- Research Article
- Open Access
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.
- Bounded Domain
- Unit Ball
- Spherical Harmonic
- Radial Function
- Strong Version
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.
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.
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