An optimal pointwise Morrey-Sobolev inequality

Let $\mathrm{\Omega }$ be a bounded, smooth domain of ${\mathbb{R}}^N,$ $N\ge 1.$ For each $p>N$ we study the optimal function $s=s_p$ in the pointwise inequality $$\mid v(x)\mid \le s(x){\parallel \mathrm{\nabla }v\parallel }_{L^p(\mathrm{\Omega })},\mathrm{\forall }(x,v)\in \bar{\mathrm{\Omega }}\times W_0^{1,p}(\mathrm{\Omega })\mathrm{.}$$ We show that $s_p\in C_0^{0,1-(N\mathrm{/}p)}(\bar{\mathrm{\Omega }})$ and that $s_p$ converges pointwise to the distance function to the boundary, as $p\to \mathrm{\infty }\mathrm{.}$ Moreover, we prove that if $\mathrm{\Omega }$ is convex, then $s_p$ is concave and has a unique maximum point.