## Optimal Hardy–Rellich inequalities, maximum principle and related eigenvalue problem.(English)Zbl 1109.31005

The Hardy-Rellich inequality states that for all $$u \in H_0^2(\Omega)$$
$\int_\Omega | \Delta u| ^2\,dx -\frac{n^2(n-4)^2}{16}\int_\Omega \frac{u^2}{| x| ^4}\,dx \geq 0, \;n \geq 5,$
where $$\Omega \subset \mathbb{R}^n$$ is a smooth bounded domain and $$0 \in \Omega$$. Topic of the present paper is the Hardy-Rellich inequality in the critical dimension $$n=4$$. The basic result is the following:
Let $$n\geq 4$$ and $$B$$ be the unit ball centered at zero. Then $$\forall \;u \in H_{0,r}^2(B)$$ or $$\forall \;u \in H_r^2(B)\cap H_{0,r}^1(B)$$ $\int_B | \Delta u| ^2\, dx \geq \frac{n^2}{4}\int_B\frac{\nabla u| ^2}{| x| ^2}\,dx$ and equality holds if and only if $$u \equiv 0.$$
The paper contains also conditions for the existence or non-existence of the first eigenvalue of the Hardy-Rellich operator
$\Delta^2 -\frac{n^2(n-4)^2}{16}\frac{q(x)}{| x| ^4}$
depending on $$q(x)$$. Furthermore, a maximum principle is given which implies $$u \geq 0$$ for the solution $$u$$ of $\Delta^2u-Vu =f \;\text{in} \;\Omega, \quad u=\phi \;\text{on} \;\partial \Omega, \quad -\Delta u = \psi \;\text{ on } \;\partial \Omega.$

### MSC:

 31B30 Biharmonic and polyharmonic equations and functions in higher dimensions 35P15 Estimates of eigenvalues in context of PDEs 26D10 Inequalities involving derivatives and differential and integral operators
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### References:

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