- 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]).
and is the unit ball in . Our main concern in this note is to improve (1.4). In fact we have the following theorem.
Inequality (1.7) is optimal in case is a ball centered at zero.
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.
inequality (1.5) is implied by (1.9) in case of .
To prove the main results, we first need the following preliminary result.
Our next step is to prove the following. If is not a radial function, inequality (2.6) also holds.
Using equality (2.10), we have that (see, e.g., [6, page 452])
An immediate consequence of the inequalities (2.13) and Lemma 2.2 is the following result. For ,
Using inequalities (2.18) and Lemma 2.1, we have that, since , for ,
Proof of Theorem 1.1.
Using Lemma 2.1 and inequality (1.2), we have that . On the other hand, we have, by inequality (2.21), . Thus . The proof is complete.
Proof of Theorem 1.3.
A scaling argument shows that we may assume and .
is the radial Laplacian in .
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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