Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Factorizations and Hardy-Rellich-Type Inequalities   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 precisly,
introducing the two-parameter $n$-dimensional homogeneous scalar differential
expressions $T_{\alpha,\beta} := - \Delta + \alpha |x|^{-2} x \cdot \nabla +
\beta |x|^{-2}$, $\alpha, \beta \in \mathbb{R}$, $x \in \mathbb{R}^n \backslash
\{0\}$, $n \in \mathbb{N}$, $n \geq 2$, and its formal adjoint, denoted by
$T_{\alpha,\beta}^+$, we show that nonnegativity of $T_{\alpha,\beta}^+
T_{\alpha,\beta}$ on $C_0^{\infty}(\mathbb{R}^n \backslash \{0\})$ implies the
fundamental inequality, \begin{align} \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 \notag \\ & \quad - \alpha (\alpha - 4) \int_{\mathbb{R}^n}
|x|^{-4} |x \cdot (\nabla f)(x)|^2 \, d^n x \notag \\ & \quad + \beta [(n - 4)
(\alpha - 2) - \beta] \int_{\mathbb{R}^n} |x|^{-4} |f(x)|^2 \, d^n x, \notag
\end{align} for $f \in C^{\infty}_0(\mathbb{R}^n \backslash \{0\})$. 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 $f \in C^{\infty}_0(\Omega \backslash
\{0\})$.
  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
situations.

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The categories of this paper are: eess
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-fin
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The categories of this paper are: chem-ph
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The categories of this paper are: funct-an
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The categories of this paper are: stat
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The categories of this paper are: cs
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-th
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The categories of this paper are: adap-org
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The categories of this paper are: physics
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The categories of this paper are: q-bio