An optimal pointwise Morrey-Sobolev inequality

Let $\Omega$ be a bounded, smooth domain of $\mathbb{R}^{N},$ $N\geq1.$ For each $p>N$ we study the optimal function $s=s_{p}$ in the pointwise inequality $$\left\vert v(x)\right\vert \leq s(x)\left\Vert \nabla v\right\Vert _{L^{p}(\Omega)},\quad\forall\,(x,v)\in\overline{\Omega}\times W_{0}$$ We show that $s_{p}\in C_{0}^{0,1-(N/p)}(\overline{\Omega})$ and that $s_{p}$ converges pointwise to the distance function to the boundary, as $p\rightarrow\infty.$ Moreover, we prove that if $\Omega$ is convex, then $s_{p}$ is concave and has a unique maximum point.