Marco Barchiesi

Stability of the Steiner symmetrization of convex sets

The isoperimetric inequality for Steiner symmetrization of any codimension is investigated and the equality cases are characterized. Moreover, a quantitative version of this inequality is proven for convex sets.

Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality

We provide a full quantitative version of the Gaussian isoperimetric inequality: the difference between the Gaussian perimeter of a given set and a half-space with the same mass controls the gap between the norms of the corresponding barycenters. In particular, it controls the Gaussian measure of the symmetric difference between the set and the half-space oriented so to have the barycenter in the same direction of the set. Our estimate is independent of the dimension, sharp on the decay rate with respect to the gap and with optimal dependence on the mass.

A Bridging Mechanism in the Homogenization of Brittle Composites with Soft Inclusions

We provide a homogenization result for the energy-functional associated with a purely brittle composite whose microstructure is characterized by soft periodic inclusions embedded in a stiffer matrix. We show that the two constituents as above can be suitably arranged on a microscopic scale

Robustness of the Gaussian concentration inequality and the Brunn–Minkowski inequality

\varepsilon

Toughening by Crack Deflection in the Homogenization of Brittle Composites with Soft Inclusions

to obtain, in the limit as

MULTISCALE HOMOGENIZATION OF CONVEX FUNCTIONALS WITH DISCONTINUOUS INTEGRAND

\varepsilon

Local Invertibility in Sobolev Spaces with Applications to Nematic Elastomers and Magnetoelasticity

tends to zero, a homogeneous macroscopic energy-functional explicitly depending on the opening of the crack.

Loss of polyconvexity by homogenization: a new example

We provide a sharp quantitative version of the Gaussian concentration inequality: for every

Symmetry of minimizers of a Gaussian isoperimetric problem

r>0