Perron's Method with a Weighted Penalty for Operators Convex in the Value, Gradient and Matrix Variables: Existence and Uniqueness of a Viscosity Solution of Given Weight

In the setting of Second-Order Equations on Euclidean Open Sets, let $n\ge1$ be a natural number, let $D\subseteq\mathbb{R}^{n}$ be open and nonempty, let $P$ be a penalty on $D$, let $w\in\mathbb{R}$, let $\gamma\in\mathbb{R}$ be positive, and let $F$ be a second-order equation operator on $D$ that is continuous, strictly proper with constant $\gamma$ and convex in $(r,p,X)$, and satisfies the structure condition on the sublevel sets of $P$ and has a classical subsolution $sP-K$ with $s<w$. We say that $f:D\to\mathbb{R}$ is of weight $w$ if $f-wP$ has $P$-subordinate growth from above and from below.

Assume that there are a viscosity subsolution $\underline{u}:D\to\mathbb{R}$ and a viscosity supersolution $\overline{u}:D\to\mathbb{R}$ of $F$ on $D$, both of weight $w$.

Then the following hold.

1. (Existence)¶ There is a function $u:D\to\mathbb{R}$ of weight $w$ that is both a viscosity subsolution and a viscosity supersolution of $F$ on $D$.

2. (Uniqueness)¶ Any two functions with the properties of claim 1 are equal.

3. (Continuity and bounds)¶ The function $u$ of claim 1 is continuous on $D$, as a map into the real line $(\mathbb{R},d_{\mathbb{R}})$, and $\underline{u}(x)\le u(x)\le\overline{u}(x)$ for every $x\in D$.