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Atsushi Nakayasu

Publications and source records attributed to Atsushi Nakayasu.

11 recordsLinked to original sources
arXiv2026

Complex harmonic mean

We study the harmonic mean of non-zero complex-valued random variables (complex harmonic mean) and establish several geometric estimates and bounds. In contrast to the classical positive-valued case, complex harmonic means may lie outside the convex hull of the range. We prove that if the range is contained in a disk not containing the origin, then the complex harmonic mean is confined to the same disk. This result is based on the behavior of disks under inversion and convexity arguments. Further estimates involving the modulus and the real part are obtained, and the two-point case is analyzed explicitly, revealing a circular structure. Several examples are provided to illustrate the distinctive features of complex harmonic means.

math.CV↗
arXiv2025

On a calculation method of the thickness via partial differential equations

This paper presents a mathematical analysis of an elliptic partial differential equation (PDE) designed to compute the geometric thickness of a given shape. The PDE-based formulation provides a direct and systematic approach to evaluate thickness through the elliptic equation, whose solution yields a vector field from which the thickness is extracted as the divergence. While the convergence of this PDE-based thickness to the geometric thickness had been rigorously justified only for simple geometries such as intervals and straight bands, its validity for more general shapes remained open. In this work, we extend the analysis to annular domains, where curvature effects are nontrivial. We prove that the PDE-based thickness converges to the geometric thickness as the diffusion parameter tends to zero by estimating the difference between two notions of thickness with the square root of the diffusion parameter. Explicit expressions involving modified Bessel functions are obtained for annuli, together with sharp inequalities for their ratios. These results provide a rigorous mathematical foundation for the PDE-based thickness and demonstrate its potential as a reliable tool in shape analysis and topology optimization.

math.AP↗
arXiv2024

Mathematical analysis of a partial differential equation system on the thickness

This study focuses on linear partial differential equation (PDE) systems that arise in topology optimization where the thickness of a structure is constrained. The thickness derived from the PDE is a fictitious one, and the key challenge of this work is to verify its equivalence to the intuitive, geometrically defined thickness. The main difficulty lies in that while intuitive thickness is determined solely by the shape, the thickness defined by the PDE depends not only on the shape but also on the entire domain and the diffusion coefficients used in solving the PDE. In this paper, we demonstrate that the thickness of an infinite, straight film as a simple shape with constant thickness is equivalent within a general domain. The proof involves constructing a reference solution within a special domain and evaluating the difference using the maximum (modulus) principle and an interior $H^1$ estimate. Additionally, we provide an estimate of the dependence of thickness on the diffusion coefficient.

math.AP↗
arXiv2024

Stability of metric viscosity solutions under Hausdorff convergence

This study investigated the stability of Hamilton--Jacobi equation on general metric spaces with a perturbation in some whole space. This type of stability appears in the domain perturbation problem. We find that the stability holds when the set converges in the Hausdorff sense and when the metric converges in some uniform sense. Examples of the perturbed space satisfying these assumptions include network approximation of self-similar sets such as the Sierpiński gasket, junction of shrinking tubes, and lattice lines with the Manhattan distance. We also give supplemental results on time-dependent or noncompact case. Stability can be achieved when the class of test function of metric viscosity solutions is reduced to the squared distance functions, whose proof is also given.

math.AP↗
arXiv2017

Convexity preserving properties for Hamilton-Jacobi equations in geodesic spaces

We study the convexity preserving property for a class of time-dependent Hamilton-Jacobi equations in a complete geodesic space. Assuming that the Hamiltonian is nondecreasing, we show that in a Busemann space the unique metric viscosity solution preserves the geodesic convexity of the initial value at any time. We provide two approaches and also discuss several generalizations for more general geodesic spaces including the lattice graph.

math.AP↗
arXiv2017

Integrability of the derivative of solutions to a singular one-dimensional parabolic problem

We study integrability of the derivative of solutions to a singular one-dimensional parabolic equation with initial data in $W^{1,1}$. In order to avoid additional difficulties we consider only the periodic boundary conditions. The problem we study is a gradient flow of a convex, linear growth variational functional. We also prove a similar result for the elliptic companion problem, i.e. the time semidiscretization.

math.AP↗
arXiv2015

On cell problems for Hamilton-Jacobi equations with non-coercive Hamiltonians and its application to homogenization problems

We study a cell problem arising in homogenization for a Hamilton-Jacobi equation whose Hamiltonian is not coercive. We introduce a generalized notion of effective Hamiltonians by approximating the equation and characterize the solvability of the cell problem in terms of the generalized effective Hamiltonian. Under some sufficient conditions, the result is applied to the associated homogenization problem. We also show that homogenization for non-coercive equations fails in general.

math.AP↗
arXiv2014

Two approaches to minimax formula of the additive eigenvalue for quasiconvex Hamiltonians

Two different proofs for an inf-sup type representation formula (minimax formula) of the additive eigenvalues corresponding to first-order Hamilton-Jacobi equations are given for quasiconvex (level-set convex) Hamiltonians not necessarily convex. The first proof, which is similar to known proofs for convex Hamiltonians, invokes a Jensen-like inequality for quasiconvex functions instead of the standard Jensen's inequality. The second proof is completely different with elementary calculations. It is based on convergence of derivatives of mollified Lipschitz continuous functions whose proof is also given. These methods also relate to an approximation problem of viscosity solutions.

math.AP↗
arXiv2014

On metric viscosity solutions for Hamilton-Jacobi equations of evolution type

This paper studies Hamilton-Jacobi equations of evolution type defined in a general metric space. We give a notion of a solution through optimal principles and establish a unique existence theorem of the solution for initial value problems. We also note a relationship between the notion of a solution and another notion based on characterization of the modulus of the gradient in the sense of [13].

math.AP↗