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:
...are terms derived from a place outside ancient Jerusalem known in the Hebrew Bible as the Valley of the Son of Hinnom (Hebrew: גֵיא בֶן־הִנֹּם or גיא בן-הינום); one of the two principal valleys surrounding the Old City.
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:
The exact location of the Valley of Hinnom is disputed. Older commentaries give the location as below the southern wall of ancient Jerusalem, stretching from the foot of Mount Zion eastward past the Tyropoeon to the Kidron Valley. However the Tyropoeon Valley is usually no longer associated with the Valley of Hinnom because during the period of Ahaz and Manasseh, the Tyropoeon lay within the city walls and child sacrifice would have been practiced outside the walls of the city. Smith (1907), Dalman (1930), Bailey (1986) and Watson (1992) identify the Wadi er-Rababi, which fits the data of Joshua that Hinnom ran East to West and lay outside the city walls. According to Joshua, the valley began in En-rogel. If the modern Bir Ayyub is En-rogel then the Wadi er-Rababi which begins there is Hinnom.
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.