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.