Elementary Properties of the Weak Partial Derivative on the Torus

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$ and real numbers $p$ and $q$ with $1\le p$ and $1\le q$; the cell $Q$, the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$, the integral over $\mathbb{T}^{n}$, the classes $\mathcal{L}^{r}(\mathbb{T}^{n})$ and the spaces $L^{r}(\mathbb{T}^{n})$ with the class map $[\,\cdot\,]$, for a real number $r$ with $1\le r$, the periodic classes $C_{\mathrm{per}}$, $C^{1}_{\mathrm{per}}$ and $C^{\infty}_{\mathrm{per}}$, the restriction $u|_{Q}$, and the partial derivatives $\partial_{i}$ are the ones fixed there. Weak partial derivatives of functions and of classes, and the notation $\partial_{i}U$, are those of The Weak Partial Derivative on the Torus; applied to a member of $C^{1}_{\mathrm{per}}$ the symbol $\partial_{i}$ keeps its classical meaning. Let $\psi\star u$ denote the periodic convolution, let $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$ denote the inner product of $L^{2}(\mathbb{T}^{n})$, and let a sum $\sum_{j=1}^{n}$ be that of Finite Sum Notation in a Field.

Let $i\in[n]$. Then the following hold.

1. (Agreement with the classical derivative)(i) Let $u\in C^{1}_{\mathrm{per}}$. Then $u\in C_{\mathrm{per}}$ and $\partial_{i}u\in C_{\mathrm{per}}$, by claim 3 of Euclidean Space Is Open in Itself with Lattice-Periodic Functions and the Periodic Function Classes (iii) and by Elementary Properties of Lattice-Periodic Functions (ii); so $u|_{Q}$ and $(\partial_{i}u)|_{Q}$ lie in $\mathcal{L}^{r}(\mathbb{T}^{n})$ for every real number $r$ with $1\le r$, by Continuous Periodic Functions are Power-Integrable and Dense on the Torus (i). Moreover $[(\partial_{i}u)|_{Q}]$ is the $i$-th weak partial derivative of $[u|_{Q}]$, in $L^{r}(\mathbb{T}^{n})$ for every such $r$.

2. (Linearity)(ii) Let $U,U'\in L^{p}(\mathbb{T}^{n})$ have $i$-th weak partial derivatives in $L^{q}(\mathbb{T}^{n})$ and let $c\in\mathbb{R}$. Then the $i$-th weak partial derivative of $U+c\,U'$ exists in $L^{q}(\mathbb{T}^{n})$ and

$$\partial_{i}(U+c\,U')=\partial_{i}U+c\,\partial_{i}U' .$$

3. (The test class may be enlarged)(iii) Let $U\in L^{p}(\mathbb{T}^{n})$ have $i$-th weak partial derivative $G\in L^{q}(\mathbb{T}^{n})$, and let $u$ and $g$ be representatives of $U$ and $G$. Then

$$\int_{\mathbb{T}^{n}}u\,\bigl((\partial_{i}\varphi)|_{Q}\bigr)\,dx=-\int_{\mathbb{T}^{n}}g\,(\varphi|_{Q})\,dx\qquad\text{for every }\varphi\in C^{1}_{\mathrm{per}} ,$$

the integrals being real numbers by Pairing an Integrable Function on the Torus with a Continuous Periodic Function (i), since $\varphi$ and $\partial_{i}\varphi$ lie in $C_{\mathrm{per}}$.

4. (Mollification commutes with the weak derivative)(iv) Let $U$, $G$, $u$ and $g$ be as in claim 3. Let $\delta$ and $\varepsilon$ be real numbers with $0<\delta$ and $0<\varepsilon$, let $\rho$ be a mollifier kernel of radius $\delta$ on $\mathbb{R}^{n}$ and let $\rho_{\varepsilon}$ be its rescaling, a mollifier kernel of radius $\varepsilon\delta$. Then $\rho_{\varepsilon}\star u$ and $\rho_{\varepsilon}\star g$ belong to $C^{\infty}_{\mathrm{per}}$ and

$$\partial_{i}(\rho_{\varepsilon}\star u)=\rho_{\varepsilon}\star g\qquad\text{as maps on }\mathbb{R}^{n}.$$

5. (Closure under limits in the square-integrable space)(v) Let $U$, $G$ and, for every natural number $k$, $U_{k}$ and $G_{k}$ be members of $L^{2}(\mathbb{T}^{n})$ such that $G_{k}$ is the $i$-th weak partial derivative of $U_{k}$ for every $k$, and such that the sequences $(U_{k})$ and $(G_{k})$ converge to $U$ and to $G$ in the metric of $L^{2}(\mathbb{T}^{n})$. Then $G$ is the $i$-th weak partial derivative of $U$.

6. (The Sobolev inner product)(vi) Let

$$W=\bigl\{\,U\in L^{2}(\mathbb{T}^{n})\ :\ \text{for every }j\in[n]\text{ the }j\text{-th weak partial derivative of }U\text{ exists in }L^{2}(\mathbb{T}^{n})\,\bigr\}.$$

Then $W$ is a linear subspace of $L^{2}(\mathbb{T}^{n})$, and it contains $[u|_{Q}]$ for every $u\in C^{1}_{\mathrm{per}}$. Moreover the map sending $(U,V)\in W\times W$ to

$$\langle U,V\rangle_{W}=\langle U,V\rangle_{L^{2}}+\sum_{j=1}^{n}\langle\partial_{j}U,\partial_{j}V\rangle_{L^{2}}$$

is an inner product on $W$, so that $W$ together with it is a real inner product space.