Inserted: 23 mar 2009
Last Updated: 18 aug 2010
Journal: Comptes Rendus Acad. Sci.
Warning : this paper will not appear in its first form (i.e. file GammaLong). After refusal by a journal, due to the new developments that had occurred in the meanwhile, perspectives had changed. A new paper, in collaboration with E. Oudet, presenting the approximation result together with numerics and applications to Steiner Problem, is in preparation. Anyway, I summarized the contents of the ancient versions in a shorter one (file GammaShort) in order to present the results, and this shorter version is published as a CRAS note.
The $M^\alpha$ energy which is usually minimized in branched transport problems among singular 1-dimensional rectifiable vector measures with prescribed divergence is approximated (and convergence is proved) by means of a sequence of elliptic energies, defined on more regular vector fields. The procedure recalls the Modica-Mortola one for approximating the perimeter, and the double-well potential is replaced by a concave power.
Keywords: Gamma-convergence, Branched transport, singular energies