We study the asymptotic behaviour of solutions to Dirichlet problems in perforated domains for nonlinear elliptic equations associated with monotone operators. The main difference with respect to the previous papers on this subject is that no uniformity is assumed in the monotonicity condition. Under a very general hypothesis on the holes of the domains, we construct a limit equation, which is satisfied by the weak limits of the solutions. The additional term in the limit problem depends only on the local behaviour of the holes, which can be expressed in terms of suitable nonlinear capacities associated with the monotone operator.
This paper studies the Dirichlet problem for Laplace's equation in a domain $Ω_{\varepsilon, η}$ perforated with small holes, where $\varepsilon$ represents the scale of the minimal distances between holes and $η$ the ratio between the scale of sizes of holes and $\varepsilon$. We establish $W^{1, p}$ estimates for solutions with bounding constants depending explicitly on the small parameters $\varepsilon$ and $η$. We also show that these estimates are either optimal or near optimal.
In this paper we study the Dirichlet problem for Laplace's equation in a domain $ω_{ε, η}$ perforated periodically with small holes in $\mathbb{R}^d$, where $ε$ represents the scale of the minimal distances between holes and $η$ the ratio between the scale of sizes of holes and $ε$. We establish $W^{1, p}$ estimates for solutions with bounding constants depending explicitly on $ε$ and $η$. The proof relies on a large-scale Lipschitz estimate for harmonic functions in perforated domains. The results are optimal for $d\ge 2$.
We revisit the periodic homogenization of Dirichlet problems for the Laplace operator in perforated domains, and establish a unified proof that works for different regimes of hole-cell ratios, that is the ratio between the scaling factor of the holes and that of the periodic cells. The approach is then made quantitative and it yields correctors and error estimates for vanishing hole-cell ratios. For positive volume fraction of holes, the approach is just the standard oscillating test function method; for vanishing volume fraction of holes, we study asymptotic behaviors of a properly rescaled c…
We consider a sufficiently regular bounded open connected subset $Ω$ of $\mathbb{R}^n$ such that $0 \in Ω$ and such that $\mathbb{R}^n \setminus \clΩ$ is connected. Then we choose a point $w \in ]0,1[^n$. If $ε$ is a small positive real number, then we define the periodically perforated domain $T(ε) \equiv \mathbb{R}^n\setminus \cup_{z \in \mathbb{Z}^n}\cl(w+εΩ+z)$. For each small positive $ε$, we introduce a particular Dirichlet problem for the Laplace operator in the set $T(ε)$. More precisely, we consider a Dirichlet condition on the boundary of the set $w+εΩ$, and we denote the unique peri…
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 contained in a bounded region and the density of the electrostatic capacity of the balls is sufficiently small. If the capacity density is too large or the balls extend to the whole space, the method diverges, but we provide a suitable modification of the method that converges to the solution of the Dirichlet problem also in this case. We give new proofs of classi…
Let $Ω$ be a sufficiently regular bounded open connected subset of $\mathbb{R}^n$ such that $0 \in Ω$ and that $\mathbb{R}^n \setminus \mathrm{cl}Ω$ is connected. Then we take $q_{11},..., q_{nn}\in ]0,+\infty[$ and $p \in Q\equiv \prod_{j=1}^{n}]0,q_{jj}[$. If $ε$ is a small positive number, then we define the periodically perforated domain $\mathbb{S}[Ω_ε]^{-} \equiv \mathbb{R}^n\setminus \cup_{z \in \mathbb{Z}^n}\mathrm{cl}\bigl(p+εΩ+\sum_{j=1}^n (q_{jj}z_j)e_j\bigr)$, where $\{e_1,...,e_n\}$ is the canonical basis of $\mathbb{R}^n$. For $ε$ small and positive, we introduce a particular Dir…
This paper is concerned with boundary regularity estimates in the homogenization of elliptic equations with rapidly oscillating and high-contrast coefficients. We establish uniform nontangential-maximal-function estimates for the Dirichlet, regularity, and Neumann problems with $L^2$ boundary data in a periodically perforated Lipschitz domain.
