Proof of Uniqueness for the Dirichlet Problem for a Viscous Hamilton-Jacobi Equation

Let $T=\{t\in\mathbb{R}:0\le t\}$. Since $\overline{\Omega}$ is nonempty and compact by Bounded Open Domain in Euclidean Space (ii) and $f$ is continuous on $\overline{\Omega}$, A Continuous Function on a Compact Set Admits a Nondecreasing Modulus of Continuity applies with the metric space $(\mathbb{R}^{n},d_{E})$ and $K=\overline{\Omega}$ and furnishes a modulus of continuity $\omega:T\to\mathbb{R}$ which, by A Continuous Function on a Compact Set Admits a Nondecreasing Modulus of Continuity (i) and A Continuous Function on a Compact Set Admits a Nondecreasing Modulus of Continuity (ii), is nondecreasing and dominates the oscillation of $f$ in exactly the two senses required of the modulus in The Viscous Hamilton-Jacobi Operator is Continuous, Strictly Proper and Satisfies the Structure Condition. Fix such an $\omega$; the proposition may then be applied with this $\omega$.

By The Viscous Hamilton-Jacobi Operator is Continuous, Strictly Proper and Satisfies the Structure Condition (i) the operator $F$ is continuous, by The Viscous Hamilton-Jacobi Operator is Continuous, Strictly Proper and Satisfies the Structure Condition (ii) it is strictly proper with constant $\gamma$, and by The Viscous Hamilton-Jacobi Operator is Continuous, Strictly Proper and Satisfies the Structure Condition (iii) the operator $F$ and this $\omega$ satisfy the structure condition of the comparison principle.

The hypotheses of Uniqueness for the Dirichlet Problem for Second-Order Equations are therefore all met, with this $F$, this $\gamma$, this $\omega$ and the given $u$ and $v$: the domain data are those of Bounded Open Domain in Euclidean Space, the functions $u$ and $v$ are continuous on $\overline{\Omega}$ as maps into $(\mathbb{R},d_{\mathbb{R}})$, their restrictions $u|_{\Omega}$ and $v|_{\Omega}$ are viscosity solutions of $F$ on $\Omega$, and $u$ and $v$ agree on $\partial\Omega$. That corollary gives $u(x)=v(x)$ for every $x\in\overline{\Omega}$, as required.