One Quick Handwriting Transcription Challenge Every Day-for the genealogist in you
Patrick Sullivan is what I see.
Patrick Sullivan is it.
A google search does indicate the existence of the surname Sullwain. Without other evidence (which you might have) I would still go with that, considering where the "I" is dotted, since the surname does exist.
My premise was that there's not enough "humps" for the name to end in "ain." The writer doesn't seem to be too concerned with neatness so the placement of the dot over the "i" doesn't concern me either.