[proofplan] Apply the maximum principle to the difference $w=u-v$. [/proofplan] [step:Form the difference] Let $w=u-v$. Then $w\in C^2(U)\cap C(\overline U)$, and the assumptions give \begin{align*} -\Delta w\le 0 \quad\text{in }U, \qquad w\le 0 \quad\text{on }\partial U. \end{align*} Equivalently, $\Delta w\ge0$ in $U$. [/step] [step:Use the maximum principle] Since $U$ is bounded, $w\in C^2(U)\cap C(\overline U)$, and $\Delta w\ge0$ in $U$, the [Weak Maximum Principle for Elliptic Operators](/theorems/100) applied to the Laplacian gives that the maximum of $w$ on $\overline U$ is attained on $\partial U$. Since $w\le0$ on $\partial U$, it follows that \begin{align*} w\le0 \quad\text{in }U. \end{align*} Thus $u-v\le0$ in $U$, so $u\le v$ in $U$. [/step]