Viscosity Sub- and Supersolutions and Solutions 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, and write $\overline{\Omega}=\operatorname{cl}_{\mathbb{R}^{n}}(\Omega)$ and $\partial\Omega=\partial_{\mathbb{R}^{n}}\Omega$. Then $\Omega\subseteq\overline{\Omega}$ by claim 1 of The Closure is the Smallest Closed Superset, and since $\Omega$ is open, claim 4 of The Interior is the Largest Open Subset gives $\operatorname{int}_{\mathbb{R}^{n}}(\Omega)=\Omega$, so that by the definition of the boundary

$$\partial\Omega=\overline{\Omega}\setminus\Omega ;$$

in particular $\Omega$ and $\partial\Omega$ are disjoint and $\overline{\Omega}=\Omega\cup\partial\Omega$.

Let $F$ be a second-order equation operator on $\Omega$, let $g:\partial\Omega\to\mathbb{R}$, and let $u:\overline{\Omega}\to\mathbb{R}$. We call the pair $(F,g)$ the Dirichlet problem for $F$ with boundary data $g$ on $\Omega$.

1. (Subsolution)(i) We say that $u$ is a viscosity subsolution of the Dirichlet problem $(F,g)$ if $u$ is a viscosity subsolution of $F$ up to the boundary of $\Omega$ and $u(x)\le g(x)$ for every $x\in\partial\Omega$.

2. (Supersolution)(ii) We say that $u$ is a viscosity supersolution of the Dirichlet problem $(F,g)$ if $u$ is a viscosity supersolution of $F$ up to the boundary of $\Omega$ and $g(x)\le u(x)$ for every $x\in\partial\Omega$.

3. (Solution)(iii) We say that $u$ is a viscosity solution of the Dirichlet problem $(F,g)$ if it is both a viscosity subsolution and a viscosity supersolution of $(F,g)$.