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 $C^k$ Maps are Continuous and periodicity is the same condition for both classes by Lattice-Periodic Functions and the Periodic Function Classes §classes. 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 §bounded; the restriction $\varphi|_{Q}$ is measurable with respect to $\mathcal{B}_{Q}$ by Continuous Periodic Functions are Power-Integrable and Dense on the Torus §member; and the product $g\,(\varphi|_{Q})$ is measurable with respect to $\mathcal{B}_{Q}$ by claim 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. 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, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and $\int_{Q}|g|\,d\lambda_{Q}$ is finite by Integrable Function and the Lebesgue Integral, so claim 1 of Linearity and Monotonicity of the Lebesgue Integral 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 Function and the Lebesgue Integral. Each integral displayed below is therefore a real number. Then the following hold.

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

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

2. (The power-integrable case)¶ 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 §finite-measure, and suppose that

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