Sobolev inequalities with remainder terms

The usual Sobolev inequality in Rn, n ≥ 3, asserts that ∥▽f{hook}∥22 ≥ Sn ∥f{hook}∥2*2, with Sn being the sharp constant. This paper is concerned, instead, with functions restricted to bounded domains Ω ⊂ Rn. Two kinds of inequalities are established: (i) If f{hook} = 0 on ∂Ω, then ∥▽f{hook}∥22 ≥ Sn ∥f{hook}||2*2 + C(Ω) ∥f{hook}∥p,w2 with p = 2* 2 and ∥▽f{hook}∥22 ≥ Sn ∥f{hook}∥2*2 + D(Ω) ∥▽f{hook}∥q,w2 with q = n (n - 1). (ii) If f{hook} ≠ 0 on ∂Ω, then ∥▽f{hook}∥2 + C(Ω) ∥f{hook}∥q,∂Ω ≥ Sn 1 2 ∥f{hook}∥2* with q = 2(n - 1) (n - 2). Some further results and open problems in this area are also presented.