So far, the only known sufficient condition for the existence of an extremal Teichmüller mapping with given boundary values is the frame mapping criterion. It assures, under certain conditions, the existence of an extremal mapping associated with a quadratic differential of finite norm. In this preliminary report, a sufficient condition will be given for the existence of an extremal Teichmüller mapping of the disk with infinitely many distinguished boundary points (called a polygon), if the frame mapping criterion fails.