Solution of Dirichlet's problem for the equation $\Delta u=-1$ in a convex region

Let $u$ be a solution of the following boundary-value problem: $u|_\Gamma=0$, where $\Gamma$ is a closed convex curve and $\Delta u=-1$ in the region $D$ bounded by $\Gamma$. Then $u$ has only one local maximum, and all its level curves are convex.