}}$, the interior $\operatorname{int}_{\mathbb{R}^{q}}$ and the boundary $\partial_{\mathbb{R}^{q}}$ are taken in $(\mathbb{R}^{q},\mathcal{T}_{d_{E}})$.

3. (Symmetric matrices)(iii) The real matrices, with the sum $P+Q$, the difference $P-Q$, the scalar multiple $\mu P$, the product $PQ$, the transpose $P^{\top}$, the matrix-vector product $Pz$ with $z\in\mathbb{R}^{q}$, and the identity matrix $I_{q}$ of size $q$, the set $\mathcal{S}(q)$ of symmetric real $q\times q$ matrices, which is closed under sums, differences and scalar multiples, the positive semidefinite ordering $\preceq$ on $\mathcal{S}(q)$, which is reflexive, transitive and antisymmetric, is compatible with adding a fixed element of $\mathcal{S}(q)$ to both sides and with multiplying both sides by a nonnegative real number, and under which $X\preceq Y$ holds exactly when $Y-X$ is positive semidefinite, and the norm $\lVert P\rVert$ of a symmetric real matrix together with the distance $d_{\mathcal{S}(q)}$, a metric on $\mathcal{S}(q)$, are all as fixed there.

4. (Functions of class $C^{2}$, gradients and Hessians)(iv) Let $V\subseteq\mathbb{R}^{q}$ be open. For a function $\varphi:V\to\mathbb{R}$, that $\varphi$ is of class $C^{2}$ on $V$, and, for such a $\varphi$ and $x\in V$, its gradient $D\varphi(x)\in\mathbb{R}^{q}$ and Hessian matrix $D^{2}\varphi(x)$, which lies in $\mathcal{S}(q)$, are understood as fixed there.

5. (Semicontinuity, local extrema, and differences of functions)(v) Let $S\subseteq\mathbb{R}^{q}$ and let $f,g:S\to\mathbb{R}$. The upper semicontinuity and lower semicontinuity of $f$ on $S$, the local maxima and the local minima of $f$ relative to $S$, and the functions $f+g$, $f-g$ and $\mu f$ for $\mu\in\mathbb{R}$ are as fixed there, always with the ambient metric space $(\mathbb{R}^{q},d_{E})$ of clause 2. Nothing is asserted about $f+g$, $f-g$ and $\mu f$ beyond this reading of the symbols.