Let $Ω$ be a sufficiently regular bounded open connected subset of $\mathbb{R}^n$ such that $0 \in Ω$ and that $\mathbb{R}^n \setminus \mathrm{cl}Ω$ is connected. Then we take $(q_{11},\dots, q_{nn})\in ]0,+\infty[^n$ and $p \in Q\equiv \prod_{j=1}^{n}]0,q_{jj}[$. If $ε$ is a small positive number, then we define the periodically perforated domain $\mathbb{S}[Ω_{p,ε}]^{-} \equiv \mathbb{R}^n\setminus \cup_{z \in \mathbb{Z}^n}\mathrm{cl}\bigl(p+εΩ+\sum_{j=1}^n (q_{jj}z_j)e_j\bigr)$, where $\{e_1,\dots,e_n\}$ is the canonical basis of $\mathbb{R}^n$. For $ε$ small and positive, we introduce a pa…
In this paper, we develop a viscosity method for Homogenization of Nonlinear Parabolic Equations constrained by highly oscillating obstacles or Dirichlet data in perforated domains. The Dirichlet data on the perforated domain can be considered as a constraint or an obstacle. Homogenization of nonlinear eigen value problems has been also considered to control the degeneracy of the Porous medium equation in perforated domains. For the simplicity, we consider obstacles that consist of cylindrical columns distributed periodically and perforated domains with punctured balls. If the decay rate of th…
Corrigendum to "Ramification of multiple eigenvalues for the Dirichlet-Laplacian in perforated domains" published in Journal of Functional Analysis 283 (2022) 109718
Taking advantage from the so-called "Lemma on small eigenvalues" by Colin de Verdière, we study ramification for multiple eigenvalues of the Dirichlet Laplacian in bounded perforated domains. The asymptotic behavior of multiple eigenvalues turns out to depend on the asymptotic expansion of suitable associated eigenfunctions. We treat the case of planar domains in details, thanks to the asymptotic expansion of a generalization of the so-called u-capacity which we compute in dimension 2. In this case multiple eigenvalues are proved to split essentially by different rates of convergence of the pe…
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 of the holes goes to zero but their volume fraction remains fixed and positive, the limit problem is a Dirichlet boundary value problem posed in the domain without the holes, and the effective diffusion coefficients are non-trivially modified; if that volume fraction goes to zero instead, i.e. the holes are dilute, the effective operator remains the Laplacia…
In this work we analyze the behavior of the solutions to nonlocal evolution equations of the form $u_t(x,t) = \int J(x-y) u(y,t) \, dy - h_ε(x) u(x,t) + f(x,u(x,t))$ with $x$ in a perturbed domain $Ω^ε\subset Ω$ which is thought as a fixed set $Ω$ from where we remove a subset $A^ε$ called the holes. We choose an appropriated families of functions $h_ε\in L^\infty$ in order to deal with both Neumann and Dirichlet conditions in the holes setting a Dirichlet condition outside $Ω$. Moreover, we take $J$ as a non-singular kernel and $f$ as a nonlocal nonlinearity. % Under the assumption that the c…
We prove a homogenization theorem for a class of quadratic convolution energies with random coefficients. Under suitably stated hypotheses of ergodicity and stationarity we prove that the $Γ$-limit of such energy is almost surely a deterministic quadratic Dirichlet-type integral functional, whose integrand can be characterized through an asymptotic formula. The proof of this characterization relies on results on the asymptotic behaviour of subadditive processes. The proof of the limit theorem uses a blow-up technique common for local energies, that can be extended to this `asymptotically-local…
We study the Poisson equation in a perforated domain with homogeneous Dirichlet boundary conditions. The size of the perforations is denoted by $ε$ > 0, and is proportional to the distance between neighbouring perforations. In the periodic case, the homogenized problem (obtained in the limit $ε$ $\rightarrow$ 0) is well understood (see [21]). We extend these results to a non-periodic case which is defined as a localized deformation of the periodic setting. We propose geometric assumptions that make precise this setting, and we prove results which extend those of the periodic case: existence of…
We consider a general second order linear elliptic equation in a finely perforated domain. The shapes of cavities and their distribution in the domain are arbitrary and non-periodic; they are supposed to satisfy minimal natural geometric conditions. On the boundaries of the cavities we impose either the Dirichlet or a nonlinear Robin condition; the choice of the type of the boundary condition for each cavity is arbitrary. Then we suppose that for some cavities the nonlinear Robin condition is sign-definite in certain sense. Provided such cavities and ones with the Dirichlet condition are distr…
We consider a boundary value problem for a general second order linear equation in a domain with a fine perforation. The latter is made by small cavities; both the shapes of the cavities and their distribution are arbitrary. The boundaries of the cavities are subject either to a Dirichlet or a nonlinear Robin condition. On the perforation, certain rather weak conditions are imposed to ensure that under the homogenization we obtain a similar problem in a non-perforated domain with an additional potential in the equation usually called a strange term. Our main results state the convergence of th…
