Divergence form equations arising in models for inhomogeneous materials

Abstract

This paper will examine some mathematical properties and models of inhomogeneous materials. By deriving models for elastic energy and heat flow we are able to establish equations that arise in the study of divergence form uniformly elliptic partial differential equations. In the late 1950's DeGiorgi and Nash showed that weak solutions to our partial differential equation lie in the Holder class. After fixing the dimension of the space, the Holder exponent guaranteed by this work depends only on the ratio of the eigenvalues. In this paper we will look at a specific geometry and show that the Holder exponent of the actual solutions is bounded away from zero independent of the eigenvalues.