The Quartic Energy Functional on the First Sobolev Space of 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 $n\le3$; the cell $Q$, the integral over $\mathbb{T}^{n}$, and the classes $\mathcal{L}^{t}(\mathbb{T}^{n})$ and spaces $L^{t}(\mathbb{T}^{n})$ with the class map $[\,\cdot\,]$, for a real number $t$ with $1\le t$, are the ones fixed there. A representative of a class $U\in L^{t}(\mathbb{T}^{n})$ is a member $u$ of $\mathcal{L}^{t}(\mathbb{T}^{n})$ with $[u]=U$. Weak partial derivatives of classes are those of The Weak Partial Derivative on the Torus, and $H^{1}(\mathbb{T}^{n})$ is the Sobolev space fixed there, so that the weak partial derivative $\partial_{j}U$ exists in $L^{2}(\mathbb{T}^{n})$ for every $U\in H^{1}(\mathbb{T}^{n})$ and every $j\in[n]$; $\lVert\,\cdot\,\rVert_{L^{2}}$ is the norm of $L^{2}(\mathbb{T}^{n})$, a sum $\sum_{j=1}^{n}$ is the finite sum in $\mathbb{R}$, and for a map $v$ defined on $Q$ the pointwise fourth power $v^{4}$ is the map on $Q$ whose value at $y$ is the fourth power of $v(y)$. Let $a,c\in\mathbb{R}$.

1. (The quartic energy)¶ Let $U\in H^{1}(\mathbb{T}^{n})$ and let $u$ be a representative of $U$. Then $u^{4}\in\mathcal{L}^{1}(\mathbb{T}^{n})$ and the real number $\int_{\mathbb{T}^{n}}u^{4}\,dx$ is the same for every representative $u$ of $U$, by Integrability of Products of Four Sobolev Classes on the Torus in Dimensions at Most Three §quartic. The quartic energy with parameters $a$ and $c$ is the function $f:H^{1}(\mathbb{T}^{n})\to\mathbb{R}$ whose value at $U$ is

for $u$ any representative of $U$. This clause introduces no object beyond the function $f$.