In this paper, we focus on the homogenization process of the non-local elliptic boundary value problem $$\mathcal{L}_\varepsilon^s u_\varepsilon =(-\nabla\cdot (A_\varepsilon(x)\nabla))^{s}u_\varepsilon=f \mbox{ in } \mathcal O, $$ with $0<s<1$, considering non-homogeneous Dirichlet type condition outside of the bounded domain $\mathcal O\subseteq \mathbb{R}^n$. We find the homogenized problem by using the $H$-convergence method, as $\varepsilon\to 0$, under standard uniform ellipticity, boundedness and symmetry assumptions on coefficients $A_\varepsilon(x)$, with the homogenized coefficients …
We consider the spectral Dirichlet problem for the Laplace operator in the plane $Ω^{\circ}$ with double-periodic perforation but also in the domain $Ω^{\bullet}$ with a semi-infinite foreign inclusion so that the Floquet-Bloch technique and the Gelfand transform do not apply directly. We describe waves which are localized near the inclusion and propagate along it. We give a formulation of the problem with radiation conditions that provides a Fredholm operator of index zero. The main conclusion concerns the spectra $σ^{\circ}$ and $σ^{\bullet}$ of the problems in $Ω^{\circ}$ and $Ω^{\bullet},$…
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 spherical and are generated by a rescaled marked point process $(Φ, \mathcal{R})$. The point process $Φ$ generating the centres of the holes is either a Poisson point process or the lattice $\mathbb{Z}^d$; the marks $\mathcal{R}$ generating the radii are unbounded i.i.d random variables having finite $(d-2+β)$-moment, for $β> 0$. We study the rate of converge…
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 $Ω^ε\subset Ω$. Here $J$ is a non-singular kernel. We think about $Ω^ε$ as a fixed set $Ω$ from where we have removed a subset that we call the holes. We deal both with the Neumann and Dirichlet conditions in the holes and assume a Dirichlet condition outside $Ω$. In the later case we impose that $u$ vanishes in the holes but integrate in the whole $\mathbb{R}^N$ ($B=\mathbb{R}^N$) and in the former we just consid…
As a main result of the paper, we construct and justify an asymptotic approximation of Green's function in a domain with many small inclusions. Periodicity of the array of inclusions is not required. We start with an analysis of the Dirichlet problem for the Laplacian in such a domain to illustrate a method of meso scale asymptotic approximations for solutions of boundary value problems in multiply perforated domains. The asymptotic formula obtained involves a linear combination of solutions to certain model problems whose coefficients satisfy a linear algebraic system. The solvability of this…
We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem in perforated domains.
Let $\varepsilon>0$ be a small parameter. We consider the domain $Ω_\varepsilon:=Ω\setminus D_\varepsilon$, where $Ω$ is an open domain in $\mathbb{R}^n$, and $D_\varepsilon$ is a family of small balls of the radius $d_\varepsilon=o(\varepsilon)$ distributed periodically with period $\varepsilon$. Let $Δ_\varepsilon$ be the Laplace operator in $Ω_\varepsilon$ subject to the Robin condition ${\partial u\over \partial n}+γ_\varepsilon u = 0$ with $γ_\varepsilon\ge 0$ on the boundary of the holes and the Dirichlet condition on the exterior boundary. Kaizu (1985, 1989) and Brillard (1988) have sho…
For equations of order two with the Dirichlet boundary condition, as the Laplace problem, the Stokes and the Navier-Stokes systems, perforated domains were only studied when the distance between the holes $d_{\varepsilon}$ is equal or much larger than the size of the holes $\varepsilon$. Such a diluted porous medium is interesting because it contains some cases where we have a non-negligible effect on the solution when $(\varepsilon,d_{\varepsilon})\to (0,0)$. Smaller distance was avoided for mathematical reasons and for theses large distances, the geometry of the holes does not affect -- or f…
For several different boundary conditions (Dirichlet, Neumann, Robin), we prove norm-resolvent convergence for the operator $-Δ$ in the perforated domain $Ω\setminus \bigcup_{ i\in 2\varepsilon\mathbb Z^d }B_{a_\varepsilon}(i),$ $a_\varepsilon\ll\varepsilon,$ to the limit operator $-Δ+μ_ι$ on $L^2(Ω)$, where $μ_ι\in\mathbb C$ is a constant depending on the choice of boundary conditions. This is an improvement of previous results [Cioranescu & Murat. A Strange Term Coming From Nowhere, Progress in Nonlinear Differential Equations and Their Applications, 31, (1997)], [S. Kaizu. The Robin Problem…
We prove norm-resolvent and spectral convergence in $L^2$ of solutions to the Neumann Poisson problem $-Δu_\varepsilon = f$ on a domain $Ω_\varepsilon$ perforated by Dirichlet-holes and shrinking to a 1-dimensional interval. The limit $u$ satisfies an equation of the type $-u''+μu = f$ on the interval $(0,1)$, where $μ$ is a positive constant. As an application we study the convergence of solutions in perforated graph-like domains. We show that if the scaling between the edge neighbourhood and the vertex neighbourhood is chosen correctly, the constant $μ$ will appear in the vertex condition of…
