The Cube Map is a Monotone Nonlinearity on the Sobolev Hilbert Triple 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 measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$, 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. Let $\lVert\,\cdot\,\rVert_{t}$ denote the $L^{t}$ seminorm of $(Q,\mathcal{B}_{Q},\lambda_{Q})$. A representative of a class $U\in L^{t}(\mathbb{T}^{n})$ is a member $u$ of $\mathcal{L}^{t}(\mathbb{T}^{n})$ with $[u]=U$; two representatives of one class agree almost everywhere by The Lebesgue Space Lᵖ(μ) (i). 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 $\langle\,\cdot\,,\cdot\,\rangle_{H^{1}}$ and norm $\lVert\,\cdot\,\rVert_{H^{1}}$, 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 $L^{2}(\mathbb{T}^{n})$, so that $\lVert[v]\rVert_{L^{2}}=\lVert v\rVert_{2}$ for $v\in\mathcal{L}^{2}(\mathbb{T}^{n})$ by that clause. For $s\in\mathbb{R}$ and a natural number $m$, $s^{m}$ is the $m$-th power of $s$, and for $v:Q\to\mathbb{R}$ the pointwise power $v^{m}$ is the function whose value at $x$ is $(v(x))^{m}$.

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 $\langle\,\cdot\,,\cdot\,\rangle_{H}$ is $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$ and $|\cdot|_{H}$ is $\lVert\,\cdot\,\rVert_{L^{2}}$, $\langle\,\cdot\,,\cdot\,\rangle_{V}$ is $\langle\,\cdot\,,\cdot\,\rangle_{H^{1}}$ and $|\cdot|_{V}$ is $\lVert\,\cdot\,\rVert_{H^{1}}$, and $D(A)\subseteq H^{1}(\mathbb{T}^{n})$ is the domain of the form operator, as fixed in Hilbert Triples: Standing Notation and Background (iii).

Let $b\in\mathbb{R}$ satisfy $0\le b$, and write $5=1+1+1+1+1$ and $125=5^{3}$. Then the following hold.

1. (The cube map)(i) Let $U\in H^{1}(\mathbb{T}^{n})$ and let $u$ be a representative of $U$. Then $u\in\mathcal{L}^{6}(\mathbb{T}^{n})$ with $\lVert u\rVert_{6}\le5\,\lVert U\rVert_{H^{1}}$, by The Sobolev Embedding of the First Sobolev Space of the Torus into the Sixth Lebesgue Space in Dimensions at Most Three (ii); consequently $u^{3}\in\mathcal{L}^{2}(\mathbb{T}^{n})$ and

$$\lVert u^{3}\rVert_{2}\le125\,\bigl(\lVert U\rVert_{H^{1}}\bigr)^{3},$$

and the class $[u^{3}]\in L^{2}(\mathbb{T}^{n})$ is the same for every representative $u$ of $U$. Accordingly $B$ denotes the map from $H^{1}(\mathbb{T}^{n})$ to $L^{2}(\mathbb{T}^{n})$ whose value at $U$ is

$$B(U)=b\,[u^{3}],$$

for $u$ any representative of $U$.

2. (Monotonicity)(ii) For all $U,W\in H^{1}(\mathbb{T}^{n})$,

$$0\le\bigl\langle B(U)-B(W),U-W\bigr\rangle_{L^{2}} .$$

Thus $B$ is monotone.

3. (Boundedness on bounded sets of the Sobolev space)(iii) Let $R\in\mathbb{R}$ satisfy $0<R$ and let $U\in H^{1}(\mathbb{T}^{n})$ satisfy $\lVert U\rVert_{H^{1}}\le R$. Then

$$\lVert B(U)\rVert_{L^{2}}\le125\,b\,R^{3} .$$

Thus $B$ is bounded on $V$-bounded sets.

4. (Monotonicity relative to the form operator)(iv) For every $U\in D(A)$,

$$0\le\bigl\langle B(U),AU\bigr\rangle_{L^{2}} .$$

Thus $B$ is $A$-monotone.

5. (A monotone nonlinearity)(v) $B$ is a monotone nonlinearity for $(H,V,A)$.