On the Ambrosio-Tortorelli approximation of the Mumford Shah functional for S^1 valued maps

Abstract:

We consider the classical Ambrosio-Tortorelli functional $AT_eps(u,v)$ for maps u with values in $S^1$, and $v$ scalar.

The $Gamma$-convergence of this as $epsrightarrow 0$ is then investigated and it is proved that the limit functional

depends on the jump set of a lifting of the limiting map $u$. This analysis is related with an optimal transport problem

and with the analysis of Ginzburg-Landau type energies.