Extremal functions for Morrey's inequality in convex domains

For a bounded domain Omega subset of R-n and p > n, Morrey's inequality implies that there is c > 0 such that c parallel to u parallel to(p)(infinity) <= integral(Omega) vertical bar Du vertical bar(p) dx for each u belonging to the Sobolev space W-0(1,p) (Omega). We show that the ratio of any two extremal functions is constant provided that Omega is convex. We also show with concrete examples why this property fails to hold in general and verify that convexity is not a necessary condition for a domain to have this feature. As a by product, we obtain the uniqueness of an optimization problem involving the Green's function for the p-Laplacian