Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity

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 and closed superlevel sets of functions on $U$, on $V\cap U$, and in $(U,d_{H})$ are as fixed there. For a function $v$ on $U$ or on $V\cap U$, $-v$ denotes the function with value $-v(x)$ at $x$. Then the following hold.

1. (Semicontinuity)(i) If $u$ is bounded above near each point of $U$, then $u^{-}_{\delta}$ is upper semicontinuous on $V\cap U$, is bounded above near each point of $V\cap U$, and satisfies $u(x)-\delta h(x)\le u^{-}_{\delta}(x)$ for every $x\in V\cap U$. If $u$ is bounded below near each point of $U$, then $u^{+}_{\delta}$ is lower semicontinuous on $V\cap U$, is bounded below near each point of $V\cap U$, and satisfies $u^{+}_{\delta}(x)\le u(x)+\delta h(x)$ for every $x\in V\cap U$.

2. (Duality)(ii) The function $u$ is bounded above near each point of $U$ if and only if $-u$ is bounded below near each point of $U$, and in that case $(-u)^{+}_{\delta}(x)=-u^{-}_{\delta}(x)$ for every $x\in V\cap U$. Likewise $u$ is bounded below near each point of $U$ if and only if $-u$ is bounded above near each point of $U$, and in that case $(-u)^{-}_{\delta}(x)=-u^{+}_{\delta}(x)$ for every $x\in V\cap U$.

3. (Closed superlevel sets)(iii) If $u$ is bounded above near each point of $U$, then $u^{-}_{\delta}$, as a function on the subset $V\cap U$ of the metric space $(U,d_{H})$, has closed superlevel sets in $(U,d_{H})$; that is, for every $t\in\mathbb{R}$ the set $\{x\in V\cap U: t\le u^{-}_{\delta}(x)\}$ is closed in $(U,d_{H})$. If $u$ is bounded below near each point of $U$, then $-u^{+}_{\delta}$ has closed superlevel sets in $(U,d_{H})$; that is, for every $t\in\mathbb{R}$ the set $\{x\in V\cap U: u^{+}_{\delta}(x)\le t\}$ is closed in $(U,d_{H})$.

4. (Global bounds)(iv) If $C\in\mathbb{R}$ satisfies $u(x)\le C$ for every $x\in U$, then $u$ is bounded above near each point of $U$ and

$$u^{-}_{\delta}(x)\le C-\delta h(x)\le C-\tfrac{\delta}{2}|x|_{H}^{2}\qquad\text{for every }x\in V\cap U .$$

If $C\in\mathbb{R}$ satisfies $-C\le u(x)$ for every $x\in U$, then $u$ is bounded below near each point of $U$ and $-C+\tfrac{\delta}{2}|x|_{H}^{2}\le -C+\delta h(x)\le u^{+}_{\delta}(x)$ for every $x\in V\cap U$.

5. (Monotonicity)(v) Let $v:U\to\mathbb{R}$ satisfy $u(x)\le v(x)$ for every $x\in U$, and let $\delta'\in\mathbb{R}$ satisfy $\delta\le\delta'$. If $u$ and $v$ are bounded above near each point of $U$, then $u^{-}_{\delta}(x)\le v^{-}_{\delta}(x)$ and $u^{-}_{\delta'}(x)\le u^{-}_{\delta}(x)$ for every $x\in V\cap U$. If $u$ and $v$ are bounded below near each point of $U$, then $u^{+}_{\delta}(x)\le v^{+}_{\delta}(x)$ and $u^{+}_{\delta}(x)\le u^{+}_{\delta'}(x)$ for every $x\in V\cap U$.

6. (Semicontinuous functions)(vi) If $u$ is upper semicontinuous on $U$, then $u$ is bounded above near each point of $U$ and $u^{-}_{\delta}(x)=u(x)-\delta h(x)$ for every $x\in V\cap U$. If $u$ is lower semicontinuous on $U$, then $u$ is bounded below near each point of $U$ and $u^{+}_{\delta}(x)=u(x)+\delta h(x)$ for every $x\in V\cap U$. In particular both conclusions hold when $u$ is continuous on $U$.

7. (Monotonicity in $\delta$ with the penalty gap)(vii) Let $\delta'\in\mathbb{R}$ satisfy $\delta\le\delta'$. If $u$ is bounded above near each point of $U$, then

$$u^{-}_{\delta'}(x)+(\delta'-\delta)\,h(x)\le u^{-}_{\delta}(x)\qquad\text{for every }x\in V\cap U .$$

If $u$ is bounded below near each point of $U$, then $u^{+}_{\delta}(x)+(\delta'-\delta)\,h(x)\le u^{+}_{\delta'}(x)$ for every $x\in V\cap U$.