Dependence of Friedrichs' constant on boundary integrals

Abstract

This note extends the results in ‘Optimal coercivity inequalities in W1,p(Ω)’ (G. Auchmuty, Proc. R. Soc. Edinb. A 135, 915–933.) describing the dependence of the optimal constant in the p-version of Friedrichs' inequality on the boundary integral term. In particular, it is shown that this constant is continuous, increasing, concave and increases to the optimal constant for the Dirichlet problem as s → ∞.