Detection of cavities in a nonlinear model arising from cardiac electrophysiology via $Gamma$-convergence

In this paper we deal with the problem of determining perfectly insulating regions (cavities) from boundary measurements in a nonlinear elliptic equation arising from cardiac electrophysiology. With minimal regularity assumptions on the cavities, we first show well-posedness of the direct problem and then prove uniqueness of the inverse problem. Finally, we propose a new reconstruction algorithm by means of a phase-field approach rigorously justified via $Gamma$-convergence.