## ESAIM: Control, Optimisation and Calculus of Variations

### On the Representation of Effective Energy Densities

Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, U.S.A.; cjlarsen@wpi.edu.

Abstract

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.

(Online publication August 15 2002)

Key Words:

• Relaxation;
• quasiconvexity;
• integral representation.

Mathematics Subject Classification:

• 49Q15;
• 49Q20;
• 73M25;
• 73V25