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

Throughout we work in the setting of Second-Order Equations on Euclidean Open Sets and of Bounded Open Domain in Euclidean Space, whose notation, including that of Euclidean Space and Real Matrices: Standing Notation on which the former rests, is in force in the dimension $n$, a natural number with $1\le n$. In addition $\operatorname{tr}$ denotes the trace of a square real matrix; we abbreviate $\lVert z\rVert^{2}=\lVert z\rVert\lVert z\rVert$; and $\tfrac{s}{2}$ denotes the product of $s\in\mathbb{R}$ with the multiplicative inverse of $2=1+1$, which exists by claim 8 of Elementary Order Arithmetic in an Ordered Field.

Let $\gamma\in\mathbb{R}$ be positive, let $\kappa\in\mathbb{R}$ be nonnegative, let $f:\overline{\Omega}\to\mathbb{R}$ be continuous on $\overline{\Omega}$, as a map into the metric space $(\mathbb{R},d_{\mathbb{R}})$ of The Absolute Value Metric on the Real Numbers, and let $F$ be the second-order equation operator on $\Omega$ given by

$$F(x,r,p,X)=\gamma r-\tfrac{\kappa}{2}\operatorname{tr}(X)+\tfrac{1}{2}\lVert p\rVert^{2}-f(x),$$

as in The Viscous Hamilton-Jacobi Operator is Continuous, Strictly Proper and Satisfies the Structure Condition.

Let $u,v:\overline{\Omega}\to\mathbb{R}$ be continuous on $\overline{\Omega}$, as maps into $(\mathbb{R},d_{\mathbb{R}})$, and let $u|_{\Omega},v|_{\Omega}:\Omega\to\mathbb{R}$ be the functions whose values at $x\in\Omega$ are $u(x)$ and $v(x)$. Assume that $u|_{\Omega}$ and $v|_{\Omega}$ are viscosity solutions of $F$ on $\Omega$ and that

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

Then $u(x)=v(x)$ for every $x\in\overline{\Omega}$.