The Fundamental Lemma of the Calculus of Variations 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 a real number $p$ with $1\le p$; the cell $Q$, the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$, the integral over $\mathbb{T}^{n}$, the classes $\mathcal{L}^{p}(\mathbb{T}^{n})$ and $L^{p}(\mathbb{T}^{n})$ with the class map $[\,\cdot\,]$, the periodic class $C^{\infty}_{\mathrm{per}}$ and the restriction $\varphi|_{Q}$ are the ones fixed there.

Every member of $C^{\infty}_{\mathrm{per}}$ belongs to $C_{\mathrm{per}}$, since a smooth map on $\mathbb{R}^{n}$ is continuous by claim 3 of Euclidean Space Is Open in Itself and periodicity is the same condition for both classes by Lattice-Periodic Functions and the Periodic Function Classes (iii). Consequently, for every $\varphi\in C^{\infty}_{\mathrm{per}}$ and every $g:Q\to\mathbb{R}$ integrable with respect to $\lambda_{Q}$: the map $\varphi$ is bounded by Elementary Properties of Lattice-Periodic Functions (iii); the restriction $\varphi|_{Q}$ is measurable with respect to $\mathcal{B}_{Q}$ by Continuous Periodic Functions are Power-Integrable and Dense on the Torus (i); and the product $g\,(\varphi|_{Q})$ is measurable with respect to $\mathcal{B}_{Q}$ by claim 3 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps. Fixing a real number $M$ with $0\le M$ and $|\varphi(x)|\le M$ for every $x\in\mathbb{R}^{n}$, one has $|g\,(\varphi|_{Q})|\le M\,|g|$ pointwise on $Q$, by claim 4 of Properties of the Absolute Value in an Ordered Field and claim 5 of Elementary Arithmetic in an Ordered Field; both $|g\,(\varphi|_{Q})|$ and $|g|$ are measurable with respect to $\mathcal{B}_{Q}$ by claim 4 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps, and $\int_{Q}|g|\,d\lambda_{Q}$ is finite by Integrable Functions and Their Lebesgue Integral, so claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion gives $\int_{Q}|g\,(\varphi|_{Q})|\,d\lambda_{Q}\le M\int_{Q}|g|\,d\lambda_{Q}<\infty$, so $g\,(\varphi|_{Q})$ is integrable with respect to $\lambda_{Q}$, again by Integrable Functions and Their Lebesgue Integral. Each integral displayed below is therefore a real number. Then the following hold.

1. (Vanishing against every smooth periodic test function)(i) Let $v:Q\to\mathbb{R}$ be integrable with respect to $\lambda_{Q}$ and suppose that

$$\int_{\mathbb{T}^{n}}v\,(\varphi|_{Q})\,dx=0\qquad\text{for every }\varphi\in C^{\infty}_{\mathrm{per}}.$$

Then $v=0$ $\lambda_{Q}$-almost everywhere on $Q$.

2. (The power-integrable case)(ii) Let $u\in\mathcal{L}^{p}(\mathbb{T}^{n})$, which is integrable with respect to $\lambda_{Q}$ by The Periodic Extension of a Function on the Unit Cell (iv), and suppose that

$$\int_{\mathbb{T}^{n}}u\,(\varphi|_{Q})\,dx=0\qquad\text{for every }\varphi\in C^{\infty}_{\mathrm{per}}.$$

Then $u=0$ $\lambda_{Q}$-almost everywhere on $Q$, and $[u]$ is the zero element of $L^{p}(\mathbb{T}^{n})$.