Proof of The Gradient of the Quartic Energy on the First Sobolev Space of the Torus

Each result cited is universally quantified over the data in its own statement and is applied here to the data at hand; the notation is that of the statement, and the measure space throughout is $(Q,\mathcal{B}_{Q},\lambda_{Q})$ of The Flat Torus: Standing Notation (iii).

Step 0 (Membership in $\mathcal{L}^{1}(\mathbb{T}^{n})$ and integrability). Let $h:Q\to\mathbb{R}$ be measurable. By Properties of Real Powers (ii) the power of a nonnegative real number with exponent $1$ is that number itself, so the map $|h|^{1}$ of Power-Integrable Functions and the p-Seminorm (i) is $|h|$; hence by Power-Integrable Functions and the p-Seminorm (ii) and the last sentence of Integrable Functions and Their Lebesgue Integral, $h\in\mathcal{L}^{1}(\mathbb{T}^{n})$ if and only if $h$ is integrable.

Step 1 (Squares). Let $Z\in L^{2}(\mathbb{T}^{n})$ and let $z$ be a representative of $Z$. Then the pointwise square $z^{2}$ lies in $\mathcal{L}^{1}(\mathbb{T}^{n})$ and

$$\int_{\mathbb{T}^{n}}z^{2}\,dx=\langle Z,Z\rangle_{L^{2}}=\bigl(\lVert Z\rVert_{L^{2}}\bigr)^{2}.$$

Indeed $z^{2}=zz$ valuewise by claim 1 of Properties of Natural Number Powers in a Field; L²(μ) Is a Real Hilbert Space (i), applied to $z$ and $z$, gives that $zz$ is integrable — hence in $\mathcal{L}^{1}(\mathbb{T}^{n})$ by Step 0 — with $\int_{\mathbb{T}^{n}}zz\,dx=\langle[z],[z]\rangle_{L^{2}}=\langle Z,Z\rangle_{L^{2}}$; and the norm of Real Inner Product Spaces (ii) is the nonnegative square root of $\langle Z,Z\rangle_{L^{2}}$, whose square is $\langle Z,Z\rangle_{L^{2}}$.

Claim 1. Let $U$, $u$ and $g_{1},\dots,g_{n}$ be as in the claim. By The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus (i), $U\in L^{2}(\mathbb{T}^{n})$ and $\partial_{j}U\in L^{2}(\mathbb{T}^{n})$ for every $j\in[n]$, so Step 1 gives $u^{2},g_{1}^{2},\dots,g_{n}^{2}\in\mathcal{L}^{1}(\mathbb{T}^{n})$ with

$$\int_{\mathbb{T}^{n}}u^{2}\,dx=\bigl(\lVert U\rVert_{L^{2}}\bigr)^{2},\qquad\int_{\mathbb{T}^{n}}g_{j}^{2}\,dx=\bigl(\lVert\partial_{j}U\rVert_{L^{2}}\bigr)^{2}\quad(j\in[n]);$$

and $u^{4}\in\mathcal{L}^{1}(\mathbb{T}^{n})$ by Integrability of Products of Four Sobolev Classes on the Torus in Dimensions at Most Three (ii). All these maps are integrable by Step 0. Apply Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions with the $n+2$ integrable maps $g_{1}^{2},\dots,g_{n}^{2},u^{4},u^{2}$ and the coefficients $1,\dots,1,a,-c$: by the recursion in claim 1 of Properties of Finite Sums and the multiplicative identity axiom, the resulting map is the pointwise combination $\sum_{j=1}^{n}g_{j}^{2}+a\,u^{4}-c\,u^{2}$ and the resulting sum of integrals regroups in the same way, so that map is integrable, hence lies in $\mathcal{L}^{1}(\mathbb{T}^{n})$ by Step 0, and

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

Substituting the three identities displayed above, the right-hand side is $\sum_{j=1}^{n}(\lVert\partial_{j}U\rVert_{L^{2}})^{2}+a\int_{\mathbb{T}^{n}}u^{4}\,dx-c(\lVert U\rVert_{L^{2}})^{2}$, which is $f(U)$ by The Quartic Energy Functional on the First Sobolev Space of the Torus (i). This is claim 1.

