Optimal convergence rates for the finite element approximation of the Sobolev constant

Abstract: We establish optimal convergence rates for the P1 finite element approximation of the Sobolev constant in arbitrary dimensions N\geq 2 and for Lebesgue exponents 1<p<N. Our analysis relies on a refined study of the Sobolev deficit in suitable quasi-norms, which have been introduced and utilized in the context of finite element approximations of the p- Laplacian. The proof further involves sharp estimates for the finite element approximation of Sobolev minimizers.