A. Corbo Esposito

Korn's inequality for periodic solids and convergence rate of homogenization

In a three-dimensional solid with arbitrary periodic Lipschitz perforation the Korn inequality is proved with a constant independent of the perforation size. The convergence rate of homogenization as a function of the Sobolev–Slobodetskii smoothness of data is also estimated. We improve foregoing results in elasticity dropping customary restrictions on the shape of the periodicity cell and superfluous smoothness and smallness assumptions on the external forces and traction.

Refinable functions, functionals, and iterated function systems

Invariant measures of iterated function systems and refinable functions are mathematical tools of vast usage, that, although usually considered in different contexts, are deeply linked. We outline these links, in particular with respect to existence and regularity of these objects.

Complete representation of some functionals showing the Lavrentieff phenomenon

The functional F ( u ) = ∫ B f ( x, Du ) is considered, where B is the unit ball in R 2 , u varies in the set of the locally Lipschitz functions on R 2 and f belongs to a family containing, as model case, the following integrand: The computation of the relaxed functional F is provided yielding an explicit representation formula. This formula nevertheless is not integral, because F is not a measure and does not coincide with the obvious extension of F over all W 1, p ( B ). This phenomenon is essentially due to the non-standard growth behaviour of f ( x, z ) in the variable z .

A one-dimensional variational problem with gradient constraint

In this note we give an explicit characterization of the minimum value of one-dimensional integral variational problem with gradient constraint by a positive measurable function.