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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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