Proof of Elementary Properties of the Weak Partial Derivative on the Torus

Each result cited below is universally quantified over the data appearing in its own statement, and is applied here to the data named in the statement of this lemma. Throughout, 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 Pairing an Integrable Function on the Torus with a Continuous Periodic Function (i) applies to it; and a member of $C^{\infty}_{\mathrm{per}}$ lies in $C^{1}_{\mathrm{per}}$, and a member of $C^{1}_{\mathrm{per}}$ in $C_{\mathrm{per}}$, by claims 2 and 3 of Euclidean Space Is Open in Itself together with Lattice-Periodic Functions and the Periodic Function Classes (iii). For $\varphi\in C^{1}_{\mathrm{per}}$ we have $\partial_{i}\varphi\in C_{\mathrm{per}}$, and for $\varphi\in C^{\infty}_{\mathrm{per}}$ we have $\partial_{i}\varphi\in C^{\infty}_{\mathrm{per}}$, by Elementary Properties of Lattice-Periodic Functions (ii).

Step 1. Proof of claim 1.

Let $u\in C^{1}_{\mathrm{per}}$ and let $\varphi\in C^{\infty}_{\mathrm{per}}$, so that $\varphi\in C^{1}_{\mathrm{per}}$. By Vanishing Mean of a Derivative and Integration by Parts on the Torus (ii), applied with $g=u$ and $h=\varphi$,

$$\int_{\mathbb{T}^{n}}(\partial_{i}u)\,\varphi\,dx=-\int_{\mathbb{T}^{n}}u\,(\partial_{i}\varphi)\,dx .$$

In the abbreviation fixed in that lemma each side is the integral over $\mathbb{T}^{n}$ of the restriction to $Q$ of the displayed product, and the restriction of a pointwise product is the product of the restrictions, so the display says

$$\int_{\mathbb{T}^{n}}u|_{Q}\,\bigl((\partial_{i}\varphi)|_{Q}\bigr)\,dx=-\int_{\mathbb{T}^{n}}(\partial_{i}u)|_{Q}\,(\varphi|_{Q})\,dx .$$

As $\varphi$ was arbitrary, $(\partial_{i}u)|_{Q}$ is an $i$-th weak partial derivative of $u|_{Q}$ in the sense of The Weak Partial Derivative on the Torus (i); since $u|_{Q}$ and $(\partial_{i}u)|_{Q}$ lie in $\mathcal{L}^{r}(\mathbb{T}^{n})$ for every real $r$ with $1\le r$, as recorded in the claim, The Weak Partial Derivative on the Torus (ii) gives that $[(\partial_{i}u)|_{Q}]$ is the $i$-th weak partial derivative of $[u|_{Q}]$ in $L^{r}(\mathbb{T}^{n})$.

Step 2. Proof of claim 2.

Let $u,u'$ represent $U,U'$ and let $g,g'$ represent $\partial_{i}U,\partial_{i}U'$. Then $u+c\,u'\in\mathcal{L}^{p}(\mathbb{T}^{n})$ and $g+c\,g'\in\mathcal{L}^{q}(\mathbb{T}^{n})$ by Minkowski's Inequality (ii), and they represent $U+c\,U'$ and $\partial_{i}U+c\,\partial_{i}U'$ by The Lebesgue Space Lᵖ(μ) (ii), under which the vector operations on classes are induced by the pointwise ones. Let $\varphi\in C^{\infty}_{\mathrm{per}}$. Using Pairing an Integrable Function on the Torus with a Continuous Periodic Function (ii) twice and the defining identities for $U$ and for $U'$,

$$\int_{\mathbb{T}^{n}}(u+c\,u')\,\bigl((\partial_{i}\varphi)|_{Q}\bigr)\,dx =\int_{\mathbb{T}^{n}}u\,\bigl((\partial_{i}\varphi)|_{Q}\bigr)dx+c\int_{\mathbb{T}^{n}}u'\,\bigl((\partial_{i}\varphi)|_{Q}\bigr)dx =-\int_{\mathbb{T}^{n}}(g+c\,g')\,(\varphi|_{Q})\,dx .$$

As $\varphi$ was arbitrary, claim 2 follows by The Weak Partial Derivative on the Torus (ii), which also supplies the uniqueness that makes the displayed identification legitimate.

Step 3. Proof of claim 3.

Let $\varphi\in C^{1}_{\mathrm{per}}$ and put

$$D=\int_{\mathbb{T}^{n}}u\,\bigl((\partial_{i}\varphi)|_{Q}\bigr)\,dx+\int_{\mathbb{T}^{n}}g\,(\varphi|_{Q})\,dx , \qquad K=\int_{\mathbb{T}^{n}}|u|\,dx+\int_{\mathbb{T}^{n}}|g|\,dx ,$$

both real numbers, $K$ being nonnegative by claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion. We show that $|D|\le\theta$ for every real $\theta$ with $0<\theta$; then $D=0$ by Comparison of Real Numbers with Arbitrary Positive Slack (iii), which is claim 3.

