arXiv subjects

Hiroto Ishida

Publications and source records attributed to Hiroto Ishida.

3 recordsLinked to original sources

arXiv2024

Inverse homogenization problem for the Drichlet problem for Poisson equation for $W^{-1,\infty}$ potential

We consider Poisson problems $-Δu^\varepsilon=f$ on perforated domains, and characterize the limit of $u^\varepsilon$ as the solution to $(-Δ+μ)u=f$ on domain $Ω\subset\mathbb{R}^d$ with some potential $μ\in W^{-1,\infty}(Ω).$ It is known that $μ$ is related to the capacity of holes when $μ\in L^\infty(Ω).$ In this paper, we characterize $μ$ as the limit of the density of the capacity of holes also for many $μ\in W^{-1,\infty}(Ω).$ We apply the result for the inverse homogenization problem, i.e. we construct holes corresponding to the given potential $μ\in L^d(Ω)+L^\infty(δ_S)$ where $δ_S$ is a surface measure.

math.AP↗

arXiv2023

Convergence rate of Dirichlet Laplacians on domains with holes to the Schrödinger operator with $L^p$ potential

We consider the Dirichlet Laplacian $\mathcal{A}_\varepsilon=-Δ$ in the domain $Ω\setminus\bigcup_i K_{i\varepsilon}\subset\mathbb{R}^n$ with holes $K_{i\varepsilon}$ and the Schrödinger operator $\mathcal{A}=-Δ+V$ in $Ω$ where $V$ is the $L^n(Ω)$ limit of the density of the capacities $\operatorname{cap}(K_{i\varepsilon}).$ Strong resolvent convergence for many $V\in W^{-1,\infty}(Ω)$ was studied by the author. In this paper, we study about convergence rate for $\mathcal{A}_\varepsilon\to\mathcal{A}$ in norm resolvent sense. The case for which $V$ is a constant is studied by Andrii Khrabustovskyi and Olaf Post.

math.SP↗

arXiv2023

Poisson equation in domains with concentrated holes

We consider solutions $u^\varepsilon$ of Poisson problems with the Dirichlet condition on domains $Ω_\varepsilon$ with holes concentrated at subsets of a domain $Ω$ non-periodically. We show $u^\varepsilon$ converges to a solution of a Poisson problem with a simple function potential. This is a generalized result of a sample model given by Cioranescu and Murat (1997). They showed a result for case that holes are distributed at $Ω$ periodically.

math.AP↗