Note on Friedrichs' inequality in N-star-shaped domains

We establish an explicit constant in the Friedrichs inequality‖u−1|S|∫Su(x)dx‖Lp(Ω)≤C(Ω,S,B)‖∇u‖Lp(Ω) for u∈W1,p(Ω)u∈W1,p(Ω), 1≤p<∞1≤p<∞, a domain Ω⊂RnΩ⊂Rn star-shaped with respect to a convex set B  , and a measurable subset S⊂ΩS⊂Ω with |S|>0|S|>0, n≥2n≥2. This result will be generalized to so-called N-star-shaped domains which can be written as union of N domains star-shaped with respect to convex sets. A second generalization concerns the use of higher order moments when replacing the integral mean of u over S by the mean of higher order Taylor polynomials. The proofs are direct and elementary.