A Viscosity Solution of the Dirichlet Problem is Continuous and Attains the Boundary Data

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 let $\overline{\Omega}$ and $\partial\Omega$ be its closure and its boundary, as in Viscosity Sub- and Supersolutions and Solutions of the Dirichlet Problem. Let $F$ be a second-order equation operator on $\Omega$, let $g:\partial\Omega\to\mathbb{R}$, and let $u:\overline{\Omega}\to\mathbb{R}$ be a viscosity solution of the Dirichlet problem $(F,g)$. Let $u|_{\Omega}:\Omega\to\mathbb{R}$ be the function whose value at $z\in\Omega$ is $u(z)$, let $\overline{\Omega}$ be regarded as a subset of the metric space $(\mathbb{R}^{n},d_{E})$, and let $\mathbb{R}$ carry the metric $d_{\mathbb{R}}$ of The Absolute Value Metric on the Real Numbers.

Then the following hold.

1. (The boundary data are attained)(i) $u(x)=g(x)$ for every $x\in\partial\Omega$.

2. (Continuity)(ii) $u$ is continuous at every point of $\overline{\Omega}$ relative to $\overline{\Omega}$, as a map into $(\mathbb{R},d_{\mathbb{R}})$.

3. (The interior equation)(iii) $u|_{\Omega}$ is a viscosity solution of $F$ on $\Omega$.