Elliptic systems and material interpenetration (Q1032736)

The authors deal with generalizations of two classical results from the theory of planar harmonic mappings, the Radó-Kneser-Choquet theorem and the Lewy theorem, see [ \textit{P. Duren,} Harmonic mappings in the plane. Cambridge Tracts in Mathematics 156. Cambridge: Cambridge University Press (2004; Zbl 1055.31001)], for mappings whose components satisfy a linear elliptic equation. These results are motivated, besides the theoretical background of planar harmonic mappings, also by requirements imposed by certain applications in physics, e.g., no interpenetration of matter is permitted. In the main results, a characterization of the second order linear two by two systems for which these two theorems hold is given. They are the systems which can be written in diagonal form with the same operator on both diagonal blocks. Moreover, the authors show by an example that the Lamé system of isotropic, linearized elasticity in the plane, with constant Lamé coefficients, may lead to physically unacceptable solutions, because interpenetration of matter occurs.