Gradient type estimates for linear elliptic systems from composite materials

TL;DR

This paper develops gradient estimates for solutions to linear elliptic systems in composite materials with discontinuous coefficients, establishing local Hölder continuity of the gradient and a related function.

Contribution

It introduces a new function related to the gradient that allows for local Hölder continuity estimates in composite materials with discontinuous coefficients.

Findings

Gradient of solutions can be estimated by the derived function.

Proves local piecewise gradient Hölder continuity.

Establishes a new approach for gradient estimates in composite materials.

Abstract

In this paper, we consider linear elliptic systems from composite materials where the coefficients depend on the shape and might have the discontinuity between the subregions. We derive a function which is related to the gradient of the weak solutions and which is not only locally piecewise H\"{o}lder continuous but locally H\"{o}lder continuous. The gradient of the weak solutions can be estimated by this derived function and we also prove local piecewise gradient H\"{o}lder continuity which was obtained by the previous results.