General Hardy inequalities with optimal constants and remainder terms
- PDF / 614,849 Bytes
- 13 Pages / 468 x 680 pts Page_size
- 18 Downloads / 180 Views
One-dimensional Hardy inequalities with weights and remainder terms are studied. The corresponding optimal constants are discussed. Then by the process of symmetrization, Hardy inequalities with remainder terms in high-dimensional Sobolev spaces are obtained. This result gives a positive answer to the Br´ezis-V´azquez conjecture. 1. Introduction In 1919, Hardy [7] proved the following inequality: ∞ 0
p
p u(t) dt ≤ p t p−1
p ∞ 0
p u (t) dt,
u ∈ C01 (0, ∞),
(1.1)
where 1 < p < +∞. The readers can refer to [8] for the proof of this inequality. The best constant (p/(p − 1)) p in the above inequality was given by Landau [10]. It is pointed out in [9] that, in 1933, Leray [11] proved the following two inequalities:
|u|2
|Du|2 dx, 2 R2 \B1 (0) |x|2 ln |x| |u|2 2 2 dx ≤ |Du|2 dx, |x|2 n−2 Rn
dx ≤ 4
R2 \B1 (0)
Rn
(1.2) (1.3)
where u ∈ H01 . Shen [13] obtained (1.2) for a bounded domain Ω ⊂ BR (0) with ln2 |x| replaced by ln2 R/ |x|. In 1995, Peral and V´azquez [12] showed that (2/(n − 2))2 is the best constant in (1.3). In 1980, Shen [14] proved if ψ and φ satisfy (φ1/ p ψ 1−1/ p ) = (p − 1)ψ, then ∞ 0
p
ψ(t)u(t) dt ≤
p
p ∞
p−1
0
p
φ(t)u (t) dt
(1.4)
for u ∈ C01 (0, ∞). Moreover, if ψ and φ also satisfy φ(0)ψ p−1 (0) = 0, then the above inequality is also true for u ∈ C 1 (0, ∞), see [16]. Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:3 (2005) 207–219 DOI: 10.1155/JIA.2005.207
208
General Hardy inequalities
It is proved in [15] that for p > 1, Rn
|u| p dx ≤ |x| p
p
p n− p
Rn
1,p
|Du| p dx,
u ∈ W0 (Rn ).
(1.5)
Garc´ıa Azorero and Peral Alonso [5] proved (1.5) by using a different method. Similar to [12], it is showed that (p/(n − p)) p is the best constant. For Hardy inequalities with remainder terms, Br´ezis and V´azquez [4] proved recently that there exists a constant C > 0, depending only on n and Ω, such that
n−2 |Du| dx ≥ 2 Ω
2
2
u2 dx + C 2 Ω |x |
Ω
|u|2 dx,
∀u ∈ H01 (Ω).
(1.6)
They asked whether the two terms on the right-hand side of (1.6) are just two terms of a series. Recently, Gazzola et al. [6] generalized (1.6) to the case of n > p. They proved that Ω
|Du| p dx ≥
n− p p
p
|u| p dx + C p Ω |x |
Ω
|u| p dx,
1,p
∀u ∈ W0 (Ω).
(1.7)
Another generalized form of (1.6) given by Adimurthi et al. [1] is Ω
|Du| p dx ≥
n− p p
p
k |u| p |u| p dx + C p |x| p Ω |x | j =1 Ω
j i=1
ln(i)
R |x|
2
dx.
(1.8)
Our paper is organized as follows. In Section 1, we study one-dimensional Hardy inequalities with any weights and the corresponding optimal constants. We prove that the constant (p/(p − 1)) p (p > 1) is the best constant in the inequality. Meanwhile, we give the relation between the weights in the Hardy inequalities, from which we can determine the other weight if one of the weights is given. In Section 2, we deal with one-dimensional Hardy inequalities involving any weights and remainder terms (p ≥ 2). We also study
Data Loading...
