Functions with constant Laplacian satisfying homogeneous Robin boundary conditions

Abstract

The authors study properties of real-valued functions u defined over Ω, a simply-connected domain in RN for which the Laplacian of u is constant in Ω, and which satisfy, on the boundary of Ω, the Robin boundary condition u + β (∂ u /∂ n )=0. Here n is the outward normal and β ≥0. When N =2 and β =0, this is the classical St Venant torsion problem, but the concern in this paper is with N ≥2 and β ≥0. Results concerning the magnitude um and location zm of the maximum value of u , and estimates for the functional S β =∫ Ωu , and the maxima pm and qm of |∇ u | and |∂u/∂n|, respectively, are established using comparison theorems and variational arguments.