Proof of Uniqueness for the Dirichlet Problem for Second-Order Equations

By claim 2 of Semicontinuity Under Negation and Characterization of Continuity, applied at each point of $\overline{\Omega}$, the function $u$ is both upper semicontinuous and lower semicontinuous on $\overline{\Omega}$, and likewise $v$.

By Viscosity Solution of a Second-Order Equation, $u|_{\Omega}$ is both a viscosity subsolution and a viscosity supersolution of $F$ on $\Omega$, and likewise $v|_{\Omega}$. Hence, by Viscosity Subsolution and Supersolution up to the Boundary, each of $u$ and $v$ is both a viscosity subsolution and a viscosity supersolution of $F$ up to the boundary of $\Omega$.

The operator $F$, the constant $\gamma$ and the modulus $\omega$ satisfy conditions 1, 2 and 3 of Comparison Principle for the Dirichlet Problem for Second-Order Equations by hypothesis. Applying that theorem to the viscosity subsolution $u$ and the viscosity supersolution $v$, whose boundary hypothesis $u(x)\le v(x)$ for $x\in\partial\Omega$ follows from $u(x)=v(x)$ there, gives

$$u(x)\le v(x)\qquad\text{for every }x\in\overline{\Omega}.$$

Applying it to the viscosity subsolution $v$ and the viscosity supersolution $u$, whose boundary hypothesis $v(x)\le u(x)$ for $x\in\partial\Omega$ follows in the same way, gives

$$v(x)\le u(x)\qquad\text{for every }x\in\overline{\Omega}.$$

By the antisymmetry of $\le$, part of the total order structure of $\mathbb{R}$, we conclude that $u(x)=v(x)$ for every $x\in\overline{\Omega}$.