A note on an overdetermined problem with non-constant boundary condition

We review some results about a variant of the Saint-Venant problem and about a related overdetermined problem. The latter is a generalization of the Serrin problem where the overdetermination reads |∇u(x)| = g(x) on the boundary of the unknown domain Ω, and g : R^N → [0, ∞) is a given function. We analyze some geometric properties of the solution Ω in relation with g and we prove some new results about the continuity of Ω with respect to g, assuming g is an homogeneous function.