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The general theory of homogenization. A personalized introduction. (English) Zbl 1188.35004

Lecture Notes of the Unione Matematica Italiana 7. Berlin: Springer (ISBN 978-3-642-05194-4/pbk; 978-3-642-05195-1/ebook). xvii, 470 p. (2009).
The book is divided in 34 short chapters including the introduction and a conclusion. It gathers some mathematical theories and tools which can be involved in problems where a homogenization process occurs. Although mainly dealing with linear situations, the author presents some of the classical but deep concepts he developed alone or with some colleagues: the div-curl convergence result, the H-convergence, the Young measures, effective bounds in homogenization theory, among others. More than forty years of intense mathematical activities in the field of homogenization theory are gathered here. In many parts of the book, the author complains about French colleagues who use his ideas without mentioning his name or rejecting his recruitment in France. The whole book is based on results obtained by the author, frequently together with F. Murat, his indefectible friend for 40 years.
The first three chapters are historical and personal notes from the author on the interaction between applied mathematics and other sciences and on the contributions of the author possibly with some colleagues and/or to other works dealing with homogenization.
The presentation of the mathematical material contained in the book really begins with chapter 4. The author here recalls the original question raised by [J. L. Lions, Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Paris: Dunod; Paris: Gauthier-Villars (1968; Zbl 0179.41801)] which consists in finding a minimizer for an \(L^{2}\) -cost functional with an underlying state equation and an admissible set of controls. F. Murat proved that oscillations may occur for a minimizing sequence which does not lead to the existence of a minimizer. The author explains the notion of relaxation within this context.
The short chapter 5 is devoted to a weak convergence result proved by F. Murat. In a rectangle \(\omega =I_{1}\times I_{2}\) of \(\mathbb{R}^{2}\) the elliptic equation \(-\frac{\partial }{\partial x_{1}}(a_{n}\frac{\partial y_{n}}{\partial x_{1}})--\frac{\partial }{\partial x_{2}}(a_{n}\frac{ \partial y_{n}}{\partial x_{2}})=f\) is considered with homogeneous Dirichlet boundary conditions. The coefficients \(a_{n}\) are supposed to depend only on \(x_{1}\). If \(a_{n}\) weakly converges to \(a_{+}\) and \(1/a_{n}\) weakly converges to \(1/a_{-}\) in \(L^{\infty }(I_{1})\)-weak and \(y_{n}\) weakly converges to \(y_{\infty }\) in \(H_{0}^{1}(\omega )\)-weak then \(\frac{\partial y_{n}}{\partial x_{\alpha }}\) weakly converges to \(\frac{\partial y_{\infty } }{\partial x_{\alpha }}\) in \(L^{2}(\omega )\), for \(\alpha =1,2\). This can be generalized in \(\mathbb{R}^{N}\).
Chapter 6 starts with the description of the G-convergence as introduced by E. De Giorgi and S. Spagnolo in [Boll. Unione Mat. Ital., IV. Ser. 8, 391–411 (1973; Zbl 0274.35002)] within the context of linear elliptic operators. The author presents the classical compactness result associated to this notion of convergence.
Chapter 7 presents the div-curl convergence lemma which is a key tool when dealing with many homogenization problems. Beside the original div-curl lemma, the author gathers various extensions.
Chapter 8 gives some insights on the link between the mathematical framework associated to homogenization and other sciences such as continuum mechanics, or physics and specially electrostatics. The author describes the link between the div-curl lemma presented in the previous chapter and the notion of equirepartition of hidden energy in the case of a scalar wave equation.
In chapter 9, the author explains how the div-curl lemma can be proved in higher dimensions using the Hodge decomposition of differential forms.
Chapter 10 describes some properties of the H-convergence in the framework of linear elliptic equations. Once again, the chapter starts with a short historical and personal introduction. Then the author presents the notion of H-convergence and its properties among which is a lower-semicontinuity property of the associated energy with respect to the weak \(H^{1}\)-topology.
