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This monograph surveys the theory of quantitative homogenization for second-order linear elliptic systems in divergence form with rapidly oscillating periodic coefficients in a bounded domain. It begins with a review of the classical qualitative homogenization theory, and addresses the problem of convergence rates of solutions. The main body of the monograph investigates various interior and boundary regularity estimates that are uniform in the small parameter e>0. Additional topics include convergence rates for Dirichlet eigenvalues and asymptotic expansions of fundamental solutions, Green functions, and Neumann functions.

The monograph is intended for advanced graduate students and researchers in the general areas of analysis and partial differential equations. It provides the reader with a clear and concise exposition of an important and currently active area of quantitative homogenization.



Chapter 1. Introduction

Partial differential equations and systems with rapidly oscillating coefficients are used to model various physical phenomena in inhomogeneous or heterogeneous media, such as composite and perforated materials. Let ε > 0 be a small parameter, representing the inhomogeneity scale – the scale of the microstructure of an inhomogeneous medium. The local characteristics of the medium are described by functions of the form $$\mathit{A}(\mathit{x}/\varepsilon)$$ , which vary rapidly with respect to the space variables. Since ε is much smaller than the linear size of the domain where the physical process takes place, solving the corresponding boundary value problems for the partial differential equations directly by numerical methods may be costly.

Zhongwei Shen

Chapter 2. Second-Order Elliptic Systems with Periodic Coefficients

In this monograph we shall be concerned with a family of second-order linear elliptic operators in divergence form with rapidly oscillating periodic coefficients, 2.0.1 $$\mathcal{L}_\varepsilon = - \mathrm{div}(\mathit{A}(\mathit{x}/\varepsilon)\nabla), \,\,\, \varepsilon > 0,$$ in $$\mathbb{R}^\mathit{d}$$ . The coefficient matrix (tensor) $${A}$$ in (2.0.1) is given by $$\mathit{A}(\mathit{y})\,\,= ({\mathit{a}_\mathit{ij}^{\alpha\beta}} \, (\mathit{y})), \,\,\,\mathrm{with\,\,1}\,\leq \mathit{i,j} \leq \mathit{d}\,\, \mathrm{and\,\,1}\,\leq \alpha,\beta \leq \mathit{m}.$$

Zhongwei Shen

Chapter 3. Convergence Rates, Part I

Let $$\mathcal{L}_\varepsilon\,\,=\,\,\mathrm{-div}(\mathit{A}(\mathit{x}/\varepsilon) \nabla )$$ for ε > 0, where $$\mathit{A}(\mathit{y})\,\,=\,\,(\mathit{a}_\mathit{i,j}^{\alpha\beta} \, (\mathit{y})) $$ is 1-periodic and satisfies a certain ellipticity condition. Let $$\mathcal{L}_0\,\,=\,\,\mathrm{-div}(\mathit{\hat{A}}\nabla), $$ , where $$\mathit{\hat{A}}\,\,=\,\,(\hat{\mathit{a}}_{\mathit{ij}}^{\alpha\beta}) $$ denotes the matrix of effective coefficients, given by (2.2.16).

Zhongwei Shen

Chapter 4. Interior Estimates

In this chapter we establish interior Hölder (C0,α) estimates, W1,p estimates, and Lipschitz (C0,1) estimates, that are uniform in ε > 0, for solutions of $$\mathcal{L}_\varepsilon (\mathit{u}_\varepsilon)= \mathit{F}$$ , where $$\mathcal{L}_\varepsilon = - \mathrm{div}(\mathit{A}(\mathit{x}/\varepsilon)\nabla)$$ . As a result, we obtain uniform size estimates of $$ \Gamma_\varepsilon(\mathit{x, y}), \nabla_\mathit{x}\Gamma_\varepsilon (\mathit{x,y}), \nabla_\mathit{y}\Gamma_\varepsilon (\mathit{x,y})$$ , and $$ \nabla_\mathit{x}\nabla_\mathit{y}\Gamma_\varepsilon(\mathit{x,y})$$ , where $$\Gamma_\varepsilon(\mathit{x,y})$$ denotes the matrix of fundamental solutions for $$ \mathcal{L}_\varepsilon \,\,\mathrm{in}\,\, \mathbb{R}^\mathit{d}$$ . This in turn allows us to derive asymptotic expansions, as $$ \varepsilon \rightarrow\,0,\,\, \mathrm{of} \,\,\Gamma_\varepsilon (\mathit{x,y}),\,\,\nabla_\mathit{x}\Gamma(\mathit{x,y}),\, \nabla\mathit{y}\Gamma_\varepsilon(\mathit{x,y}), \,\, \mathrm{and}\,\, \nabla\mathit{x}\nabla\mathit{y}\Gamma_\varepsilon(\mathit{x,y})$$ . Note that if $$ \mathit{u}_\varepsilon(\mathit{x})\,\,=\,\,\mathit{P}_\mathit{j}^{\beta}\,(\mathit{x})\,\,+ \,\,\varepsilon\chi_\mathit{j}^\beta\,(\mathit{x}/\varepsilon), \,\,\mathrm{then}\,\,\nabla\mathit{u}_\varepsilon\,\,=\,\,\nabla\mathit{P}_\mathit{j}^\beta\,\,+\,\,\nabla\chi_\mathit{j}^\beta\,(\mathit{x}/\varepsilon)\,\,\mathrm{and}\,\,\mathcal{L}\varepsilon(\mathit{u}_\varepsilon)\,\,=\,\,0\,\,\mathrm{in}\,\,\mathbb{R}^\mathit{d}$$ . Thus no uniform regularity beyond Lipschitz estimates should be expected (unless div(A) = 0, which would imply $$ \chi_\mathit{j}^\beta\,\,=\,\,0$$ ).

