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 $C_{\mathrm{per}}$ and $C^{\infty}_{\mathrm{per}}$, the restriction $\varphi|_{Q}$, and the partial derivatives $\partial_{i}$ are the ones fixed there.

Every member of $C^{\infty}_{\mathrm{per}}$ belongs to $C_{\mathrm{per}}$, a smooth map on $\mathbb{R}^{n}$ being continuous by claim 3 of Euclidean Space Is Open in Itself and periodicity being the same condition for the two classes by Lattice-Periodic Functions and the Periodic Function Classes (iii); and $\partial_{i}\varphi\in C^{\infty}_{\mathrm{per}}$ whenever $\varphi\in C^{\infty}_{\mathrm{per}}$ and $i\in[n]$, by Elementary Properties of Lattice-Periodic Functions (ii). Consequently, for $v\in\mathcal{L}^{1}(\mathbb{T}^{n})$ and $\varphi\in C^{\infty}_{\mathrm{per}}$, the products $v\,(\varphi|_{Q})$ and $v\,((\partial_{i}\varphi)|_{Q})$ lie in $\mathcal{L}^{1}(\mathbb{T}^{n})$ and the integrals written below are real numbers, by Pairing an Integrable Function on the Torus with a Continuous Periodic Function (i). Finally, every member of $\mathcal{L}^{r}(\mathbb{T}^{n})$ lies in $\mathcal{L}^{1}(\mathbb{T}^{n})$ by The Periodic Extension of a Function on the Unit Cell (iv), so the same applies to representatives of members of $L^{r}(\mathbb{T}^{n})$.

Let $i\in[n]$.

1. (Weak partial derivative of a function)(i) Let $u,g\in\mathcal{L}^{1}(\mathbb{T}^{n})$. Then $g$ is called an $i$-th weak partial derivative of $u$ when

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

2. (Weak partial derivative of a class)(ii) Let $U\in L^{p}(\mathbb{T}^{n})$ and $G\in L^{q}(\mathbb{T}^{n})$. Then $G$ is called the $i$-th weak partial derivative of $U$ when some representative $u$ of $U$ and some representative $g$ of $G$ stand in the relation of claim 1; by Pairing an Integrable Function on the Torus with a Continuous Periodic Function (iii) this holds for one such pair exactly when it holds for every such pair, so the condition depends only on $U$ and $G$. For a given $q$ there is at most one such $G$, by Pairing an Integrable Function on the Torus with a Continuous Periodic Function (iv) applied to representatives of two candidates. When such a $G$ exists we write $\partial_{i}U$ for it and say that the $i$-th weak partial derivative of $U$ exists in $L^{q}(\mathbb{T}^{n})$.

3. (The weak gradient)(iii) Suppose that for every $i\in[n]$ the $i$-th weak partial derivative of $U$ exists in $L^{q}(\mathbb{T}^{n})$. Then $\nabla U$ denotes the $n$-tuple $(\partial_{1}U,\dots,\partial_{n}U)$ of members of $L^{q}(\mathbb{T}^{n})$. This clause introduces notation only.