Modica-type estimates and curvature results for overdetermined elliptic problems

In this paper, we establish a Modica-type estimate on bounded solutions to the overdetermined elliptic problem [Formula: see text] where [Formula: see text] [Formula: see text]. As we will see, the presence of the boundary changes the usual form of the Modica estimate for entire solutions. We will also discuss the equality case. From such estimates, we will deduce information about the curvature of [Formula: see text] under a certain condition on [Formula: see text] and [Formula: see text]. The proof uses the maximum principle together with scaling arguments and a careful passage to the limit in the arguments by contradiction.