相关论文: Poisson equation in domains with concentrated hole…

Let $\Omega$ be a sufficiently regular bounded open connected subset of $\mathbb{R}^n$ such that $0 \in \Omega$ and that $\mathbb{R}^n \setminus \mathrm{cl}\Omega$ is connected. Then we take $(q_{11},\dots, q_{nn})\in ]0,+\infty[^n$ and $p…

This work showcases level set estimates for weak solutions to the $p$-Poisson equation on a bounded domain, which we use to establish Lebesgue space inclusions for weak solutions. In particular we show that if $\Omega\subset\mathbb{R}^n$ is…

We consider the Poisson equation with homogeneous Dirichlet conditions in a family of domains in $R^{n}$ indexed by a small parameter $\epsilon$. The domains depend on $\epsilon$ only within a ball of radius proportional to $\epsilon$ and,…

This paper studies the Dirichlet problem for Laplace's equation in a domain $\Omega_{\varepsilon, \eta}$ perforated with small holes, where $\varepsilon$ represents the scale of the minimal distances between holes and $\eta$ the ratio…

We consider Poisson problems $-\Delta u^\varepsilon=f$ on perforated domains, and characterize the limit of $u^\varepsilon$ as the solution to $(-\Delta+\mu)u=f$ on domain $\Omega\subset\mathbb{R}^d$ with some potential $\mu\in…

In this paper we establish $W^{1,p}$ estimates for solutions $u_\varepsilon$ to Laplace's equation with the Dirichlet condition in a bounded and perforated, not necessarily periodically, $C^1$ domain $\Omega_{\varepsilon, \eta}$ in…

We consider a Dirichlet problem for the Poisson equation in a periodically perforated domain. The geometry of the domain is controlled by two parameters: a real number $\epsilon>0$ proportional to the radius of the holes and a map $\phi$,…

We take an open regular domain $\Omega$ in $\mathbb{R}^n$ with $n\ge 3$. We introduce a pair of positive parameters $\epsilon_1$ and $\epsilon_2$ and we set $\epsilon\equiv(\epsilon_1,\epsilon_2)$. Then we define the perforated domain…

We consider the homogenization of a Poisson problem or a Stokes system in a randomly punctured domain with Dirichlet boundary conditions. We assume that the holes are spherical and have random centres and radii. We impose that the average…

This paper deals with the homogenization of the Poisson equation in a bounded domain of $\mathbb{R}^d$, $d>2$, which is perforated by a random number of small spherical holes with random radii and positions. We show that for a class of…

We revisit the homogenization problem for the Poisson equation in periodically perforated domains with zero Neumann data at the boundary of the holes and prescribed Dirichlet data at the outer boundary. It is known that, if the periodicity…

We study the Dirichlet problem in a domain with a small hole close to the boundary. To do so, for each pair $\boldsymbol\varepsilon = (\varepsilon_1, \varepsilon_2 )$ of positive parameters, we consider a perforated domain…

In this paper we study the Dirichlet problem for Laplace's equation in a domain $\omega_{\epsilon, \eta}$ perforated periodically with small holes in $\mathbb{R}^d$, where $\epsilon$ represents the scale of the minimal distances between…

A common approach is present concerning the problem of Dirichlet, both for bounded 3D domains and their (unbounded) complements, regarding the fractional (3D) Poisson equation.

We study the Poisson equation in a perforated domain with homogeneous Dirichlet boundary conditions. The size of the perforations is denoted by $\epsilon$ > 0, and is proportional to the distance between neighbouring perforations. In the…

We investigate the behavior of the solution of a mixed problem in a domain with two moderately close holes. We introduce a positive parameter $\epsilon$ and we define a perforated domain $\Omega_{\epsilon}$ obtained by making two small…

We study the convergence of the Method of Reflections for the Dirichlet problem of the Poisson and the Stokes equations in perforated domains which consist in the exterior of balls. We prove that the method converges if the balls are…

In this paper we provide converge rates for the homogenization of the Poisson problem with Dirichlet boundary conditions in a randomly perforated domain of $\mathbb{R}^d$, $d \geq 3$. We assume that the holes that perforate the domain are…

We consider a sufficiently regular bounded open connected subset $\Omega$ of $\mathbb{R}^n$ such that $0 \in \Omega$ and such that $\mathbb{R}^n \setminus \cl\Omega$ is connected. Then we choose a point $w \in ]0,1[^n$. If $\epsilon$ is a…

In this paper we analyze nonlocal equations in perforated domains. We consider nonlocal problems of the form $f(x) = \int_{B} J(x-y) (u(y) - u(x)) dy$ with $x$ in a perforated domain $\Omega^\epsilon \subset \Omega$. Here $J$ is a…