The Bump Construction on a Hilbert Triple: the Maximum of a Viscosity Subsolution and a Penalised $C^2$ Function

In the setting of Hilbert Triples: Standing Notation and Background, let $U\subseteq H$ be nonempty and open in $H$, with $W=D(A)\cap U$ and the class $C^{2}(U)$ as in Hilbert Triples: Standing Notation and Background (v), and let $h$ be the penalty function. Let $F$ be a second-order equation operator on $U$ relative to $(H,V,A)$ that is degenerate elliptic, with $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$. Let $\mu\in\mathbb{R}$ satisfy $0<\mu$, let $\psi\in C^{2}(U)$, with gradient $D\psi(x)\in H$ and Hessian $D^{2}\psi(x)\in\mathrm{Sym}(H)$, and let $v:U\to\mathbb{R}$ be a viscosity subsolution of $F$ on $U$. For a real $\delta>0$, $(\cdot)^{-}_{\delta}$ denotes the $\delta$-envelope of a function on $U$ that is bounded above near each point of $U$. For $a,b\in\mathbb{R}$, $\max\{a,b\}$ is their maximum, the order of $\mathbb{R}$ being a total order.

Let $w:U\to\mathbb{R}$ be the function given by

$$w(x)=\max\{\psi(x)-\mu h(x),\,v(x)\}\ \ \text{for }x\in V\cap U,\qquad w(x)=v(x)\ \ \text{for }x\in U\setminus V ,$$

which is well defined because $h(x)$ is a real number for $x\in V$.

Assume the following.

(Test condition on the classical piece)(i) For every $x\in W$ with $v(x)<\psi(x)-\mu h(x)$,

$$F^{+}_{\mu}\bigl(x,\psi(x),D\psi(x),D^{2}\psi(x)\bigr)\le 0 .$$

Then the following hold.

1. (Local bounds)(ii) The function $w$ is bounded above near each point of $U$, and $v(x)\le w(x)$ for every $x\in U$.

2. (The $\delta$-envelope of $w$)(iii) For every real $\delta>0$ and every $x\in V\cap U$,

$$w^{-}_{\delta}(x)=\max\bigl\{\psi(x)-(\mu+\delta)h(x),\ v^{-}_{\delta}(x)\bigr\} .$$

3. (The maximum is a viscosity subsolution)(iv) The function $w$ is a viscosity subsolution of $F$ on $U$.