On a Non-Standard Convex Regularization and the Relaxation of Unbounded Integral Functionals of the Calculus of Variations

A new convex regularization procedure for a function, appearing in the study of some sequential relaxation processes for integral functionals of the Calculus of Variations, is introduced and studied. The connections with the usual convex, lower semicontinuous enveloping are investigated, together with examples showing that they are different. Applications to integral representation problems, and to sequential relaxation processes are also given