  We consider solutions of the nonlinear Poisson equation $$-\Delta u = f(u)$$
in convex domains $\Omega \subset \mathbb{R}^2$ with Dirichlet boundary
conditions on $\partial \Omega$. Solutions of such equations are not usually
concave. We show (for monotonically increasing concave $f$) that if the
curvature $\kappa$ of the boundary is not too large, then $u$ is concave. For
example, if $\Omega \subset \mathbb{R}^2$ has measure $|\Omega| = \pi$ and
$\kappa \leq \sqrt{2}$, then the solution of $-\Delta u = 1$ is concave. Or, if
$|\Omega| = \pi$ and $\kappa \leq 1.3$, then the solution of $-\Delta u =
\sqrt{1+u}$ is concave. It would be interesting to understand under which
conditions (on the domain $\Omega$ and the nonlinearity $f$) are solutions of
the Poisson equation concave once the domain is `sufficiently round'.
