Zheng, Hao2014-08-272014-08-272014-08-27http://hdl.handle.net/2097/18279Under the guidance of Dr. Ivan Blank, I study the obstacle problem with an elliptic operator in divergence form. First, I give all of the nontrivial details needed to prove a mean value theorem, which was stated by Caffarelli in the Fermi lectures in 1998. In fact, in 1963, Littman, Stampacchia, and Weinberger proved a mean value theorem for elliptic operators in divergence form with bounded measurable coefficients. The formula stated by Caffarelli is much simpler, but he did not include the proof. Second, I study the obstacle problem with an elliptic operator in divergence form. I develop all of the basic theory of existence, uniqueness, optimal regularity, and nondegeneracy of the solutions. These results allow us to begin the study of the regularity of the free boundary in the case where the coefficients are in the space of vanishing mean oscillation (VMO).en-US© the author. This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).http://rightsstatements.org/vocab/InC/1.0/Obstacle ProblemsEllipticDivergence FormDissertationMathematics (0405)