Stochastic homogenization of nonconvex integral functionals with non-standard convex growth conditions

One of the main unsolved problems in the field of homogenization of multiple integrals concerns integrands which are not bounded polynomially from above. This is typically the case when incompressible (or quasi-incompressible) materials are considered, although this is still currently a major open problem.

In this talk I will present recent progress on the stochastic homogenization of nonconvex integral functionals in view of the derivation of nonlinear elasticity from polymer physics, and consider integrands which satisfy very mild convex growth conditions from above.

I will first treat convex integrands and prove homogenization by combining approximation arguments in physical space with the Fenchel duality theory in probability. In a second part I will generalize this homogenization result to the case of nonconvex integrands which can be written in the form of a convex part (with mild growth condition from above) and a nonconvex part (that satisfies a standard polynomial growth condition). This decomposition is particularly relevant for the derivation of nonlinear elasticity from polymer physics.

This is joint work with Mitia Duerinckx (ULB).