An optimal pointwise Morrey-Sobolev inequality

Abstract

Let $\Omega$ 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 \[ \left\vert v(x)\right\vert \leq s(x)\left\Vert \nabla v\right\Vert _{L^{p}(Ω)},\quad\forall\,(x,v)\in\overlineΩ\times W_{0}% ^{1,p}(Ω). \] 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 $p\to \infty .$ Moreover, we prove that if $\Omega$ is convex, then $s_p$ is concave and has a unique maximum point.

15 pages