Brinkman's term for the steady-state Navier-Stokes equations with prescribed flux rate or pressure drop in perforated pipes
Fakultät für Mathematik, Universität Regensburg（雷根斯堡大学）；Institut de Mathématiques de Jussieu - Paris Rive Gauche, Université Paris Cité（巴黎西岱大学）；Facultad de Matemáticas, Pontificia Universidad Católica de Chile（智利天主教Pontificia大学）
We take an open regular domain $Ω$ in $\mathbb{R}^n$ with $n\ge 3$. We introduce a pair of positive parameters $ε_1$ and $ε_2$ and we set $ε\equiv(ε_1,ε_2)$. Then we define the perforated domain $Ω_ε$ by making in $Ω$ a small hole of size $ε_1ε_2$ at distance $ε_1$ from the boundary. When $ε\rightarrow(0, 0)$, the hole approaches the boundary while its size shrinks at a faster rate. In $Ω_ε$ we consider a Dirichlet problem for the Laplace equation and we denote its solution by $u_ε$. By an approach based on functional analysis and on the introduction of special layer potentials we show that th…
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 $ε>0$ proportional to the radius of the holes and a map $ϕ$, which models the shape of the holes. So, if $g$ denotes the Dirichlet boundary datum and $f$ the Poisson datum, we have a solution for each quadruple $(ε,ϕ,g,f)$. Our aim is to study how the solution depends on $(ε,ϕ,g,f)$, especially when $ε$ is very small and the holes narrow to points. In contrast with previous works, we don't introduce the assumption that $f$ has z…
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 distance between the balls is of size $\eps$ and their average radius is $\varepsilon^α$, $α\in (1; 3)$. We prove that, as in the periodic case [G. Allaire, ``Homogenization of the Navier-Stokes equations in domains perforated with tiny holes. II''], the solutions converge to the solution of Darcy's law (or its scalar analogue in the case of Poisson). In the …
In this paper, we consider the Dirichlet problem of the three-dimensional Laplace equation in the unit ball with a shrinking hole. The problem typically arises from homogenization problems in domains perforated with tiny holes. We give an almost complete description concerning the uniform $W^{1,p}$ estimates: for any $3/2<p<3$ there hold the uniform $W^{1,p}$ estimates; for any $1<p<3/2$ or $3<p<\infty $, there are counterexamples indicating that the uniform $W^{1,p}$ estimates do not hold. The results can be generalized to higher dimensions.
We present several applications of mode matching methods in spectral and scattering problems. First, we consider the eigenvalue problem for the Dirichlet Laplacian in a finite cylindrical domain that is split into two subdomains by a "perforated" barrier. We prove that the first eigenfunction is localized in the larger subdomain, i.e., its $L_2$ norm in the smaller subdomain can be made arbitrarily small by setting the diameter of the "holes" in the barrier small enough. This result extends the well known localization of Laplacian eigenfunctions in dumbbell domains. We also discuss an extensio…
We consider the Dirichlet problem of the Stokes equations in a domain with a shrinking hole in $\mathbb{R}^d, \ d\geq 2$. A typical observation is that, the Lipschitz norm of the domain goes to infinity as the size of the hole goes to zero. Thus, if $p\neq 2$, the classical results indicate that the $W^{1,p}$ estimate of the solution may go to infinity as the size of the hole tends to zero. In this paper, we give a complete description for the uniform $W^{1,p}$ estimates of the solution for all $1<p<\infty$. We show that the uniform $W^{1,p}$ estimate holds if and only if $d'<p<d$ ($p=2$ when …
In this paper we study we study a Dirichlet optimal control prob- lem associated with a linear elliptic equation the coefficients of which we take as controls in the class of integrable functions. The characteristic feature of this control object is the fact that the skew-symmetric part of matrix-valued control A(x) belongs to L2-space (rather than Linfinty). In spite of the fact that the equations of this type can exhibit non-uniqueness of weak solutions, the corresponding OCP, under rather general assumptions on the class of admissi- ble controls, is well-posed and admits a nonempty set of s…
A Dirichlet problem for the Laplace operator in a domain with a small hole close to the boundary
École normale supérieure, CNRS, PSL Research University（巴黎高等师范学院、法国国家科学研究中心、PSL研究大学）；The University of Tulsa（塔尔萨大学）；Aberystwyth University（阿伯里斯特威斯大学）；CNRS, Univ Pau & Pays Adour, Laboratoire de mathématiques et de leur applications de Pau - Fédération IPRA, UMR 5142（法国国家科学研究中心、波城与阿杜尔河地区大学、波数学及其应用实验室）
This work is devoted to the development and analysis of a linearization algorithm for microscopic elliptic equations, with scaled degenerate production, posed in a perforated medium and constrained by the homogeneous Neumann-Dirichlet boundary conditions. This technique plays two roles: to guarantee the unique weak solvability of the microscopic problem and to provide a fine approximation in the macroscopic setting. The scheme systematically relies on the choice of a stabilization parameter in such a way as to guarantee the strong convergence in $H^1$ norm for both the microscopic and macrosco…
