Calculus of Variations and Geometric Measure Theory

A. Braides - M. Briane - J. Casado-Diaz

Homogenization of non-uniformly bounded periodic diffusion energies in dimension two

created by braidesa on 26 Oct 2008
modified on 27 Oct 2009


Published Paper

Inserted: 26 oct 2008
Last Updated: 27 oct 2009

Journal: Nonlinearity
Volume: 22
Number: 6
Pages: 1459-1480
Year: 2009


This paper deals with the homogenization of two-dimensional oscillating convex functionals, the densities of which are equicoercive but not uniformly bounded from above. Using a uniform-convergence result for the minimizer, which holds for this type of scalar problems in dimension two, we prove in particular that the limit energy is local and recover the validity of the analog of the well-known periodic homogenization formula in this degenerate case. However, in the present context the classical argument leading to integral representation based on the use of cut-off functions is useless due to the unboundedness of the densities. In its place we build sequences with bounded energy, which converge uniformly to piecewise-affine functions, taking pointwise extrema of recovery sequences for affine functions

Keywords: Homogenization, Gamma-convergence, local functionals, degenerate growth