Sobolev Inequalities with Remainder Terms

Abstract

The usual Sobolev inequality in ℝⁿ, n≥3, asserts that

$$\left\| {\nabla f} \right\|_2^2 \geqslant {S_n}\left\| f \right\|_{2*}^2$$

(1)

, with S _n 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 = 0 on ∂ Ω, then

$$\left\| {\nabla f} \right\|_2^2 \geqslant {S_n}\left\| f \right\|_{2*}^2 + C(\Omega )\left\| f \right\|_{p,w}^2$$

(2)

with p=2*/2. Some further results and open problems in this area are also presented.