The Square-Integrable and Sobolev Spaces of the Torus Form a Hilbert Triple

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$; the cell $Q$, the class $\mathcal{L}^{2}(\mathbb{T}^{n})$ and the space $L^{2}(\mathbb{T}^{n})$ with the class map $[\,\cdot\,]$, the periodic classes $C_{\mathrm{per}}$, $C^{1}_{\mathrm{per}}$ and $C^{2}_{\mathrm{per}}$, the restriction $u|_{Q}$, and the partial derivatives $\partial_{i}$, the iterated partial derivatives and the classes $C^{k}$ on a Euclidean open set are the ones fixed there. Weak partial derivatives of classes are those of The Weak Partial Derivative on the Torus; $H^{1}(\mathbb{T}^{n})$, together with its inner product $\langle\,\cdot\,,\cdot\,\rangle_{H^{1}}$ and distance $d_{H^{1}}$, is the Sobolev space fixed there; and $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$ is the inner product of $L^{2}(\mathbb{T}^{n})$. The Laplacian $\Delta u$ of a map $u$ of class $C^{2}$ on $\mathbb{R}^{n}$ is that of The Laplacian of a Twice Continuously Differentiable Function (i), the set $\mathbb{R}^{n}$ being open in itself by claim 1 of Euclidean Space Is Open in Itself; for maps $v,w:\mathbb{R}^{n}\to\mathbb{R}$, $v-w$ denotes the pointwise difference. Then the following hold.

1. (The Hilbert triple)(i) $L^{2}(\mathbb{T}^{n})$ is a real Hilbert space by L²(μ) Is a Real Hilbert Space (ii), and $H^{1}(\mathbb{T}^{n})$ is a linear subspace of it by The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus (i) which, equipped with $\langle\,\cdot\,,\cdot\,\rangle_{H^{1}}$, is itself a real Hilbert space by The Sobolev Space on the Torus is a Separable Hilbert Space Densely Embedded in the Square-Integrable Space (ii). The condition Hilbert Triple: a Densely and Continuously Embedded Hilbert Space and Its Form Operator (ii) holds for these data by the first inequality of The Sobolev Space on the Torus is a Separable Hilbert Space Densely Embedded in the Square-Integrable Space (i), and the condition Hilbert Triple: a Densely and Continuously Embedded Hilbert Space and Its Form Operator (iii) holds by The Sobolev Space on the Torus is a Separable Hilbert Space Densely Embedded in the Square-Integrable Space (ix). Let $A$ be the form operator determined by these data and let $D(A)\subseteq H^{1}(\mathbb{T}^{n})$ be its domain, so that by that clause

$$\langle X,Y\rangle_{H^{1}}=\langle AX,Y\rangle_{L^{2}}\qquad\text{for all }X\in D(A)\text{ and all }Y\in H^{1}(\mathbb{T}^{n}).$$

Then $L^{2}(\mathbb{T}^{n})$, $H^{1}(\mathbb{T}^{n})$ with $\langle\,\cdot\,,\cdot\,\rangle_{H^{1}}$, and $A$ form a Hilbert triple. Moreover the metric space $(H^{1}(\mathbb{T}^{n}),d_{H^{1}})$ is separable by The Sobolev Space on the Torus is a Separable Hilbert Space Densely Embedded in the Square-Integrable Space (viii), so that this Hilbert triple satisfies the standing hypothesis Hilbert Triples: Standing Notation and Background (ii).

2. (The form operator on twice continuously differentiable periodic functions)(ii) Let $u\in C^{2}_{\mathrm{per}}$. Then $u$ lies in $C^{1}_{\mathrm{per}}$ and in $C_{\mathrm{per}}$, a map of class $C^{2}$ on $\mathbb{R}^{n}$ being of class $C^{1}$ and continuous by claims 2 and 3 of Euclidean Space Is Open in Itself while periodicity is the same condition for all three classes by Lattice-Periodic Functions and the Periodic Function Classes (iii); each iterated partial derivative $\partial_{i}\partial_{i}u$ lies in $C_{\mathrm{per}}$ by two applications of Elementary Properties of Lattice-Periodic Functions (ii); and hence $\Delta u\in C_{\mathrm{per}}$ and $u-\Delta u\in C_{\mathrm{per}}$ by Elementary Properties of Lattice-Periodic Functions (i). Consequently $(u-\Delta u)|_{Q}\in\mathcal{L}^{2}(\mathbb{T}^{n})$ by Continuous Periodic Functions are Power-Integrable and Dense on the Torus (i). Moreover $[u|_{Q}]\in H^{1}(\mathbb{T}^{n})$ by The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus (i), and

$$\bigl\langle[u|_{Q}],V\bigr\rangle_{H^{1}}=\bigl\langle[(u-\Delta u)|_{Q}],V\bigr\rangle_{L^{2}}\qquad\text{for every }V\in H^{1}(\mathbb{T}^{n}).$$

Consequently $[u|_{Q}]\in D(A)$ and

$$A\,[u|_{Q}]=[(u-\Delta u)|_{Q}].$$