ESAIM-Control Optimisation and Calculus of Variations
We consider the question raised in  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.
Larsen, C. J. (2000). On the Representation of Effective Energy Densities. ESAIM-Control Optimisation and Calculus of Variations, 5, 529-538. http://dx.doi.org/10.1051/cocv:2000120
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© 2000, EDP Sciences. Available on publisher's site at http://dx.doi.org/10.1051/cocv:2000120.