Complete representation of some functionals showing the Lavrentieff phenomenon

The functional F(u) = ∫Bf(x, Du) is considered, where B is the unit ball in R2, u varies in the set of the locally Lipschitz functions on R2 and f belongs to a family containing, as model case, the following integrand: The computation of the relaxed functional F̄ is provided yielding an explicit representation formula.This formula nevertheless is not integral, because F̄ is not a measure and does not coincide with the obvious extension of F over all W1,p(B).This phenomenon is essentially due to the non-standard growth behaviour of f(x, z) in the variable z.

The Lavrentieff phenomenon for quadratic functionals

The paper provides an example of an integral functional in more than two dimensions, with a symmetric and positively defined quadratic integrand q, exhibiting the Lavrentieff phenomenon on a ball B and on a linear boundary datum u0, i.e. for whichThe example is also utilised to discuss nonidentity between some relaxation procedures for a quadratic integral functional and to provide a weighted Sobolev space in the Hilbert case in which smooth functions are not dense.