Zhongwei Shen

Chapter 5. Regularity for the Dirichlet Problem

This chapter is devoted to the study of uniform boundary regularity estimates for the Dirichlet problem 5.0.1 $$ \left\{\begin{array}{rc}{\mathcal{L}_\varepsilon(\mathit{u}_\varepsilon)\,=\,\mathit{F}} & \mathrm{in}\,\,\Omega, \\ \mathit{u}_\varepsilon\,=\,\mathit{g} & \mathrm{on}\,\,\partial\Omega, \end{array}\right.$$ where $$ \mathcal{L}_\varepsilon\,\,=\,\,\mathrm{-div}(\mathit{A}(\mathit{x}/\varepsilon)\nabla)$$ . Assuming that the coefficient matrix $$ \mathit{A}\,\,=\,\,\mathit{A}(\mathit{y}) $$ is elliptic, periodic, and belongs to VMO( $$ \mathbb{R}^\mathit{d}$$ ), we establish uniform boundary Hölder and W1, p estimates in C1 domains Ω. We also prove uniform boundary Lipschitz estimates in C1, α domains under the assumption that A is elliptic, periodic, and Hölder continuous. As in the previous chapter for interior estimates, boundary Hölder and Lipschitz estimates are proved by a compactness method. The boundaryW1, p estimates are obtained by combining the boundary Hölder estimates with the interior W1, p estimates, via the real-variable method introduced in Section 4.2.

Zhongwei Shen

Chapter 6. Regularity for the Neumann Problem

In this chapter we study uniform regularity estimates for the Neumann boundary value problem (6.0.1) $$\left\{ \begin{array}{lll}\mathcal{L_\varepsilon}(u_\varepsilon)=F & \mathrm{in}\;\Omega, \\ \left(\frac{\partial u_\varepsilon}{\partial v_\varepsilon}\right)^\alpha=g & \mathrm{on}\;\partial\Omega,\end{array}\right.$$ , where $$\mathcal{L}_\varepsilon = -\mathrm{div}\left(A\left(x/\varepsilon\right)\nabla\right)$$ and $$\frac{\partial u_\varepsilon}{\partial v_\varepsilon}$$ denotes the conormal derivative of $$\partial u_\varepsilon$$ defined by (6.0.2) $$\left(\frac{\partial u_\varepsilon}{ \partial v_\varepsilon}\right)^\alpha = n_ia_{ij}^{\alpha\beta}\left(x/\varepsilon\right)\frac{\partial u_\varepsilon^\beta}{\partial x_j}.$$

Zhongwei Shen

Chapter 7. Convergence Rates, Part II

In Chapter 3 we establish the $${O}(\sqrt{\varepsilon})$$ error estimates for some two-scale expansions in H1 and the $${O}({\varepsilon})$$ convergence rate for solutions $${u}_{\varepsilon}$$ in L2. The results are obtained without any smoothness assumption on the coefficient matrix A. In this chapter we return to the problem of convergence rates and prove various results under some additional smoothness assumptions, using uniform regularity estimates obtained in Chapters 4–6. We shall be mainly interested in the sharp $${O}({\varepsilon})$$ or near sharp rates of convergence.

Zhongwei Shen

Chapter 8. L2 Estimates in Lipschitz Domains

In this chapter we study L2 boundary value problems for $${\mathcal{L}}_{\varepsilon}({u}_{\varepsilon}) = 0$$ in a bounded Lipschitz domain Ω.

Zhongwei Shen


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