In chapter 11, the author extends the convergence result of the preceding chapter in the case of monotone operators of the kind \(-\text{div}(A(x,\text{ grad}u))\) where \(A\) is a Carathéodory function defined in \(\Omega \times \mathbb{R}^{N}\) and which satisfies monotonicity properties.
Chapter 12 is devoted to an extension of the one-dimensional homogenization theory for an elliptic operator of the kind \(-\frac{d}{dx}(a_{n}\frac{du_{n} }{dx}+b_{n}u_{n})+c_{n}\frac{du_{n}}{dx}+d_{n}u_{n}=f\) in \(\Omega =(x_{-},x_{+})\). The convergence result is obtained assuming some weak convergences of the coefficients and of some of their quotients.
Chapter 13 describes the notion of correctors within the context of linear homogenization and that of H-convergence for a sequence \(A^{n}\) of matrices. The main result of this chapter proves the existence of correctors as a sequence of operators \(P^{n}\in L^{2}(\Omega ;\mathcal{L}(\mathbb{R}^{N}; \mathbb{R}^{N}))\) which converges to the identity matrix in the weak topology of this space and such that \(\text{grad}(u_{n})-P^{n}\text{grad} (u_{\infty })\) converges to 0 in the strong topology of \(L^{1}(\Omega ; \mathbb{R}^{N})\).
Chapter 14 extends the notion of corrector defined in the preceding chapter to the case of monotone operators.
Chapter 15 moves to a classical problem in homogenization. A bounded and open subset \(\Omega \) of \(\mathbb{R}^{N}\) contains a closed subset \(T_{n}\) corresponding to holes which are not necessarily distributed in a periodic way, but such that \(\chi _{T_{n}}\) converges to some \(\theta \) in the weak* topology of \(L^{\infty }(\Omega )\). The elliptic problem under consideration is now \(-\text{div}(A^{n}\text{grad}(u_{n}))=f_{n}\) in \(\Omega _{n}=\Omega \backslash T_{n}\), with homogeneous Dirichlet boundary conditions on \( \partial \Omega _{n}\). Because of these homogeneous Dirichlet boundary conditions it is obvious to extend the solution \(u_{n}\) by 0 on the holes \( T_{n}\). The first step of the convergence result consists to derive estimates on the solution and its extension. The author introduces the best Poincaré constant \(\gamma (\Omega _{n})\) in \(H_{0}^{1}(\Omega _{n})\). In some \(\varepsilon _{n}\)-parallelepiped periodic case this \(\gamma (\Omega _{n})\) is of order \(\varepsilon _{n}^{2}\). The main result of the chapter is obtained in a periodic case, the author deriving a scalar Darcy law.
Chapter 16 considers the same problem as in chapter 15 but with the homogeneous Neumann boundary conditions \((A^{n}\text{grad}(u_{n}),\nu )=0\) on \(\partial T_{n}\). The main tool of the convergence result is the construction of an extension operator \(P_{n}\) which brings a function \(v\in H^{1}(\Omega _{n})\) which vanishes on \(\partial \Omega \) into a function \( P_{n}v\in H_{0}^{1}(\Omega )\) and such that \(\int_{\Omega }|\text{grad} (P_{n}v)|^{2}dx\leq C_{\ast }\int_{\Omega _{n}}|\text{grad}(v)|^{2}dx\) for a constant \(C_{\ast }\) independent of \(v\) and of \(n\).
Chapter 17 presents a compensated compactness result in the case where a quadratic form is involved. This extends the div-curl lemma already introduced in chapter 7.
Chapter 18 addresses estimates which are necessary when considering boundary layers. The author considers the set \(\Omega =\omega \times (0,\infty )\) in \(\mathbb{R}^{N}\) and the problem \(-\text{div}(A\text{grad}(u))=f\) in \(\Omega \), with homogeneous Neumann (resp. Dirichlet) boundary conditions on \( \partial \omega \times (0,\infty )\) (resp. \(\omega \times 0\)). An exponential decay of the solution can be proved.
