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Green’s kernels and meso-scale approximations in perforated domains. (English) Zbl 1273.35007

Lecture Notes in Mathematics 2077. Berlin: Springer (ISBN 978-3-319-00356-6/pbk; 978-3-319-00357-3/ebook). xvii, 258 p. (2013).
The book is devoted to the description of the behaviour of Green’s kernels for elliptic boundary value problems posed in domains containing one or many holes, or rigid inclusions, or defects. The book is divided in three parts and ten chapters. In the first part, the authors indicate the behaviour of Green’s kernel for the Laplace operator in a singularly perturbed domain. In the second part, they consider the case of linear elastic materials filling in domains which contain small defects or rigid inclusions or voids. In the last third part, they consider Poisson’s equation posed in domains with many inclusions or voids of different sizes and under different kinds of boundary conditions.
Chapter 1 describes the behaviour of Greens’ function for the Laplace operator in a domain \(\Omega \) of \(\mathbb{R}^{n}\), \(n\geq 2\), containing a hole \(F_{\varepsilon }\) associated to a contractible and compact subset \(F\) of positive harmonic capacity through the process \(F_{\varepsilon }=\{x:\varepsilon ^{-1}x\in F\}\). Here the authors assume that \(F\) and \( \Omega \) contain the origin \(O\) of \(\mathbb{R}^{n}\), that the minimum distance of \(O\) to the boundary of \(\Omega \) is equal to 1 and that the maximum distance of \(O\) to the boundary of \(F^{c}=\mathbb{R}^{n}\setminus F\) is equal to 1 too. They introduce the Green kernels \(G\) and \(g\) of \(\Omega \) and \(F^{c}\) respectively for the Laplace operator with Dirichlet boundary conditions, and their regular parts \(H\) and \(h\). Considering the small hole \( F_{\varepsilon }\) and \(\Omega _{\varepsilon }=\Omega \setminus F\), they introduce the Green function \(G_{\varepsilon }\) of the Laplace operator in \( \Omega _{\varepsilon }\) with Dirichlet boundary conditions and they indicate its behaviour in terms of \(G\), \(g\), \(H\) and of the capacitary potential \(P\) of \(F\) in \(\mathbb{R}^{n}\), that is the solution of \(-\Delta P=0\) in \(F^{c}\) , \(P=1\) on \(\partial F^{c}\) and \(P(\xi )\rightarrow \infty \) as \(\left| \xi \right| \rightarrow \infty \). The authors distinguish the cases \( n>2 \) and \(n=2\).
In Chapter 2, the case of Neumann boundary conditions imposed on a part of the boundary of \(\Omega _{\varepsilon }\) is considered. The authors start with the 2D case. They introduce the Green’s function for the Dirichlet problem in \(\Omega \) and its regular part, the Neumann function defined by \( \mathcal{N}(\xi ,\eta )=(2\pi )^{-1}\log \left| \xi -\eta \right| ^{-1}-h_{N}(\xi ,\eta )\) where \(h_{N}\) is the regular part of this Neumann function and a dipole vector \(\mathcal{D}(\xi )=(\mathcal{D}_{1}(\xi ), \mathcal{D}_{2}(\xi ))\) defined as the solution of the exterior Neumann problems \(\Delta \mathcal{D}_{j}(\xi )=0\) in \(\mathbb{R}^{2}\setminus F\), \( \frac{\partial \mathcal{D}_{j}}{\partial n}(\xi )=n_{j}\) on \(\partial F\) and \(\mathcal{D}_{j}(\xi )\rightarrow \infty \) as \(\left| \xi \right| \rightarrow \infty \). The authors first give an estimate on the difference \( h_{N}(\xi ,\eta )-\frac{\mathcal{D}(\eta )\cdot \xi }{2\pi \left| \xi \right| ^{2}}\) for \(\left| \xi \right| >2\) and \(\eta \in \mathbb{ R}^{2}\setminus F\). They then describe the behaviour of \(G_{\varepsilon }^{(N)}(x,y)\) in terms of \(G\) and of \(H\). They finally consider other situations, changing the boundary conditions.
