Sobolev - Poincaré Implies

Ω |∇u| dx )1/p holds for 1 ≤ p < n whenever u is smooth, uΩ = |Ω|−1 ∫ Ω u dx, and Ω ⊂ R is bounded and satisfies the cone condition. By the density of smooth functions, (1.1) then holds for all functions in the Sobolev space W (Ω) consisting of all functions in L(Ω) whose distributional gradients belong to L(Ω). For 1 < p < n, inequality (1.1) was proved by Sobolev [S1], [S2], and, for p = 1, this result is due to Gagliardo [G] and Nirenberg [N]. In fact, the above inequality for 1 < p < n can be deduced from the case p = 1 using Hölder’s inequality. The case p = 1 has a geometric interpretation: the above inequality holds if and only if the domain Ω satisfies a relative isoperimetric inequality. This observation is due to Maz’ya [M1], [M2]; also see the paper of Federer and Fleming [FF]. Recently Bojarski [B] has verified the above Sobolev-Poincaré inequality for so-called John domains; also see [C] for related results. In [BK], the current authors establish Sobolev-Poincaré inequalities on John domains in the full range 0 < p < n for solutions to certain elliptic equations,