Gradient estimates for parabolic and elliptic systems from linear laminates

TL;DR

This paper derives gradient estimates for second-order parabolic and elliptic systems with coefficients that are H"older or Dini continuous in most variables, using a novel Campanato approach, relevant to elastic laminates and composite materials.

Contribution

It introduces new gradient estimates for systems with partial regularity assumptions, extending previous results to more complex materials models.

Findings

Established gradient bounds for systems with partial regularity.

Extended analysis to non-divergence type equations.

Applied Campanato's method in a novel way.

Abstract

We establish several gradient estimates for second-order divergence type parabolic and elliptic systems. The coefficients and data are assumed to be H\"older or Dini continuous in the time variable and all but one spatial variables. This type of systems arises from the problems of linearly elastic laminates and composite materials. For the proof, we use Campanato's approach in a novel way. Non-divergence type equations under a similar condition are also discussed.