Nonautonomous functionals, borderline cases and related function classes

The class of nonautonomous functionals under study is characterized by the fact that the energy density changes its ellipticity and growth properties according to the point; some regularity results are proved for related minimisers. These results are the borderline counterpart of analogous ones previously derived for nonautonomous functionals with $(p,q)$-growth. Also, similar functionals related to Musielak–Orlicz spaces are discussed, in which basic properties like the density of smooth functions, the boundedness of maximal and integral operators, and the validity of Sobolev type inequalities are related naturally to the assumptions needed to prove the regularity of minima.