Extremal functions for Morrey's inequality in convex domains

For a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ and $p>n$, Morrey’s inequality implies that there is $c>0$ such that $$c\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}^p\le {\int }_{\mathrm{\Omega }}\mathrm{\mid }Du{\mathrm{\mid }}^{p}dx$$ for each $u$ belonging to the Sobolev space $W_0^{1,p}(\mathrm{\Omega })$. We show that the ratio of any two extremal functions is constant provided that $\mathrm{\Omega }$ is convex. We also explain why this property fails to hold in general and verify that convexity is not a necessary condition for a domain to have this property. As a by product, we obtain the uniqueness of an optimization problem involving the Green’s function for the $p$-Laplacian.