Let $\theta$ be a real number with $0<\theta$ and put $\eta=\theta\,(1+K)^{-1}$, a positive real number by claims 7 and 5 of Elementary Order Arithmetic in an Ordered Field, since $0<1+K$ by claims 6 and 3 of that lemma, applied with $0<1$ and $0\le K$. By Existence of Mollifier Kernels of Every Radius there is a mollifier kernel $\rho$ of radius $1$ on $\mathbb{R}^{n}$. By Transfer of a Partial Derivative in Periodic Convolution (ii), applied with that kernel, the function $\varphi$ and the tolerance $\eta$, there is a positive real $\varepsilon_{0}$ such that for every real $\varepsilon$ with $0<\varepsilon<\varepsilon_{0}$ the map $\varphi_{\varepsilon}=\rho_{\varepsilon}\star(\varphi|_{Q})$ satisfies

$$|\varphi_{\varepsilon}(x)-\varphi(x)|\le\eta\qquad\text{and}\qquad|\partial_{i}\varphi_{\varepsilon}(x)-\partial_{i}\varphi(x)|\le\eta\qquad(x\in\mathbb{R}^{n}).$$

Fix such an $\varepsilon$, for instance $\varepsilon_{0}/2$, positive and smaller than $\varepsilon_{0}$ by claim 8 of Elementary Order Arithmetic in an Ordered Field. Here $\varphi|_{Q}\in\mathcal{L}^{1}(\mathbb{T}^{n})$ by Continuous Periodic Functions are Power-Integrable and Dense on the Torus (i) and $\rho_{\varepsilon}$ is smooth by Mollifier Kernel of Radius $\delta$ on $\mathbb{R}^n$, so $\varphi_{\varepsilon}\in C^{\infty}_{\mathrm{per}}$ by Properties of Periodic Convolution on the Torus (i). The defining identity for $U$, applied with the test function $\varphi_{\varepsilon}$, gives

$$\int_{\mathbb{T}^{n}}u\,\bigl((\partial_{i}\varphi_{\varepsilon})|_{Q}\bigr)\,dx+\int_{\mathbb{T}^{n}}g\,(\varphi_{\varepsilon}|_{Q})\,dx=0 .$$

The maps $\partial_{i}\varphi_{\varepsilon}-\partial_{i}\varphi$ and $\varphi_{\varepsilon}-\varphi$ lie in $C_{\mathrm{per}}$ by Elementary Properties of Lattice-Periodic Functions (i), used with the scalar $-1$, and are bounded in absolute value by $\eta$; so Pairing an Integrable Function on the Torus with a Continuous Periodic Function (ii) and Pairing an Integrable Function on the Torus with a Continuous Periodic Function (i) give

$$\Bigl|\int_{\mathbb{T}^{n}}u\,\bigl((\partial_{i}\varphi_{\varepsilon})|_{Q}\bigr)dx-\int_{\mathbb{T}^{n}}u\,\bigl((\partial_{i}\varphi)|_{Q}\bigr)dx\Bigr|\le\eta\int_{\mathbb{T}^{n}}|u|\,dx ,$$

and likewise with $g$, $\varphi_{\varepsilon}$, $\varphi$ in place of $u$, $\partial_{i}\varphi_{\varepsilon}$, $\partial_{i}\varphi$. Subtracting the vanishing identity from $D$ and applying the triangle inequality, claim 5 of Properties of the Absolute Value in an Ordered Field,

$$|D|\le\eta\int_{\mathbb{T}^{n}}|u|\,dx+\eta\int_{\mathbb{T}^{n}}|g|\,dx=\eta\,K\le\eta\,(1+K)=\theta ,$$

the last inequality by claim 5 of Elementary Arithmetic in an Ordered Field with the nonnegative factor $\eta$, since $K\le1+K$.

Step 4. Proof of claim 4.

By Mollifier Kernel of Radius $\delta$ on $\mathbb{R}^n$ the map $\rho_{\varepsilon}$ is smooth and vanishes at every $y$ with $\varepsilon\delta<\lVert y\rVert$, so it satisfies the hypotheses placed on a kernel in The Periodised Kernel of a Periodic Convolution, with the radius $\varepsilon\delta$; and $\partial_{i}\rho_{\varepsilon}$ is continuous and vanishes at every such $y$ by Convolution of a Locally Integrable Function with a Compactly Supported Kernel (iv), so it does too. Let $\Psi$ be the periodised kernel of $\rho_{\varepsilon}$ and $\Psi^{(i)}$ that of $\partial_{i}\rho_{\varepsilon}$, and for $x\in\mathbb{R}^{n}$ let $\Psi_{x}$ and $\Psi^{(i)}_{x}$ be the restrictions to $Q$ of $\Psi(x,\cdot)$ and $\Psi^{(i)}(x,\cdot)$. Both $\rho_{\varepsilon}\star u$ and $\rho_{\varepsilon}\star g$ lie in $C^{\infty}_{\mathrm{per}}$ by Properties of Periodic Convolution on the Torus (i).

