The Gradient of the Quartic Energy 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 $H^{1}(\mathbb{T}^{n})$, with its inner product $\langle\,\cdot\,,\cdot\,\rangle_{H^{1}}$, norm $\lVert\,\cdot\,\rVert_{H^{1}}$ and distance $d_{H^{1}}$, is the Sobolev space fixed there; $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$ and $\lVert\,\cdot\,\rVert_{L^{2}}$ are the inner product and norm of $L^{2}(\mathbb{T}^{n})$. A sum $\sum_{j=1}^{n}$ is the finite sum in $\mathbb{R}$; for $s\in\mathbb{R}$ and a natural number $m$, $s^{m}$ is the $m$-th power of $s$, and for a map $v$ defined on $Q$ the pointwise power $v^{m}$ is the map on $Q$ whose value at $y$ is $(v(y))^{m}$, pointwise sums, differences, real multiples and products of maps on $Q$ being formed valuewise.

We work also in the setting of Hilbert Triples: Standing Notation and Background, used here with the Hilbert triple $(H,V,A)$ taken to be the one of The Square-Integrable and Sobolev Spaces of the Torus Form a Hilbert Triple (i), that is, with $H=L^{2}(\mathbb{T}^{n})$, with $V=H^{1}(\mathbb{T}^{n})$ carrying $\langle\,\cdot\,,\cdot\,\rangle_{H^{1}}$, and with $A$ the form operator determined by these data; its standing hypothesis Hilbert Triples: Standing Notation and Background (ii) holds for this triple by that clause. Accordingly $D(A)\subseteq H^{1}(\mathbb{T}^{n})$ is the domain of the form operator and $J:L^{2}(\mathbb{T}^{n})\to H^{1}(\mathbb{T}^{n})$ is the Riesz map, both as fixed in Hilbert Triples: Standing Notation and Background (iii), so that

$$\langle JZ,W\rangle_{H^{1}}=\langle Z,W\rangle_{L^{2}}\qquad\text{for all }Z\in L^{2}(\mathbb{T}^{n})\text{ and }W\in H^{1}(\mathbb{T}^{n}).$$

Let $a,c\in\mathbb{R}$ and let $f:H^{1}(\mathbb{T}^{n})\to\mathbb{R}$ be the quartic energy with parameters $a$ and $c$.

Differentiability and the gradient below are those of Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space, applied with the real inner product space written $E$ there taken to be $H^{1}(\mathbb{T}^{n})$ carrying $\langle\,\cdot\,,\cdot\,\rangle_{H^{1}}$, and with the open subset written $U$ there taken to be $H^{1}(\mathbb{T}^{n})$ itself, which is open in the metric space $(H^{1}(\mathbb{T}^{n}),d_{H^{1}})$. The gradient $Df(X)$ of Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space (ii) is therefore a member of $H^{1}(\mathbb{T}^{n})$ and is taken with respect to $\langle\,\cdot\,,\cdot\,\rangle_{H^{1}}$; it is not the gradient of Hilbert Triples: Standing Notation and Background (v), which is taken in $L^{2}(\mathbb{T}^{n})$ and is not used here. Then the following hold.

1. (The energy as a single integral)(i) Let $U\in H^{1}(\mathbb{T}^{n})$, let $u$ be a representative of $U$ and, for each $j\in[n]$, let $g_{j}$ be a representative of the weak partial derivative $\partial_{j}U$. Then the pointwise powers $g_{j}^{2}$, $u^{2}$ and $u^{4}$ lie in $\mathcal{L}^{1}(\mathbb{T}^{n})$, as does the pointwise combination $\sum_{j=1}^{n}g_{j}^{2}+a\,u^{4}-c\,u^{2}$, and

$$f(U)=\int_{\mathbb{T}^{n}}\Bigl(\sum_{j=1}^{n}g_{j}^{2}+a\,u^{4}-c\,u^{2}\Bigr)dx .$$

2. (The gradient)(ii) The function $f$ is differentiable on $H^{1}(\mathbb{T}^{n})$. Let $X\in H^{1}(\mathbb{T}^{n})$ and let $x$ be a representative of $X$. Then $[x^{3}]\in L^{2}(\mathbb{T}^{n})$ by Integrability of Products of Four Sobolev Classes on the Torus in Dimensions at Most Three (iii), so that $4a\,[x^{3}]-2(c+1)X\in L^{2}(\mathbb{T}^{n})$, and

$$Df(X)=2X+J\bigl(4a\,[x^{3}]-2(c+1)X\bigr);$$

equivalently, for every $W\in H^{1}(\mathbb{T}^{n})$ with representative $w$,

$$\bigl\langle Df(X),W\bigr\rangle_{H^{1}}=2\sum_{j=1}^{n}\bigl\langle\partial_{j}X,\partial_{j}W\bigr\rangle_{L^{2}}+4a\int_{\mathbb{T}^{n}}x^{3}w\,dx-2c\,\langle X,W\rangle_{L^{2}} .$$

3. (The gradient on the domain of the form operator)(iii) Let $X\in D(A)$ and let $x$ be a representative of $X$. Then

$$Df(X)=2\,J\bigl(AX+2a\,[x^{3}]-(c+1)X\bigr),$$

and $AX+2a\,[x^{3}]-(c+1)X$ is the unique $Z\in L^{2}(\mathbb{T}^{n})$ satisfying $Df(X)=2JZ$. If in addition $0\le a$, and the real numbers $b$ and $\kappa$ of Well-Posedness of the Allen-Cahn Hamilton-Jacobi Equation on the Torus (i) are taken to be $b=2a$, which is then nonnegative, and $\kappa=c+1$, then this $Z$ is the drift $AX+B(X)+L(X)$ of that clause.