Univalent solutions of an elliptic system of partial differential equations arising in homogenization

Abstract

In this paper we prove that solutions to an elliptic system of partial diVerential equations in divergence form whose boundary values are the restriction of a diVeomorphism of de- gree one onto a convex domain in two dimensions are mappings whose diVerential has a nonnegative determinant. Under appro- priate regularity assumptions on the domain, the boundary val- ues, and the coeYcients of the elliptic system, it is shown that so- lutions are diVeomorphisms whose diVerential has a strictly pos- itive determinant. We also describe applications of our results to problems arising in homogenization.