Perron's Method: Existence of a Viscosity Solution of the Dirichlet Problem

Throughout we work in the setting of Second-Order Equations on Euclidean Open Sets, whose notation, including that of Euclidean Space and Real Matrices: Standing Notation and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation on which it rests, is in force in a dimension $n$, a natural number with $1\le n$.

Let $\Omega\subseteq\mathbb{R}^{n}$ be open and nonempty, with closure $\overline{\Omega}$ and boundary $\partial\Omega$ as in Viscosity Sub- and Supersolutions and Solutions of the Dirichlet Problem. Let $F$ be a second-order equation operator on $\Omega$ that is continuous and degenerate elliptic, and let $g:\partial\Omega\to\mathbb{R}$. For a function $h:\overline{\Omega}\to\mathbb{R}$ that is bounded above and below near each point of $\overline{\Omega}$, we write $h^{*}$ and $h_{*}$ for its upper and lower semicontinuous envelopes on $\overline{\Omega}$.

Assume the following two hypotheses.

1. (Comparison holds for the Dirichlet problem)(i) Every viscosity subsolution $\underline{w}$ and every viscosity supersolution $\overline{w}$ of the Dirichlet problem $(F,g)$ satisfy $\underline{w}(x)\le\overline{w}(x)$ for every $x\in\overline{\Omega}$.

2. (Bounded barriers attaining the boundary data)(ii) There are a viscosity subsolution $\underline{u}$ and a viscosity supersolution $\overline{u}$ of $(F,g)$ and a real number $M$ such that $|\underline{u}(x)|\le M$ and $|\overline{u}(x)|\le M$ for every $x\in\overline{\Omega}$, and such that

$$\underline{u}_{*}(x)=g(x)=\overline{u}^{*}(x)\qquad\text{for every }x\in\partial\Omega .$$

Let $\mathcal{G}$ be the set of all viscosity subsolutions $\underline{w}$ of $(F,g)$ satisfying $\underline{u}(x)\le\underline{w}(x)$ and $\underline{w}(x)\le\overline{u}(x)$ for every $x\in\overline{\Omega}$. By hypothesis 1 above, applied to $\underline{u}$ and $\overline{u}$ we have $\underline{u}(x)\le\overline{u}(x)$ for every $x\in\overline{\Omega}$, so $\underline{u}\in\mathcal{G}$ and $\mathcal{G}$ is nonempty; and for $x\in\overline{\Omega}$ the set $\{\underline{w}(x):\underline{w}\in\mathcal{G}\}$ is nonempty and bounded above by $\overline{u}(x)$, so it has a least upper bound by Least Upper Bound Property of the Real Numbers, unique by Uniqueness of the Supremum and of the Infimum. Let $W:\overline{\Omega}\to\mathbb{R}$ be given by

$$W(x)=\sup\{\underline{w}(x):\underline{w}\in\mathcal{G}\}\qquad\text{for }x\in\overline{\Omega}.$$

Then $W$ is a viscosity solution of the Dirichlet problem $(F,g)$.(iii)