Linear precoding exploits the spatial degrees of freedom offered by multi-antenna transmitters to serve multiple users over the same frequency resources. The present work focuses on simultaneously serving multiple groups of users, each with its own channel, by transmitting a stream of common symbols to each group. This scenario is known as physical layer multicasting to multiple co-channel groups. Extending the current state of the art in multigroup multicasting, the practical constraint of a maximum permitted power level radiated by each antenna is tackled herein. The considered per antenna power constrained system is optimized in a maximum fairness sense. In other words, the optimization aims at favoring the worst user by maximizing the minimum rate. This Max-Min Fair criterion is imperative in multicast systems, where the performance of all the receivers listening to the same multicast is dictated by the worst rate in the group. An analytic framework to tackle the Max-Min Fair multigroup multicasting scenario under per antenna power constraints is therefore derived. Numerical results display the accuracy of the proposed solution and provide insights to the performance of a per antenna power constrained system.