An Elementary Derivation of the Generalized Kohn-Strang Relaxation Formulae

We give elementary proofs of the quasiconvex relaxation for the generalized Kohn-Strang functions [see G. Allaire and G. Francfort, Anal. Non-Lin. H. Poincare Inst. 15 (1998) 301--339; G. Allaire and V. Lods, Proc. Royal Soc. Edin. 129A (1999) 439--466] originally studied in an optimal design problem [R. V. Kohn and D. Strang, Comm. Pure Appl. Math. 39 (1986) 113--137, 139--183, 353--377]. We show that by using the translation method, we can recover the relaxations without using the homogenization method and the G-closure theory as in the papers of Allaire [loc. cit.]. Our calculations give further geometric insight of the relaxation and connections to other related areas