Extremal functions for Morrey’s inequality in convex domains

For a bounded domain Ω ⊂ R[superscript n] and p>n , Morrey’s inequality implies that there is c>0 such that c∥u∥p[subscript ∞]≤∫[subscript Ω]|Du|p[subscript dx] for each u belonging to the Sobolev space W[superscript 1,p][subscript 0](Ω) . We show that the ratio of any two extremal functions is constant provided that Ω 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