Minimal surface like equations and strain limmiting models in continuum mechanics

Elastic solids with strain-limiting response to external loading represent an interesting class of material models, capable of describing stress concentration at strains with small magnitude. A theoretical justification of this class of models comes naturally from implicit constitutive theory.
We investigate mathematical properties of static deformations for such strain-limiting nonlinear models. We obtain results concerning existence, uniqueness and regularity of weak solutions, and existence of renormalized solutions for the full range of the positive scalar parameter featuring in the model. Moreover, we show that in special geometries the problem can be reduced to the minimal surface-like equations and we can obtain the existence result on nonconvex domains.