Who owns Gehenem (also known as Hell)?
Hashem is the Ribon Kol HaOlamim, the Master of all Worlds.
While in other religions the "netherworld" etc. is seen as under the control of malevolent forces, Judaism doesn't believe that there is a power/force which serves as an opponent of G-d. Evil is necessary for free will and when that need has run its course it will no longer exist. HaSatan (The adversary), the Yetzer Hara (Evil Inclination) etc are only agents G-d created to allow the possibility of choice (and our job is to ignore them), not chas v'Shalom independent forces.
Geheinom is the place where (after death) God punishes a person for violating mitzvot lo ta'aseh (prohibitions phrased in the Torah in the form of "Don't do such and such") in the world to come. The Zohar tells us that the maximum time one's soul will spend there is one year. After that, God brings one's soul to the Gan Eden (the garden of Eden), where God rewards for the mitzvot aseh that he followed (commandments phrased in the Torah in the form of "You must do such and such").
To give a metaphor, Geheinom would be like a washing machine to purify your soul before entering Gan Eden.
Your reward in Gan Eden is in accordance with the mitzvot that you did, so someone who does more mitzvot will receive more reward, and someone who does less mitzvot will be eternally jealous of someone who gets more reward for doing more mitzvot.
God is in charge of both places, and both are important parts of God's plan for rewarding people according to their actions.
Gehennam is derived from gei Ben Hinnom, the valley of the son of Hinnom. Thus:
I suppose Mr. Ben Hinnom might have some claim to it, but he is long dead. I am not sure who his descendants are, but maybe they would receive it as a sort of inheritance.
But in terms of modern Israeli land law, we could just look it up in the land records. It depends where it is:
Why is this tagged as theology, by the way?
In other religions Hell is place. In Judaism it is a spiritual process of cleansing the soul. The Alter Rebbe's Sefer Tanya sheds light on different ways a person's soul is cleansed for sins and wrong choices.
Tanya Ch. 8 mentions different methods of cleansing the soul including Gehenom Shel Sheleg (Gehenom of Snow) which a person is purified because he neglected positive commandments or did them by route with coolness".
Sins done in the "heat of passion" are cleansed in a “Gehinnom of Fire,”
 AriZal, Likutei Torah, Shmot and Zohar I, 62b; 237b; II, 150a-b.
Everything is apart of Hashem. There is nothing outside of our world that is able to defy Hashem's will. All of the spiritual and supernal beings follow direct orders from Hashem. The reason that "evil" exists as stated above is to give the Jewish people the ability to rise up and fulfill Hashem's will with "free choice" which is the purpose of creation.