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 {Delta u+f(u)=0 in Omega, u>0 in Omega, u=0 on partial derivative Omega, partial derivative(nu)u=-kappa on partial derivative Omega, where Omega subset of R-n,n >= 2. 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 partial derivative Omega under a certain condition on kappa and f. The proof uses the maximum principle together with scaling arguments and a careful passage to the limit in the arguments by contradiction.