Invertibility versus Lagrange equation for traction free energy-minimal deformations

Let X Y ⊂ R² be bounded Jordan domains of the same topological type and a traction free minimal mapping for the Dirichlet energy integral. It is shown that (Formula presented.), restricted to any subdomain (Formula presented.) , is injective if and only if it is harmonic in (Formula presented.) . This result appears pertinent to other energy integrals and, in greater generality, may be interpreted as saying that the interpenetration of matter (under hyperelastic deformations of thin plates) is inevitable precisely in the localities where the Lagrange equation fails.