Claim 2. Let $X\in H^{1}(\mathbb{T}^{n})$, let $x$ be a representative of $X$, and put

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

By Integrability of Products of Four Sobolev Classes on the Torus in Dimensions at Most Three (iii), $[x^{3}]\in L^{2}(\mathbb{T}^{n})$; and $X\in L^{2}(\mathbb{T}^{n})$ by The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus (i), so the argument of $J$ lies in $L^{2}(\mathbb{T}^{n})$ and $p\in H^{1}(\mathbb{T}^{n})$.

(a) The pairing identity. Let $W\in H^{1}(\mathbb{T}^{n})$ and let $w$ be a representative of $W$. By claim 1 of Elementary Identities in a Real Inner Product Space together with the symmetry axiom of Real Inner Product Spaces (i), and by the property of $J$ recorded in the statement,

$$\langle p,W\rangle_{H^{1}}=2\,\langle X,W\rangle_{H^{1}}+\bigl\langle 4a\,[x^{3}]-2(c+1)X,\;W\bigr\rangle_{L^{2}} .$$

Now $\langle X,W\rangle_{H^{1}}=\langle X,W\rangle_{L^{2}}+\sum_{j=1}^{n}\langle\partial_{j}X,\partial_{j}W\rangle_{L^{2}}$ by The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus (ii), and by claim 1 of Elementary Identities in a Real Inner Product Space again together with Integrability of Products of Four Sobolev Classes on the Torus in Dimensions at Most Three (iii),

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

Since $2\langle X,W\rangle_{L^{2}}-2(c+1)\langle X,W\rangle_{L^{2}}=\bigl(2-2(c+1)\bigr)\langle X,W\rangle_{L^{2}}=-2c\,\langle X,W\rangle_{L^{2}}$ by distributivity and the field axioms of The Real Numbers: Standing Notation and Background (i), and since $2\sum_{j=1}^{n}\langle\partial_{j}X,\partial_{j}W\rangle_{L^{2}}$ is the finite sum of the doubled terms by claim 3 of Properties of Finite Sums, we obtain

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

which is the pairing formula asserted in claim 2.

(b) The expansion. Let $W\in H^{1}(\mathbb{T}^{n})$ with representative $w$. Then $x+w$ is a representative of $X+W$ by The Lebesgue Space Lᵖ(μ) (ii), and $\partial_{j}(X+W)=\partial_{j}X+\partial_{j}W$ for $j\in[n]$ by Elementary Properties of the Weak Partial Derivative on the Torus (ii). Claim 5 of Elementary Identities in a Real Inner Product Space, applied in $L^{2}(\mathbb{T}^{n})$, gives

$$\bigl(\lVert\partial_{j}X+\partial_{j}W\rVert_{L^{2}}\bigr)^{2}=\bigl(\lVert\partial_{j}X\rVert_{L^{2}}\bigr)^{2}+2\langle\partial_{j}X,\partial_{j}W\rangle_{L^{2}}+\bigl(\lVert\partial_{j}W\rVert_{L^{2}}\bigr)^{2}$$

for each $j\in[n]$, and $(\lVert X+W\rVert_{L^{2}})^{2}=(\lVert X\rVert_{L^{2}})^{2}+2\langle X,W\rangle_{L^{2}}+(\lVert W\rVert_{L^{2}})^{2}$. Summing the former over $j\in[n]$ and using claims 2 and 3 of Properties of Finite Sums,

$$\sum_{j=1}^{n}\bigl(\lVert\partial_{j}(X+W)\rVert_{L^{2}}\bigr)^{2}=\sum_{j=1}^{n}\bigl(\lVert\partial_{j}X\rVert_{L^{2}}\bigr)^{2}+2\sum_{j=1}^{n}\langle\partial_{j}X,\partial_{j}W\rangle_{L^{2}}+\sum_{j=1}^{n}\bigl(\lVert\partial_{j}W\rVert_{L^{2}}\bigr)^{2}.$$

Next, claim 2 of The Expansion of the Square and the Fourth Power of a Sum of Real Numbers, applied at each $y\in Q$ with $s=x(y)$ and $t=w(y)$, gives valuewise on $Q$

$$(x+w)^{4}=x^{4}+4\,x^{3}w+6\,x^{2}w^{2}+4\,x\,w^{3}+w^{4},$$

each term on the right being a real multiple of a pointwise product of four of the maps $x$ and $w$, by claim 1 of Properties of Natural Number Powers in a Field. By Integrability of Products of Four Sobolev Classes on the Torus in Dimensions at Most Three (i), applied with the four classes chosen among $X$ and $W$ to match each term, the maps $x^{4}$, $x^{3}w$, $x^{2}w^{2}$, $xw^{3}$ and $w^{4}$ all lie in $\mathcal{L}^{1}(\mathbb{T}^{n})$ and hence are integrable by Step 0; so Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions, applied to them with the coefficients $1,4,6,4,1$, gives

