The young woman pictured here is unknown, and in fact, the artist of the work is also in question. Though distictly in the style of Jacques-Louis David, and of the appropriate age, the work can only be assumed to be from "the circle of Louis David" — meaning from the master himself, or from one of his numerous students and protégés.
Works in the public domain are those whose intellectual property rights have expired, have been forfeited, or are inapplicable. Artworks and writings enter the public domain when their copyright expires. Determining when a work's copyright expires is complex, but in general a work is in the public domain in the U.S. when it was created before 1923, and in most countries when the artist/author has been dead for 70 years. For these works, unlimited copy, distribution, modification, and reuse are allowed for free. More About Public Domain