Josephson coupling between a disk of one superconductor and a surrounding superconducting film of different symmetry

A cylindrical Josephson junction with a spatially dependent Josephson coupling averaging to zero over the length of the junction is studied as a model for the physics of a disk of a d-wave superconductor embedded in a superconducting film of a different symmetry. It is found that the system always introduces Josephson vortices in order to gain energy at the junction. The critical current is calculated. It is argued that a recent experiment that was claimed to provide evidence for s-wave superconductivity in ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_7$ may also be consistent with d-wave superconductivity.