$$\int_{\mathbb{T}^{n}}(x+w)^{4}dx=\int_{\mathbb{T}^{n}}x^{4}dx+4\int_{\mathbb{T}^{n}}x^{3}w\,dx+6\int_{\mathbb{T}^{n}}x^{2}w^{2}dx+4\int_{\mathbb{T}^{n}}xw^{3}dx+\int_{\mathbb{T}^{n}}w^{4}dx .$$

Evaluating $f(X+W)$ by The Quartic Energy Functional on the First Sobolev Space of the Torus (i) with the representative $x+w$ and substituting the three expansions,

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

where

$$S(W)=\sum_{j=1}^{n}\bigl(\lVert\partial_{j}W\rVert_{L^{2}}\bigr)^{2}-c\,\bigl(\lVert W\rVert_{L^{2}}\bigr)^{2}+a\Bigl(6\int_{\mathbb{T}^{n}}x^{2}w^{2}dx+4\int_{\mathbb{T}^{n}}xw^{3}dx+\int_{\mathbb{T}^{n}}w^{4}dx\Bigr).$$

By (a) the terms between $f(X)$ and $S(W)$ are exactly $\langle p,W\rangle_{H^{1}}$, so

$$f(X+W)-f(X)-\langle p,W\rangle_{H^{1}}=S(W).$$

(c) The remainder bound. Write $\rho=\lVert W\rVert_{H^{1}}$ and $\beta=\lVert X\rVert_{H^{1}}$, both nonnegative by Real Inner Product Spaces (ii). By The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus (ii) and Real Inner Product Spaces (ii),

$$\rho^{2}=\bigl(\lVert W\rVert_{L^{2}}\bigr)^{2}+\sum_{j=1}^{n}\bigl(\lVert\partial_{j}W\rVert_{L^{2}}\bigr)^{2},$$

and both summands are nonnegative (the second by claim 5 of Properties of Finite Sums), so claim 3 of Elementary Arithmetic in an Ordered Field gives $\sum_{j=1}^{n}(\lVert\partial_{j}W\rVert_{L^{2}})^{2}\le\rho^{2}$ and $(\lVert W\rVert_{L^{2}})^{2}\le\rho^{2}$. By Integrability of Products of Four Sobolev Classes on the Torus in Dimensions at Most Three (i),

$$\Bigl|\int_{\mathbb{T}^{n}}x^{2}w^{2}dx\Bigr|\le625\,\beta^{2}\rho^{2},\qquad\Bigl|\int_{\mathbb{T}^{n}}xw^{3}dx\Bigr|\le625\,\beta\rho^{3},\qquad\Bigl|\int_{\mathbb{T}^{n}}w^{4}dx\Bigr|\le625\,\rho^{4}.$$

Hence, by claims 4 and 5 of Properties of the Absolute Value in an Ordered Field (multiplicativity and the triangle inequality) together with claim 5 of Elementary Arithmetic in an Ordered Field and distributivity,

$$|S(W)|\le(1+|c|)\rho^{2}+625\,|a|\bigl(6\beta^{2}\rho^{2}+4\beta\rho^{3}+\rho^{4}\bigr).$$

Suppose moreover that $\rho\le1$. The powers $\rho^{2}$ and $\rho^{3}$ are nonnegative by claim 5 of Properties of Natural Number Powers in a Field, so claim 5 of Elementary Arithmetic in an Ordered Field together with claim 1 of Properties of Natural Number Powers in a Field gives $\rho^{3}=\rho^{2}\rho\le\rho^{2}$ and $\rho^{4}=\rho^{3}\rho\le\rho^{3}\le\rho^{2}$. Applying claim 5 of Elementary Arithmetic in an Ordered Field once more with the nonnegative multipliers and collecting by distributivity,

$$|S(W)|\le K\rho^{2},\qquad\text{where }K=1+|c|+625\,|a|\,\bigl(6\beta^{2}+4\beta+1\bigr),$$

and $1\le K$, every further summand being nonnegative by claims 1, 2 and 5 of Elementary Arithmetic in an Ordered Field and claim 1 of Properties of the Absolute Value in an Ordered Field.

