Elliptic systems with measurable coefficients of the type of Lam\'e system in three dimensions

TL;DR

This paper investigates the regularity and fundamental solutions of a class of elliptic systems with measurable coefficients similar to the Lamé system in three dimensions, establishing Hölder continuity of solutions.

Contribution

It proves Hölder continuity of solutions under minimal conditions and explores applications, including estimates of Green's functions and heat kernels for these systems.

Findings

Weak solutions are Hölder continuous under minimal assumptions.

Provides estimates for Green's functions of the systems.

Discusses heat kernel properties related to the elliptic systems.

Abstract

We study the $3 \times 3$ elliptic systems $\nabla \times (a(x) \nabla\times u)-\nabla (b(x) \nabla \cdot u)=f$, where the coefficients $a(x)$ and $b(x)$ are positive scalar functions that are measurable and bounded away from zero and infinity. We prove that weak solutions of the above system are H\"older continuous under some minimal conditions on the inhomogeneous term $f$. We also present some applications and discuss several related topics including estimates of the Green's functions and the heat kernels of the above systems.