Riccardo De Arcangelis, Dipartimento di Matematica e Applicazioni "Renato Caccioppoli", University of Naples Federico II
"Relaxation of Non-Convex Pointwise Gradient Constrained Energyes"
The talk is concerned with some relaxation problems for an
integral energy of the type
when the admissible configurations are scalar-valued, and satisfy the pointwise gradient constraint for a.e. . Here is a smooth bounded open subset of , is Borel, and is a fixed subset of .
Some recent results on the integral representation problem for the lower semicontinuous envelope of are described when no convexity assumptions on and are assumed. In this case, it is proved that can be expressed on the whole as an integral with density given by the convex lower semicontinuous envelope of .
The lack of convexity of forestalls the use of the standard integral representation techniques. The novelty of the result relies in the new approach proposed in order to avoid the analysis of the measure theoretic properties of as a function of the open set .
As corollaries, some applications to differential inclusions are provided.
Some results on the same problem in the case when depends also on the space variable, and too has a true dependence on are discussed, showing that regularization processes occur, beside convexification, in the construction of the relaxed densities.
THURSDAY, February 2, 2006