(d) Differentiability. Let $\varepsilon\in\mathbb{R}$ be positive. Since $1\le K$ the number $K$ is positive, so $\tfrac{\varepsilon}{K}$ is positive by claims 7 and 5 of Elementary Order Arithmetic in an Ordered Field; let $\delta$ be the lesser of $1$ and $\tfrac{\varepsilon}{K}$, which exists by claim 9 of that lemma and is positive, being one of the two. Let $Z\in H^{1}(\mathbb{T}^{n})$ satisfy $\lVert Z\rVert_{H^{1}}<\delta$. Then $X+Z\in H^{1}(\mathbb{T}^{n})$, this being a linear subspace of $L^{2}(\mathbb{T}^{n})$ by The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus (i), and $\lVert Z\rVert_{H^{1}}\le1$; so (b) and (c), applied with $W=Z$ and $\rho=\lVert Z\rVert_{H^{1}}$, give

$$\bigl|f(X+Z)-f(X)-\langle p,Z\rangle_{H^{1}}\bigr|=|S(Z)|\le K\rho^{2}=(K\rho)\,\rho\le(K\delta)\,\rho\le\varepsilon\,\rho ,$$

the last two steps by claim 5 of Elementary Arithmetic in an Ordered Field, using $\rho\le\delta$ and $K\delta\le K\cdot\tfrac{\varepsilon}{K}=\varepsilon$. Since $\varepsilon$ was an arbitrary positive real number, $f$ is differentiable at $X$ with gradient $p$ in the sense of Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space (i), so $Df(X)=p$ by Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space (ii). As $X$ was an arbitrary member of $H^{1}(\mathbb{T}^{n})$, $f$ is differentiable on $H^{1}(\mathbb{T}^{n})$ by Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space (iii). Together with (a) this proves claim 2.

Claim 3. Let $X\in D(A)$ and let $x$ be a representative of $X$. Put

$$Z_{0}=AX+2a\,[x^{3}]-(c+1)X ,$$

a member of $L^{2}(\mathbb{T}^{n})$, since $AX\in L^{2}(\mathbb{T}^{n})$ by Hilbert Triples: Standing Notation and Background (iii), $[x^{3}]\in L^{2}(\mathbb{T}^{n})$ by Integrability of Products of Four Sobolev Classes on the Torus in Dimensions at Most Three (iii) and $X\in L^{2}(\mathbb{T}^{n})$ by The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus (i). The map $J$ is linear by Elementary Properties of a Hilbert Triple: the Embedding, the Riesz Map and the Form Operator (ii), and $J(AX)=X$ by Elementary Properties of a Hilbert Triple: the Embedding, the Riesz Map and the Form Operator (iii). Hence, writing $Y=2a\,[x^{3}]-(c+1)X$ so that $Z_{0}=AX+Y$ and $2Y=4a\,[x^{3}]-2(c+1)X$,

$$2J(Z_{0})=2J(AX)+2J(Y)=2X+J(2Y)=2X+J\bigl(4a\,[x^{3}]-2(c+1)X\bigr)=Df(X)$$

by claim 2. If $Z\in L^{2}(\mathbb{T}^{n})$ also satisfies $Df(X)=2JZ$, then $2JZ=2JZ_{0}$; the real number $2$ is positive, hence nonzero and invertible, by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, so multiplying both sides by $2^{-1}$ and using associativity of multiplication, the inverse axiom and the multiplicative identity axiom of The Real Numbers: Standing Notation and Background (i) gives $JZ=JZ_{0}$; and $Z=Z_{0}$ because $J$ is injective by Elementary Properties of a Hilbert Triple: the Embedding, the Riesz Map and the Form Operator (ii). Thus $Z_{0}$ is the unique member of $L^{2}(\mathbb{T}^{n})$ with $Df(X)=2JZ_{0}$.

Finally suppose $0\le a$ and take $b=2a$ and $\kappa=c+1$. Then $b=a+a$ by distributivity, so $0\le b$ by claim 2 of Elementary Arithmetic in an Ordered Field, and the maps $B$ and $L$ of Well-Posedness of the Allen-Cahn Hamilton-Jacobi Equation on the Torus (i) are formed for admissible data. For them, $B(X)=b\,[x^{3}]=2a\,[x^{3}]$ and $L(X)=-\kappa X=-(c+1)X$, so

$$AX+B(X)+L(X)=AX+2a\,[x^{3}]-(c+1)X=Z_{0},$$

which completes claim 3.