# zbMATH — the first resource for mathematics

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
##### Keywords:
even order PDE operators
Full Text: