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Recent developments on elliptic equations from composites

Hongjie Dong, Zhuolun Yang

Abstract

When inclusions in a composite are separated by a very small gap, high contrast between the inclusion and matrix properties can induce strong amplification of the underlying field inside the narrow region. Quantifying this field concentration phenomenon is important both for the theory of composite materials and for practical applications. This survey reviews substantial progress over the past three decades. In particular, we survey a set of elliptic equations and systems for which optimal estimates or sharp asymptotic characterizations have been obtained, and we highlight several interesting open questions.

Recent developments on elliptic equations from composites

Abstract

When inclusions in a composite are separated by a very small gap, high contrast between the inclusion and matrix properties can induce strong amplification of the underlying field inside the narrow region. Quantifying this field concentration phenomenon is important both for the theory of composite materials and for practical applications. This survey reviews substantial progress over the past three decades. In particular, we survey a set of elliptic equations and systems for which optimal estimates or sharp asymptotic characterizations have been obtained, and we highlight several interesting open questions.

Paper Structure

This paper contains 12 sections, 18 theorems, 87 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Perfect conductivity problem, elasticity with hard inclusions
  4. Linear perfect conductivity problem
  5. Lamé system with hard inclusions
  6. Perfect conductivity problem with -Laplacian
  7. Insulated conductivity problem
  8. Linear insulated conductivity problem
  9. Insulated conductivity problem with -Laplacian
  10. Conductors with different signs
  11. Conductivity problem with imperfect bonding interfaces
  12. Concluding remarks and open problems

Key Result

Theorem 2.1

Let $R_{j}$, $j=1,2$, be the reflection with respect to $\partial{D}_{j}$, $p_{1}\in{D}_{1}$ and $p_{2}\in{D}_{2}$ be the unique fixed points of $R_{1}R_{2}$ and $R_{2}R_{1}$ respectively, $\vec{n}$ be the unit vector in the direction of $p_{2}-p_{1}$, and $p$ be the middle point of the shortest lin $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Domain $\Omega$
  • Figure 2: The narrow gap between two inclusions.
  • Figure 3: Membrane of core-shell structure

Theorems & Definitions (18)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 3.1
  • Theorem 3.2
  • Proposition 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Theorem 3.7
  • ...and 8 more