The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds

In the setting of Hilbert Triples: Standing Notation and Background, let $h:V\to\mathbb{R}$ be the penalty function, $h(x)=\tfrac12|x|_{V}^{2}$. Semicontinuity of a function on $V$, or on a subset of $V$, is understood relative to that subset of the metric space $(H,d_{H})$, and closed superlevel sets are taken in the ambient space $H$. For a nonempty subset $S$ of $H$ and a function $u:S\to\mathbb{R}$, that $u$ is bounded above near each point of $S$, or bounded below near each point of $S$, refers to $S$ as a subset of the metric space $(H,d_{H})$. Claim 4 uses the standing separability of $V$. Then the following hold.

1. (Nonnegativity and coercivity)(i) For every $x\in V$, $0\le\tfrac12|x|_{H}^{2}\le h(x)$; and $h(0_{H})=0$.

2. (Expansion in $V$)(ii) For all $x,y\in V$,

$$h(y)=h(x)+\langle x,y-x\rangle_{V}+h(y-x).$$

3. (Expansion through the form operator)(iii) For all $x\in D(A)$ and $y\in V$,

$$h(y)=h(x)+\langle Ax,y-x\rangle_{H}+h(y-x).$$

4. (Closed sublevel sets)(iv) The function $-h:V\to\mathbb{R}$, whose value at $x\in V$ is $-h(x)$, has closed superlevel sets in $H$; that is, for every $c\in\mathbb{R}$ the set $\{x\in V: h(x)\le c\}$ is closed in $H$. Consequently, whenever $(x_{m})_{m\in\mathbb{N}}$ is a sequence in $V$ converging in $H$ to a point $x\in H$ and $c\in\mathbb{R}$ satisfies $h(x_{m})\le c$ for every $m\in\mathbb{N}$, one has $x\in V$ and $h(x)\le c$; and $h$ is lower semicontinuous on $V$.

5. (Density of the domain in open sets)(v) Let $U\subseteq H$ be open in $H$. Then for every $x\in U$ and every real $\varepsilon>0$ there is $y\in D(A)\cap U$ with $|y-x|_{H}<\varepsilon$. In particular, if $U$ is nonempty then so are $D(A)\cap U$ and $V\cap U$.

6. (Local bounds after penalisation)(vi) Let $U\subseteq H$ be nonempty and open in $H$, let $u:U\to\mathbb{R}$, and let $\delta\in\mathbb{R}$ satisfy $0<\delta$. If $u$ is bounded above near each point of $U$, then the function $V\cap U\to\mathbb{R}$ whose value at $x$ is $u(x)-\delta h(x)$ is bounded above near each point of $V\cap U$. If $u$ is bounded below near each point of $U$, then the function $V\cap U\to\mathbb{R}$ whose value at $x$ is $u(x)+\delta h(x)$ is bounded below near each point of $V\cap U$.