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Factorizations and Hardy-Rellich-type inequalities. (English) Zbl 1402.35014
Gesztesy, Fritz (ed.) et al., Non-linear partial differential equations, mathematical physics, and stochastic analysis. The Helge Holden anniversary volume on the occasion of his 60th birthday. Based on the presentations at the conference ‘Non-linear PDEs, mathematical physics and stochastic analysis’, NTNU, Trondheim, Norway, July 4–7, 2016. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-186-6/hbk; 978-3-03719-686-1/ebook). EMS Series of Congress Reports, 207-226 (2018).
Summary: The principal aim of this note is to illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisely, introducing the two-parameter \(n\)-dimensional homogeneous scalar differential expressions \(T_{\alpha,\beta} := - \Delta + \alpha |x|^{-2} x \cdot \nabla + \beta |x|^{-2}\), \(\alpha, \beta \in \mathbb R\), \(x \in \mathbb R^n \setminus \{0\}\), \(n \in \mathbb N\), \(n \geq 2\), and its formal adjoint, denoted by \(T_{\alpha,\beta}^+\), we show that nonnegativity of \(T_{\alpha,\beta}^+ T_{\alpha,\beta}\) on \(C_0^{\infty}(\mathbb R^n \setminus \{0\})\) implies the fundamental inequality
\[ \begin{aligned} \int_{\mathbb R^n} [(\Delta f)(x)]^2 \, d^n x & \geq [(n - 4) \alpha - 2 \beta] \int_{\mathbb R^n} |x|^{-2} |(\nabla f)(x)|^2 \, d^n x \\ & \quad - \alpha (\alpha - 4) \int_{\mathbb R^n} |x|^{-4} |x \cdot (\nabla f)(x)|^2 \, d^n x \\ & \quad + \beta [(n - 4) (\alpha - 2) - \beta] \int_{\mathbb R^n} |x|^{-4} |f(x)|^2 \, d^n x,\\ & f \in C^{\infty}_0(\mathbb R^n \setminus \{0\}). \end{aligned} \]
A particular choice of values for \(\alpha\) and \(\beta\) yields known Hardy-Rellich-type inequalities, including the classical Rellich inequality and an inequality due to Schmincke. By locality, these inequalities extend to the situation where \(\mathbb R^n\) is replaced by an arbitrary open set \(\Omega \subseteq \mathbb R^n\) for functions \(f \in C^{\infty}_0(\Omega \setminus \{0\})\).
Perhaps more importantly, we will indicate that our method, in addition to being elementary, is quite flexible when it comes to a variety of generalized situations involving the inclusion of remainder terms and higher-order operators.
For the entire collection see [Zbl 1390.35006].

MSC:
35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
35J75 Singular elliptic equations
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