Elliptic systems with measurable coefficients of the type of Lamé system in three dimensions

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 Hölder 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.

Proof of Theorem 3.1 is corrected

References in corpus (5)

• The Green function estimates for strongly elliptic systems of second order
• Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains
• Global pointwise estimates for Green's matrix of second order elliptic systems
• On the Green's matrices of strongly parabolic systems of second order
• Gaussian estimates for fundamental solutions of second order parabolic systems with time-independent coefficients

Cited by in corpus (1)

• Existence and Regularity of Weak Solutions for a Thermoelectric Model