Elliptic systems and material interpenetration

TL;DR

This paper classifies certain second order linear elliptic systems in two variables that satisfy key theorems for harmonic mappings, showing these theorems do not extend to some classical elasticity systems.

Contribution

It provides a classification of elliptic systems where classical harmonic mapping theorems hold and demonstrates their limitations for elasticity systems.

Findings

Theorems hold for systems reducible to diagonal form with identical operators.

Theorems do not extend to Lame' elasticity systems.

Theorems do not extend to elliptic systems with slightly different operators.

Abstract

We classify the second order, linear, two by two systems for which the two fundamental theorems for planar harmonic mappings, the Rado'-Kneser-Choquet Theorem and the H. Lewy Theorem, hold. They are those which, up to a linear change of variable, can be written in diagonal form with the same operator on both diagonal blocks. In particular, we prove that the aforementioned Theorems cannot be extended to solutions of either the Lame' system of elasticity, or of elliptic systems in diagonal form, even with just slightly different operators for the two components.