Tadeusz Iwaniec and Jani Onninen
Abstract: Let $X,Y\subset \mathbb R^2$ be bounded Jordan domains of the same topological type and $h: X\to Y$ a surjective traction free minimal mapping for the Dirichlet energy integral. It is shown that the restriction of $h$ to any subdomain $O\subset X$ is injective if and only if it is harmonic in $O$. 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.
Journal: Calculus of Variations and Partial Differential Equations 52 (2015), no. 3-4, 489–496.
DOI: 10.1007/s00526-014-0719-8