In Chapter 3, the authors consider the case of several inclusions placed in the domain \(\Omega \). They consider \(N\) subsets \(\omega ^{(j)}\) of \(\mathbb{R }^{n}\) and the subsets \(\omega _{\varepsilon }^{(j)}=\{x:\varepsilon ^{-1}(x-O^{(j)})\in \omega ^{(j)}\}\), where \(O^{(j)}\) is an interior point of \(\omega ^{(j)}\). They assume similar hypotheses on the distances between \( O^{(j)}\) and the points of the boundary of \(\Omega \) or of \(\omega _{\varepsilon }^{(k)}\) as in the case of a single inclusion. Considering the Laplace operator in \(\Omega _{\varepsilon }=\Omega \setminus \bigcup _{j}\omega _{\varepsilon }^{(j)}\), they introduce the Green function \( G_{\varepsilon }\) for Dirichlet boundary conditions in \(\Omega _{\varepsilon }\). The first result of the chapter describes the behaviour of the capacitary potential vector \(P_{\varepsilon }=(P_{\varepsilon }^{(j)})_{j=1,\dots ,N}\). Then the authors describe the behaviour of \( G_{\varepsilon }\).
In Chapter 4, the authors consider a 2D case for a domain containing different inclusions as in the previous chapter. They compare the asymptotic behaviour of the regular part \(H_{\varepsilon }\) of the Green function \( G_{\varepsilon }\) with some approximate solution computed using the finite element method through the software FEMLAB/COMSOL. They consider different cases: large number of small inclusions or inclusions of relatively large size. They prove that the solution computed with this software is a quite good approximation of the exact regular part \(H_{\varepsilon }\).
Chapter 5 is devoted to the description of the asymptotic behaviour of the Green function for special singularly perturbed domains: domains whose boundary contains a flat piece which is perturbed, domains with a singularly perturbed conical boundary and long rods. In each case, the authors specify the behaviour of Green’s function in terms of these specific characteristics of the domain.
For the proof of the behaviour of Green’s function in all these different situations, the authors use direct computations and tools classically used in functional analysis, such as maximum principles.
Part 2 is devoted to the case of linear elastic materials and starts with Chapter 6. The authors first recall the Lamé operator in a domain \( \Omega _{\varepsilon }\) of \(\mathbb{R}^{2}\) or \(\mathbb{R}^{3}\) which contains a small inclusion of void part. They gather some properties of this Lamé operator and introduce the elastic capacitary potential matrix and its properties. Throughout this chapter, the authors draw some computations on the capactary potentials and on the Green function.
In Chapter 7, the authors still consider the case of linear elastic materials of Lamé type but now in domains containing many rigid inclusions. They indeed consider the Lamé operator written as \(L\left( \partial _{x}u\right) =0\) in \(\Omega _{\varepsilon }=\Omega \setminus \bigcup _{j}\omega _{\varepsilon }^{(j)}\) with Dirichlet boundary conditions \( u=\varphi \) on \(\partial \Omega \) and \(u=\varphi _{\varepsilon }^{(j)}\) on \( \partial \omega _{\varepsilon }^{(j)}\), \(j=1,\dots ,N\), where \(\varphi \) and \(\varphi _{\varepsilon }^{(j)}\) are continuous vector functions. The authors first prove a uniform estimate in \(\Omega _{\varepsilon }\) for the unique solution of this problem, distinguishing the cases \(\mathbb{R}^{2}\) and \(\mathbb{R}^{3}\). The end of the chapter presents the asymptotic behaviours of the Green tensor in these cases.
Chapter 8 describes the situation of a linear elastic material filling in a domain containing a small hole with Neumann (resp. Dirichlet) boundary conditions on the boundary of the hole (resp. on the boundary of the domain). Once again, the authors prove asymptotic expansions for the Green tensor mainly using a uniform estimate for the solution of an exterior domain, with respect to the hole.
