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In this paper, we establish a Modica type estimate on bounded solutions tothe overdetermined elliptic problem \begin{equation*}  \begin{cases}  \Delta u+f(u) =0& \mbox{in $\Omega$, }\\ u>0 &\mbox{in $\Omega$, }  u=0 &\mbox{on $\partial\Omega$, }  \partial_{\nu} u=c_0 &\mbox{on $\partial\Omega$, }  \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^{n},n\geq 2$. Aswe will see, the presence of the boundary changes the usual form of the Modicaestimate for entire solutions. We will also discuss the equality case. Fromsuch estimates we will deduce information about the curvature of $\partial\Omega$ under a certain condition on $c_0$ and $f$. The proof uses the maximumprinciple together with scaling arguments and a careful passage to the limit inthe arguments by contradiction.

Modica type estimates and curvature results for overdetermined elliptic problems