Sharpness of the differentiability almost everywhere and capacitary estimates for Sobolev mappings

Abstract

We give sharp conformal conditions for the differentiability in the Sobolev space $W_{\mathrm{loc}}^{1,n-1}(\Omega ,{\mathbb{R}}^n)$. Furthermore, we show that the space $W_{\mathrm{loc}}^{1,n-1}(\Omega ,{\mathbb{R}}^n)$ can be considered as the borderline space for some capacitary inequalities.