Abstract
The usual Sobolev inequality in
R
n
,
n ⩾ 3, asserts that
∥▽ƒ∥
2
2 ⩾ S
n ∥ƒ∥
2
∗
2
, with
S
n
being the sharp constant. This paper is concerned, instead, with functions restricted to bounded domains Ω ⊂
R
n
. Two kinds of inequalities are established: (i) If ƒ = 0 on ∂Ω, then
∥▽ƒ∥
2
2 ⩾ S
n ∥ƒ||
2
∗
2 + C(Ω) ∥ƒ∥
p,w
2
with
p =
2
∗
2
and
∥▽ƒ∥
2
2 ⩾ S
n ∥ƒ∥
2
∗
2 + D(Ω) ∥▽ƒ∥
q,w
2
with
q =
n
(n − 1)
. (ii) If ƒ ≠ 0 on ∂Ω, then
∥▽ƒ∥
2 + C(Ω) ∥ƒ∥
q,∂Ω ⩾ S
n
1
2
∥ƒ∥
2
∗
with
q =
2(n − 1)
(n − 2)
. Some further results and open problems in this area are also presented.