An optimal pointwise Morrey-Sobolev inequality

Let Ω be a bounded, smooth domain of RN, N≥1. For each p>N we study the optimal function s=sp in the pointwise inequality|v(x)|≤s(x)‖∇v‖Lp(Ω),∀(x,v)∈Ω‾×W01,p(Ω). We show that sp∈C00,1−(N/p)(Ω‾) and that sp converges pointwise to the distance function to the boundary, as p→∞. Moreover, we prove that if Ω is convex, then sp is concave and has a unique maximum point.