Second-Order Equations on Euclidean Open Sets

1. (Numbers)¶ $\mathbb{N}$ denotes the natural numbers and $\mathbb{R}$ the real numbers, carrying the addition, multiplication and order $\le$ of their ordered field structure together with the notation fixed there: $a<b$ for the associated strict order, $a-b$ for $a+(-b)$, and $a^{-1}$ for the multiplicative inverse of a nonzero $a$. We write $|\cdot|$ for the absolute value on $\mathbb{R}$, and call $a\in\mathbb{R}$ positive if $0<a$ and nonnegative if $0\le a$.

2. (Euclidean space, its metric and its topology)¶ Throughout, $q$ denotes a natural number with $q\ge1$. Then $\mathbb{R}^{q}$ is Euclidean space, $\lVert\,\cdot\,\rVert$ its Euclidean norm, and $x\cdot y$ and $x-y$ the dot product and difference of $x,y\in\mathbb{R}^{q}$, while $\mu x$ denotes the scalar multiple of $x$ by $\mu\in\mathbb{R}$. The Euclidean distance $d_{E}$ is a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$, so that $(\mathbb{R}^{q},d_{E})$ is a metric space. The collection $\mathcal{T}_{d_{E}}$ of subsets open in $(\mathbb{R}^{q},d_{E})$ is a topology by Metric Open Sets Form a Topology, and a subset of $\mathbb{R}^{q}$ belongs to it precisely when 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 closure $\operatorname{cl}_{\mathbb{R}^{q}}$, the interior $\operatorname{int}_{\mathbb{R}^{q}}$ and the boundary $\partial_{\mathbb{R}^{q}}$ are taken in $(\mathbb{R}^{q},\mathcal{T}_{d_{E}})$; boundedness refers to $d_{E}$ and compactness to $\mathcal{T}_{d_{E}}$.

3. (Symmetric matrices)¶ $\mathcal{S}(q)$ denotes the set of symmetric real $q\times q$ matrices. For real matrices we write $P+Q$ for the sum, $P-Q$ for the difference, $\mu P$ for the scalar multiple, $PQ$ for the product, $P^{\top}$ for the transpose, $Pz$ for the matrix-vector product with $z\in\mathbb{R}^{q}$, and $I_{q}$ for the identity matrix of size $q$. On $\mathcal{S}(q)$ the relation $\preceq$ is the positive semidefinite ordering; by The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure it is a partial order compatible with the linear structure, and by claim 1 of that lemma a difference of elements of $\mathcal{S}(q)$ again lies in $\mathcal{S}(q)$. For $X,Y\in\mathcal{S}(q)$, $X\preceq Y$ holds if and only if $Y-X$ is positive semidefinite, by The Positive Semidefinite Ordering Compared by Differences. Finally $\lVert P\rVert$ denotes the norm of a symmetric real matrix and $d_{\mathcal{S}(q)}$ the distance between symmetric real matrices, a metric on $\mathcal{S}(q)$ by The Set of Symmetric Real Matrices is a Metric Space.

4. (Functions of class $C^{2}$, gradients and Hessians)¶ Let $V\subseteq\mathbb{R}^{q}$ be open. That a function $\varphi:V\to\mathbb{R}$ is of class $C^{2}$ on $V$ is understood in the sense of clause 3 of the definition of $C^{k}$ maps on a Euclidean open set. For such a $\varphi$ and $x\in V$ we write $D\varphi(x)\in\mathbb{R}^{q}$ for the gradient of $\varphi$ at $x$ and $D^{2}\varphi(x)$ for its Hessian matrix at $x$; by claim 2 of Equality of Mixed Second Partial Derivatives and Symmetry of the Hessian we have $D^{2}\varphi(x)\in\mathcal{S}(q)$.

5. (Semicontinuity, local extrema, and differences of functions)¶ Let $S\subseteq\mathbb{R}^{q}$ and let $f,g:S\to\mathbb{R}$. Upper semicontinuity and lower semicontinuity of $f$ on $S$, and local maxima and local minima of $f$ relative to $S$, are always understood with the ambient metric space $(\mathbb{R}^{q},d_{E})$ of clause 2. Finally, purely as notation, $f-g$ denotes the function from $S$ to $\mathbb{R}$ whose value at $x\in S$ is the difference $f(x)-g(x)$ formed in $\mathbb{R}$ as in clause 1; nothing is asserted about $f-g$ beyond this reading of the symbol.