Fix $x\in\mathbb{R}^{n}$. By that same clause and The Periodised Kernel of a Periodic Convolution (v), applied to the kernel $\partial_{i}\rho_{\varepsilon}$ and the function $u$,

$$\partial_{i}(\rho_{\varepsilon}\star u)(x)=\bigl((\partial_{i}\rho_{\varepsilon})\star u\bigr)(x)=\int_{\mathbb{T}^{n}}\Psi^{(i)}_{x}\,u\,dy .$$

By The Periodised Kernel of a Periodic Convolution (vii) we have $\partial_{i}\Psi(x,\cdot)=-\Psi^{(i)}(x,\cdot)$ as maps on $\mathbb{R}^{n}$, so $\Psi^{(i)}_{x}$ is the restriction to $Q$ of $-\partial_{i}\Psi(x,\cdot)$, and claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, with the scalar $-1$, gives

$$\int_{\mathbb{T}^{n}}\Psi^{(i)}_{x}\,u\,dy=-\int_{\mathbb{T}^{n}}u\,\bigl((\partial_{i}\Psi(x,\cdot))|_{Q}\bigr)\,dy .$$

The map $\Psi(x,\cdot)$ lies in $C^{\infty}_{\mathrm{per}}$ by The Periodised Kernel of a Periodic Convolution (iii), so the defining identity for $U$, applied with the test function $\Psi(x,\cdot)$, and then The Periodised Kernel of a Periodic Convolution (v) applied to the kernel $\rho_{\varepsilon}$ and the function $g$, give

$$\int_{\mathbb{T}^{n}}u\,\bigl((\partial_{i}\Psi(x,\cdot))|_{Q}\bigr)\,dy=-\int_{\mathbb{T}^{n}}g\,\Psi_{x}\,dy=-(\rho_{\varepsilon}\star g)(x).$$

Combining the three displays, $\partial_{i}(\rho_{\varepsilon}\star u)(x)=(\rho_{\varepsilon}\star g)(x)$. As $x$ was arbitrary, claim 4 follows.

Step 5. Proof of claim 5.

Let $\varphi\in C^{\infty}_{\mathrm{per}}$. By Elementary Properties of Lattice-Periodic Functions (iii) there are nonnegative reals $M$ and $M'$ with $|\varphi(x)|\le M$ and $|\partial_{i}\varphi(x)|\le M'$ for every $x$; put $N=M+M'$, so that $N$ is nonnegative and bounds both, by claim 2 of Elementary Arithmetic in an Ordered Field and claim 3 of that lemma. Let $u,g,u_{k},g_{k}$ be representatives of $U,G,U_{k},G_{k}$ and put

$$D=\int_{\mathbb{T}^{n}}u\,\bigl((\partial_{i}\varphi)|_{Q}\bigr)\,dx+\int_{\mathbb{T}^{n}}g\,(\varphi|_{Q})\,dx .$$

For every $k$ the defining identity for $U_{k}$ gives $\int_{\mathbb{T}^{n}}u_{k}((\partial_{i}\varphi)|_{Q})dx+\int_{\mathbb{T}^{n}}g_{k}(\varphi|_{Q})dx=0$, so by Pairing an Integrable Function on the Torus with a Continuous Periodic Function (ii), Pairing an Integrable Function on the Torus with a Continuous Periodic Function (i) and the triangle inequality, claim 5 of Properties of the Absolute Value in an Ordered Field,

$$|D|\le N\int_{\mathbb{T}^{n}}|u-u_{k}|\,dx+N\int_{\mathbb{T}^{n}}|g-g_{k}|\,dx .$$

The differences $u-u_{k}$ and $g-g_{k}$ lie in $\mathcal{L}^{2}(\mathbb{T}^{n})$ by Minkowski's Inequality (ii), used with the scalar $-1$. Now $\int_{\mathbb{T}^{n}}|u-u_{k}|\,dx=\lVert u-u_{k}\rVert_{1}$ by Power-Integrable Functions and the p-Seminorm (iii) and Properties of Real Powers (ii), and $\lVert u-u_{k}\rVert_{1}\le\lVert u-u_{k}\rVert_{2}$ by The Periodic Extension of a Function on the Unit Cell (iv); and $\lVert u-u_{k}\rVert_{2}$ is the distance from $U_{k}$ to $U$ in the metric of $L^{2}(\mathbb{T}^{n})$, by The Lebesgue Space Lᵖ(μ) (iii) and Norms and Real Normed Spaces (iii). The same holds for $g$ and $g_{k}$.

