The $\delta$-Envelopes $u^-_\delta$ and $u^+_\delta$ of a Function on an Open Subset of a Hilbert Triple

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 set $V\cap U$ 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), and local bounds and semicontinuous envelopes of functions on $U$ and on $V\cap U$ are taken for the metric $d_{H}$.

1. (The envelope $u^{-}_{\delta}$)(i) Suppose that $u$ is bounded above near each point of $U$. Then the function $u-\delta h:V\cap U\to\mathbb{R}$, whose value at $x$ is $u(x)-\delta h(x)$, is bounded above near each point of $V\cap U$ by The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds (vi), so its upper semicontinuous envelope is defined. We write

$$u^{-}_{\delta}=(u-\delta h)^{*}:\ V\cap U\to\mathbb{R}$$

for this envelope.

2. (The envelope $u^{+}_{\delta}$)(ii) Suppose that $u$ is bounded below near each point of $U$. Then the function $u+\delta h:V\cap U\to\mathbb{R}$, whose value at $x$ is $u(x)+\delta h(x)$, is bounded below near each point of $V\cap U$ by The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds (vi), so its lower semicontinuous envelope is defined. We write

$$u^{+}_{\delta}=(u+\delta h)_{*}:\ V\cap U\to\mathbb{R}$$

for this envelope.

(iii) The two functions $u^{-}_{\delta}$ and $u^{+}_{\delta}$ are together called the $\delta$-envelopes of $u$ relative to $(H,V,A)$.