- Open Access
A note on a class of Hardy-Rellich type inequalities
© Di et al.; licensee Springer 2013
- Received: 31 May 2012
- Accepted: 18 February 2013
- Published: 4 March 2013
In this note we provide simple and short proofs for a class of Hardy-Rellich type inequalities with the best constant, which extends some recent results.
- Hardy inequality
- Hardy-Rellich inequality
- Caffarelli-Kohn-Nirenberg inequality
where , the constant is optimal and not attained.
where the constant is sharp.
where the constant is sharp.
The goal of this paper is to extend the above (1.3) and (1.6) to the general case for by a different and direct approach.
In this section, we will give the proof of the main theorems.
where and the constant is sharp.
where is a positive number sequence converging to as . By direct computation and the limit process, we know the constant is sharp. □
Remark 1 When , the inequality (2.1) covers the inequality (2.4) in .
When we take special values for a, b, the following corollary holds.
- (ii)When , according to the inequality (2.1), we have(2.5)
- (iii)When and , we obtain the inequality(2.6)
By a similar method, we can prove the following case Hardy-Rellich type inequality.
where , and is the p-Laplacian operator.
We mention that we do not know whether the constant in (2.7) is optimal or not. □
- (i)when , , the inequality (2.7) is equivalent to the inequality(2.14)
- (ii)When , , we obtain the inequality(2.15)
- (iii)When , , we get(2.16)
This work is supported by NNSF of China (11001240), ZJNSF (LQ12A01023) and the foundation of the Zhejiang University of the Technology (20100229).
- Adimurthi AS: Role of the fundamental solution in Hardy-Sobolev type inequalities. Proc. R. Soc. Edinb., Sect. A 2006, 136: 1111–1130. 10.1017/S030821050000490XMathSciNetView ArticleGoogle Scholar
- Garofalo N, Lanconelli E: Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation. Ann. Inst. Fourier (Grenoble) 1990, 40: 313–356. 10.5802/aif.1215MathSciNetView ArticleGoogle Scholar
- Goldstein JA, Kombe I: Nonlinear degenerate parabolic equations on the Heisenberg group. Int. J. Evol. Equ. 2005, 1: 1–22.MathSciNetGoogle Scholar
- Goldstein JA, Zhang QS: On a degenerate heat equation with a singular potential. J. Funct. Anal. 2001, 186: 342–359. 10.1006/jfan.2001.3792MathSciNetView ArticleGoogle Scholar
- Jin Y, Han Y: Weighted Rellich inequality on H -type groups and nonisotropic Heisenberg groups. J. Inequal. Appl. 2010., 2010: Article ID 158281Google Scholar
- Jin Y, Zhang G: Degenerate p -Laplacian operators and Hardy type inequalities on h -type groups. Can. J. Math. 2010, 62: 1116–1130. 10.4153/CJM-2010-033-9MathSciNetView ArticleGoogle Scholar
- García Azorero JP, Peral Alonso I: Hardy inequalities and some critical elliptic and parabolic problems. J. Differ. Equ. 1998, 144: 441–476. 10.1006/jdeq.1997.3375View ArticleGoogle Scholar
- Caffarelli L, Kohn R, Nirenberg L: First order interpolation inequalities with weights. Compos. Math. 1984, 53: 259–275.MathSciNetGoogle Scholar
- Costa DG: Some new and short proofs for a class of Caffarelli-Kohn-Nirenberg type inequalities. J. Math. Anal. Appl. 2008, 337: 311–317. 10.1016/j.jmaa.2007.03.062MathSciNetView ArticleGoogle Scholar
- Tertikas A, Zographopoulos NB: Best constants in the Hardy-Rellich inequalities and related improvements. Adv. Math. 2007, 209: 407–459. 10.1016/j.aim.2006.05.011MathSciNetView ArticleGoogle Scholar
- Costa DG:On Hardy-Rellich type inequalities in . Appl. Math. Lett. 2009, 22: 902–905. 10.1016/j.aml.2008.02.018MathSciNetView ArticleGoogle Scholar
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