Part 3 starts with Chapter 9 where the authors consider the Poisson equation \(-\Delta u=f\) in a multiply perforated domain, the holes being not necessarily periodically distributed in the domain and not necessarily identical in size and shape. The authors claim that they indeed do not want to use homogenization tools. The domain under consideration is defined as \( \Omega _{N}=\Omega \setminus \bigcup _{j}F^{(j)}\subset \mathbb{R}^{3}\) and the authors impose homogeneous Dirichlet boundary conditions on \(\partial \Omega _{N}\). They introduce the solution \(v_{f}\) of the unperturbed problem in \( \Omega \), the capacitary potential \(P^{(j)}\) of each hole \(F^{(j)}\), \( j=1,\dots ,N\), and the regular part \(H\) of the Green tensor. They write \( u=v_{f}+R^{(1)}\), where \(R^{(1)}\) satisfies homogeneous Dirichlet boundary conditions on \(\partial \Omega \) and is sought as \(R^{(1)}(x)= \sum_{j}C_j(P^{(j)}(x)-4\pi \mathrm{ cap}(F^{(j)})H(x,O^{(j)}))\) with \(O^{(j)}\in F^{(j)}\). They prove that there exists a unique vector solution \( (C_{j})_{j=1,\dots ,N}\). The main result of this chapter proves what the authors call a meso-scale approximation of \(u\) assuming that \(\varepsilon <cd^{7/4}\), where \(\varepsilon =\inf_{j}\mathrm{diam}(F^{(j)})\) and \(d=\min_{i\neq j}\left| O^{(j)}-O^{(i)}\right| /2\). The difference \( R:=u-v_{f}-R^{(1)}\) is indeed proved to be less than \(\varepsilon \left\| \nabla v_{f}\right\| _{L_{\infty }(\omega )}+\varepsilon ^{2}d^{-7/2}\left\| v_{f}\right\| _{L_{\infty }(\omega )}\). Furthermore, if \(\varepsilon <cd^{2}\) then \(\left\| \nabla R\right\| _{L_{2}(\Omega _{N})}\leq C\frac{\varepsilon ^{2}}{d^{4}}\left\| f\right\| _{L_{\infty }(\Omega _{N})}\). The last main result of this chapter gives the behaviour of the Green tensor \(G_{N}\) of this problem in \( \Omega _{N}\) in terms of the Green tensor \(G\) in \(\Omega \), the authors giving the expression of \(G_{N}\) in terms of \(G\), of the capacitary potentials \(P^{(j)}\) and of the preceding term \(R^{(1)}\) when \(\varepsilon <cd^{2}\).
The last chapter of the book is devoted to the study of a mixed problem for Poisson’s equation in \(\mathbb{R}^{3}\): \(-\Delta u_{N}=f\) in \(\Omega _{N}=\Omega \setminus \bigcup _{j}F^{(j)}\) with Dirichlet (resp. homogeneous Neumann) boundary conditions on \(\partial \Omega \) (resp. on \(\partial F^{(j)}\)). Here \(u\mid _{\partial \Omega }\in L^{1/2,2}(\partial \Omega )\) and \(f\in L_{\infty }(\Omega _{N})\) with compact support at a positive distance from the perforations. The authors here introduce the solution \(v\) of the unperturbed problem in \(\Omega \), the regular part \(H\) of Green’s function in \(\Omega \) and the family \((\mathcal{D}^{(k)})_{k=1,\dots ,N}\) of dipole vector functions for the holes \(F^{(k)}\). The main result of this chapter says that if \(\varepsilon <cd\), for a sufficiently small constant \(c\) , \(u_{N}=v+\sum_{k}C^{(k)}\cdot \mathcal{D}^{(k)}+\mathcal{R}_{N}\), where the coefficients \(C^{(k)}\) are the unique solution of a linear algebraic system and \(\mathcal{R}_{N}\) satisfies \(\left\| \nabla \mathcal{R} _{N}\right\| _{L_{2}(\Omega _{N})}^{2}\leq C(\varepsilon ^{11}d^{-11}+\varepsilon ^{5}d^{-3})\left\| \nabla v\right\| _{L_{2}(\Omega )}\). The chapter ends with the presentation of numerical results in the case of a large number of spherical voids. The authors compare the results of finite element simulations to their asymptotic approximation. The computed solutiona is a quite good approximation.
Throughout the whole book, the authors prove their strong competencies in the computations of Green functions for scalar or vectorial steady problems and in the study of their properties. Even if the behaviour of these Green functions for perturbed problems is “what we expect”, the verification of this behaviour requires some lengthy computations and some fruitful tools.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35J08 Green’s functions for elliptic equations
35B25 Singular perturbations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
74B05 Classical linear elasticity
74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics

Software:

COMSOL
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