In chapter 19, the author considers a steady hydrodynamics problem \(-\nu \Delta u^{n}+u^{n}\times \frac{1}{\varepsilon _{n}}b(\frac{x}{\varepsilon _{n}})+\text{grad}(p_{n})=f\) , \(\text{div}(u^{n})=0\) in \(\Omega \subset \mathbb{R}^{3}\), with homogeneous Dirichlet boundary conditions on \(\partial \Omega \). The convergence result is proved building an oscillating test-function.
Chapter 20 presents the preservation of some properties through H-convergence in dimension 2.
Chapter 21 focuses on an important activity of the author dealing with bounds of the effective coefficients when considering homogenization problems. This requires the compensated compactness result already proved in chapter 17 and direct computations.
In chapter 22, the author describes some further H-convergence properties considering mixtures and functions attached to geometries. The chapter ends in the 2D case with Pick or Herglotz holomorphic functions.
In chapter 23, the author exhibits a memory effect when considering the simple evolution equation \(\frac{\partial u_{n}}{\partial t} (x,t)+a_{n}(x)u_{n}(x,t)=f(x,t)\) in \(\Omega \times (0,\infty )\), the solution starting from \(v\) at \(t=0\). The coefficients \(a_{n}\) are bounded from below and from above by positive numbers. Using the Laplace transform, the limit problem is proved to contain a further convolution term. The author here introduces the notion of Young measure.
Chapter 24 contains several generalizations of the previous chapter considering other evolution equations.
The chapter 25 describes the computations of Hashin and Shtrikman which give the effective bounds for a scalar elliptic equation. The main result proves that the bounds of chapter 21 are optimal and that they are obtained for coated spheres, with the best or worst conductor in the core.
In the quite long chapter 26, the author presents computations on effective bounds in the case of ellipsoids and spheres and for a scalar elliptic equation.
The chapter 27 presents homogenization results for quite general laminations.
In the chapter 28, the author presents some connections between the notion of propagation of singularities or of microlocal regularity, as developed by Hörmander, and that of H-measures or Young measures. This focuses on the behaviour of \(\mathcal{F}(\varphi (u_{n}))\) where \(u_{n}\) is a sequence which converges to 0 in the weak topology of \(L_{loc}^{2}(\Omega )\), \( \varphi \in C_{c}(\Omega )\) and \(\mathcal{F}\) is the Fourier transform.
Chapter 29 describes some bounds for the effective coefficients in the case where \(A^{n}\) is chosen as \(A^{n}=A+\gamma B^{n}\) with \(|\gamma |\) small enough and \(B^{n}\) converges to 0 in \(L^{\infty }(\Omega ;\mathcal{L}( \mathbb{R}^{N},\mathbb{R}^{N}))\) weak*. The author describes the structure of \(A^{eff}\) as powers of \(\gamma \) up to the order 2.
Chapter 30 puts some deeper insights on H-measures and on Young measures. The author gives some further bounds for the effective coefficients in the case of general symmetric positive matrices.
Chapter 31 applies some tools of the homogenization theory to the linear wave equation.
The long chapter 32 presents some types of H-measures. This is mainly based on results obtained by the author and P. Gerard.
The final mathematical chapter 33 describes the links between Young and H-measures.
Chapter 34 is a conclusion and, like in every chapter, the author proposes his version of creation in mathematics, the effective relationships between applied mathematics and physics. He also compares his works with ideas developed by other mathematicians.
Each chapter ends with some historical details on the mathematical authors who have been cited in the chapter.
The book is surely interesting both from the mathematical point of view (many mathematical results are presented) and for the historical background on homogenization theory, including H-convergence and Young measures. Even if it is mainly restricted to linear homogenization, many mathematicians will learn how to use the key tools and to develop the basic but deep tools which can be used within this context.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
49J45 Methods involving semicontinuity and convergence; relaxation
74Q20 Bounds on effective properties in solid mechanics
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