An Improved Hardy-Rellich Inequality with Optimal Constant
© Y.-X. Xiao and Q.-H. Yang 2009
Received: 25 May 2009
Accepted: 11 September 2009
Published: 27 September 2009
We show that a Hardy-Rellich inequality with optimal constants on a bounded domain can be refined by adding remainder terms. The procedure is based on decomposition into spherical harmonics.
where and denote the volume of the unit ball and , respectively, and is the first eigenvalue of the Dirichlet Laplacian of the unit disc in . In case is a ball centered at zero, the constant in (1.2) is sharp.
Similar improved inequalities have been recently proved if instead of (1.1) one considers the corresponding Hardy inequalities. In all these cases a correction term is added on the right-hand side (see, e.g., [2–4]).
Combining Theorem 1.1 with (1.2), we have the following.
Next we consider analogous inequality (1.5). The main result is the following theorem.
2. The Proofs
To prove the main results, we first need the following preliminary result.
Using equality (2.10), we have that (see, e.g., [6, page 452])
Proof of Theorem 1.1.
Proof of Theorem 1.3.
In getting the last equality, we used Lemma 2.1.
which demonstrates inequality (1.9).
This work was supported by National Science Foundation of China under Grant no. 10571044.
- Brezis H, Vázquez JL: Blow-up solutions of some nonlinear elliptic problems. Revista Matemática de la Universidad Complutense de Madrid 1997,10(2):443–469.MATHMathSciNetGoogle Scholar
- Adimurthi , Chaudhuri N, Ramaswamy M: An improved Hardy-Sobolev inequality and its application. Proceedings of the American Mathematical Society 2002,130(2):489–505. 10.1090/S0002-9939-01-06132-9MathSciNetView ArticleMATHGoogle Scholar
- Filippas S, Tertikas A: Optimizing improved Hardy inequalities. Journal of Functional Analysis 2002,192(1):186–233. 10.1006/jfan.2001.3900MathSciNetView ArticleMATHGoogle Scholar
- Gazzola F, Grunau H-C, Mitidieri E: Hardy inequalities with optimal constants and remainder terms. Transactions of the American Mathematical Society 2004,356(6):2149–2168. 10.1090/S0002-9947-03-03395-6MathSciNetView ArticleMATHGoogle Scholar
- Davies EB, Hinz AM: Explicit constants for Rellich inequalities in . Mathematische Zeitschrift 1998,227(3):511–523. 10.1007/PL00004389MathSciNetView ArticleMATHGoogle Scholar
- Tertikas A, Zographopoulos NB: Best constants in the Hardy-Rellich inequalities and related improvements. Advances in Mathematics 2007,209(2):407–459. 10.1016/j.aim.2006.05.011MathSciNetView ArticleMATHGoogle Scholar
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