Let $\vartheta$ be a real number with $0<\vartheta$ and put $\theta=\vartheta\,(1+2N)^{-1}$, positive by claims 7 and 5 of Elementary Order Arithmetic in an Ordered Field, since $0\le2N$ by claim 2 of Elementary Arithmetic in an Ordered Field and hence $0<1+2N$ by claims 6 and 3 of the former lemma. Since $(U_{k})$ converges to $U$ and $(G_{k})$ to $G$ in that metric, Convergent Sequence in a Metric Space provides indices beyond which each distance is at most $\theta$; choosing a $k$ beyond both and combining with the two displays,

$$|D|\le N\theta+N\theta=2N\theta\le(1+2N)\,\theta=\vartheta ,$$

by claim 5 of Elementary Arithmetic in an Ordered Field with the nonnegative factor $\theta$. As $\vartheta$ was arbitrary and $|D|$ is nonnegative, Comparison of Real Numbers with Arbitrary Positive Slack (iii) gives $D=0$. As $\varphi$ was arbitrary, $G$ is the $i$-th weak partial derivative of $U$ by The Weak Partial Derivative on the Torus (ii).

Step 6. Proof of claim 6.

The constant map with value $0$ on $\mathbb{R}^{n}$ is smooth by claim 2 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions, and it is $\mathbb{Z}^{n}$-periodic because its values at $x$ and at $x+m$ are both $0$, for every $x\in\mathbb{R}^{n}$ and every $m\in\mathbb{Z}^{n}$; hence it lies in $C^{1}_{\mathrm{per}}$. Claim 1, used with $r=2$ and applied at every index $j\in[n]$ in place of $i$, that claim being stated for an arbitrary index, gives $[u|_{Q}]\in W$ for every $u\in C^{1}_{\mathrm{per}}$; in particular the class of the constant map just described lies in $W$, so $W$ is nonempty. If $U,V\in W$ and $c\in\mathbb{R}$ then $U+c\,V\in L^{2}(\mathbb{T}^{n})$ by The Riesz–Fischer Theorem (i), and for each $j\in[n]$ its $j$-th weak partial derivative exists in $L^{2}(\mathbb{T}^{n})$ by claim 2, used with $p=q=2$ and the index $j$; so $U+c\,V\in W$ and $W$ is a linear subspace of $L^{2}(\mathbb{T}^{n})$.

For $U,V\in W$ every term of $\langle U,V\rangle_{W}$ is a real number, so the map is defined; the index set $[n]$ is nonempty because $1\le n$. By claim 1 of Properties of a Sum over a Finite Index Set the sum $\sum_{j=1}^{n}$ agrees with the sum over the finite set $[n]$ of Sum over a Finite Index Set, for which claims 3 and 4 of that lemma give additivity and homogeneity. By L²(μ) Is a Real Hilbert Space (i) the map $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$ is an inner product, hence symmetric, linear in its first argument and positive definite by Real Inner Product Spaces (i); and by claim 2 the map $U\mapsto\partial_{j}U$ is linear on $W$ for each $j$. Symmetry of $\langle\,\cdot\,,\cdot\,\rangle_{W}$ follows termwise, and linearity in the first argument follows from linearity of $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$, linearity of $U\mapsto\partial_{j}U$, and the additivity and homogeneity of the sum just cited.

For positivity, let $U\in W$. Each term $\langle\partial_{j}U,\partial_{j}U\rangle_{L^{2}}$ is nonnegative, so by induction over the members of $[n]$, using claim 2 of Elementary Arithmetic in an Ordered Field and the peeling identity of Peeling, Splitting, and Interchange for Sums over a Finite Index Set, the sum is nonnegative; hence $0\le\langle U,U\rangle_{L^{2}}\le\langle U,U\rangle_{W}$ by claim 3 of Elementary Arithmetic in an Ordered Field. If $\langle U,U\rangle_{W}=0$ then $\langle U,U\rangle_{L^{2}}=0$, so $U$ is the zero element of $L^{2}(\mathbb{T}^{n})$ by the positive definiteness of that inner product; and conversely $\langle U,U\rangle_{W}=0$ when $U$ is the zero element, by the linearity just established. So $\langle\,\cdot\,,\cdot\,\rangle_{W}$ meets the conditions of Real Inner Product Spaces (i) and $W$ with it is a real inner product space. $\blacksquare$