On Friedrichs-Poincaré-type inequalities

Friedrichs- and Poincaré-type inequalities are important and widely used in the area of partial differential equations and numerical analysis. Most of their proofs appearing in references are the argument of reduction to absurdity. In this paper, we give direct proofs of Friedrichs-type inequalities in H1(Ω) and Poincaré-type inequalities in some subspaces of W1,p(Ω). The dependencies of the inequality coefficients on the domain Ω and some sub-domains are illustrated explicitly.