Who will inherit?
Rashi: Yisrael will inherit the remnant of Edom and all Nochrim.
Radak: The verse should be re-arranged. In order that will inherit, the one on whom My name is called, the remnant of Edom. "Lema'an" explains why I will build them like of old - in order that they inherit their neighboring enemies - Edom, Pelishtim, Amon and Mo'av. "V'Chol ha'Goyim" refers to them. It says "ki'Ymei Olam" (11) because David set appointees in Edom, and made the rest submissive to him. So it will be in the future. Yisrael will inherit those nations, like it says in Ovadyah "v'Yarshu ha'Negev Es Har Esav" (1:19) and in Yeshayah "Edom u'Mo'av Mishlo'ach Yadam" (11:4).
Malbim: Melech ha'Mashi'ach.
Here it says "She'eris Edom." It says in Ovadyah "v'Lo Yihyeh Sarid l'Veis Esav" (1:18)!
Radak: Esav will not have a remnant of people, for all of them will perish in the days of Mashi'ach. Yisrael will inherit their land, and dwell in it, but not all of it - only the remnant that will not be permanent ruin. Part will be a permanent ruin, for a sign for generations that they did evil to Yisrael, and they are their brothers!
What is the significance of "Asher Nikra Shmi Aleihem"?
Rashi: Yisrael will inherit, for My name is called on Yisrael. "Asher" is like Ki (because).
Malbim: Most of Edom will fall in the war of Gog and Magog. The remnant that will remain of them and the Goyim are those who recognize Hashem and accept the true Emunah, to the point that Hashem's name is called on them.
Why does it say "Ne'um Hashem Oseh Zos"?
Radak: I will do the decree that I said, for I have the ability to do it.
