The $\delta$-Envelopes on an Open Subset, under Penalisation of a Continuous Function, and on a Closed Subset

In the setting of Hilbert Triples: Standing Notation and Background, let $U\subseteq H$ be nonempty and open in $H$, let $h$ be the penalty function, let $u:U\to\mathbb{R}$, and let $\delta\in\mathbb{R}$ satisfy $0<\delta$. The $\delta$-envelopes $u^{-}_{\delta}$ and $u^{+}_{\delta}$ are functions on $V\cap U$, defined when $u$ is bounded above, respectively below, near each point of $U$; local bounds, semicontinuity, closed superlevel sets and continuity of functions on $U$ and on its subsets are as fixed there, the ambient metric space being $(H,d_{H})$. For a nonempty $B\subseteq U$, $u|_{B}:B\to\mathbb{R}$ denotes the function whose value at $z\in B$ is $u(z)$. Then the following hold.

1. (Restriction to a smaller open set)(i) Let $U'\subseteq U$ be nonempty and open in $H$, so that $V\cap U'$ is nonempty by The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds (v). If $u$ is bounded above near each point of $U$, then $u|_{U'}$ is bounded above near each point of $U'$ and

$$\bigl(u|_{U'}\bigr)^{-}_{\delta}(x)=u^{-}_{\delta}(x)\qquad\text{for every }x\in V\cap U' .$$

If $u$ is bounded below near each point of $U$, then $u|_{U'}$ is bounded below near each point of $U'$ and $\bigl(u|_{U'}\bigr)^{+}_{\delta}(x)=u^{+}_{\delta}(x)$ for every $x\in V\cap U'$.

2. (A penalised continuous function is its own $\delta$-envelope)(ii) Let $\psi:U\to\mathbb{R}$ be continuous on $U$ and let $\lambda\in\mathbb{R}$ satisfy $0<\lambda$. Then $\psi$ is bounded above near each point of $U$ and bounded below near each point of $U$. Moreover the function $V\cap U\to\mathbb{R}$ with value $\psi(x)-\lambda h(x)$ at $x$ is upper semicontinuous on $V\cap U$ and bounded above near each point of $V\cap U$, and it coincides with its own upper semicontinuous envelope. Likewise the function $V\cap U\to\mathbb{R}$ with value $\psi(x)+\lambda h(x)$ at $x$ is lower semicontinuous on $V\cap U$ and bounded below near each point of $V\cap U$, and it coincides with its own lower semicontinuous envelope.

3. (Closed superlevel sets on the trace of a closed set)(iii) Let $K\subseteq U$ be closed in $H$ and let $\vartheta:U\to\mathbb{R}$ be continuous on $U$. If $u$ is bounded above near each point of $U$, then the function $V\cap K\to\mathbb{R}$ with value $u^{-}_{\delta}(x)-\vartheta(x)$ at $x$ has closed superlevel sets in $H$. If $u$ is bounded below near each point of $U$, then the function $V\cap K\to\mathbb{R}$ with value $\vartheta(x)-u^{+}_{\delta}(x)$ at $x$ has closed superlevel sets in $H$.