On the Representation of Effective Energy Densities

We consider the question raised in [1] of whether relaxed energy densities involving both bulk and surface energies can be written as a sum of two functions, one depending on the net gradient of admissible functions, and the other on net singular part. We show that, in general, they cannot. In particular, if the bulk density is quasiconvex but not convex, there exists a convex and homogeneous of degree 1 function of the jump such that there is no such representation. AMS Subject Classification. 49Q15, 49Q20, 73M25, 73V25. Received January 31, 2000. Revised August, 2000. Introduction This paper was motivated by a conjecture in [1] concerning the representation of effective energy densities. Though the statement of the conjecture is limited to densities corresponding to “structured deformations”, it raises an interesting question about all densities that result from relaxation in BV , as we explain below. The question can be described as follows: suppose we have an energy