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Naoki Hamamoto

Publications and source records attributed to Naoki Hamamoto.

6 recordsLinked to original sources
arXiv2023

Sharp Hardy-Leray inequality for solenoidal fields

We compute the best constant in functional integral inequality called the Hardy-Leray inequalities for solenoidal vector fields on $\mathbb{R}^N$. This gives a solenoidal improvement of the inequalities whose best constants are known for unconstrained fields, and develops of the former work by Costin-Maz'ya who found the best constant in the Hardy-Leray inequality for axisymmetric solenoidal fields. We derive the same best constant without any symmetry assumption, whose expression can be simplified in relation to the weight exponent. Moreover, it turns out that the best value cannot be attained in the space of functions satisfying the finiteness of the integrals in the inequality.

math.AP↗
arXiv2021

Sharp Uncertainty Principle inequality for solenoidal fields

This paper solves the $L^2$ version of Maz'ya's open problem (Integral Equations Operator Theory 2018) on the sharp uncertainty principle inequality \[\int_{\mathbb{R}^N}|\nabla {\bf\it u}|^2dx\int_{\mathbb{R}^N}|{\bf\it u}|^2|{\bf\it x}|^2dx\ge C_N\left(\int_{\mathbb{R}^N}|{\bf\it u}|^2dx\right)^2\] for solenoidal (namely divergence-free) vector fields ${\bf\it u}={\bf\it u}({\bf\it x})$ on $\mathbb{R}^N$. The best value of the constant turns out to be $C_N=\frac{1}{4}\left(\sqrt{N^2-4(N-3)}+2\right)^2$ which exceeds the original value $N^2/4$ for unconstrained fields. Moreover, we show the attainability of $C_N$ and specify the profiles of the extremal solenoidal fields: for $N\ge4$, the extremals are proportional to a poloidal field that is axisymmetric and unique up to the axis of symmetry; for $N=3$, there additionally exist extremal toroidal fields.

math.CA↗
arXiv2021

A curl-free improvement of the Rellich-Hardy inequality with weight

We consider the best constant in the Rellich-Hardy inequality (with a radial power weight) for curl-free vector fields on $\mathbb{R}^N$, originally found by Tertikas-Zographopoulos \cite{Tertikas-Z} for unconstrained fields. This inequality is considered as an intermediate between Hardy-Leray and Rellich-Leray inequalities. Under the curl-free condition, we compute the new explicit best constant in the inequality and prove the non-attainability of the constant. This paper is a sequel to \cite{CF_MAAN,CF_Re}.

math.AP↗
arXiv2019

Sharp Hardy-Leray inequality for three-dimensional solenoidal fields with axisymmetric swirl

In this paper, we prove Hardy-Leray inequality for three-dimensional solenoidal (i.e., divergence-free) fields with the best constant. To derive the best constant, we impose the axisymmetric condition only on the swirl components. This partially complements the former work by O. Costin and V. Maz'ya \cite{Costin-Mazya} on the sharp Hardy-Leray inequality for axisymmetric divergence-free fields.

math.AP↗
arXiv2006

Kerr-NUT-de Sitter Curvature in All Dimensions

We explicitly calculate the Riemannian curvature of D-dimensional metrics recently discussed by Chen, Lu and Pope. We find that they can be concisely written by using a single function. The Einstein condition which corresponds to the Kerr-NUT-de Sitter metric is clarified for all dimensions. It is shown that the metrics are of type D.

hep-th↗