In this paper, we study the obstacle problem with obstacles whose Laplacians are not necessarily Hölder continuous. We show that the free boundary at a regular point is C 1 if the Laplacian of the obstacle is negative and Dini continuous. We also show that this condition is sharp by giving a method to construct a counter-example when we weaken the requirement on the Laplacian of the obstacle by allowing it to have any modulus of continuity which is not Dini. In the course of proving optimal regularity we also improve some of the perturbation theory due to Caffarelli (1981). Since our methods depend on comparison principles and regularity theory, and not on linearity, our stability results apply to a large class of obstacle problems with nonlinear elliptic operators.In the case of obstacles where the Laplacian is negative and has sufficiently small oscillation, we establish measure-theoretic analogues of the alternative proven by Caffarelli (1977). Specifically, if the Laplacian is continuous, then at a free boundary point either the contact set has density zero, or the free boundary is a Reifenberg vanishing set and the contact set has density equal to one half in a neighborhood of the point. If the Laplacian is not necessarily continuous, but has sufficiently small oscillation, then at a free boundary point either the contact set has density close to zero, or the free boundary is a δ-Reifenberg set and the contact set has density close to one half in a neighborhood of the point.
In 1963, Littman, Stampacchia, and Weinberger proved a mean value theorem for elliptic operators in divergence form with bounded measurable coefficients. In the Fermi lectures in 1998, Caffarelli stated a much simpler mean value theorem for the same situation, but did not include the details of the proof. We show all of the nontrivial details needed to prove the formula stated by Caffarelli, and in the course of showing these details we establish some of the basic facts about the obstacle problem for general elliptic divergence form operators, in particular, we show a basic quadratic nondegeneracy property.
We study the obstacle problem with an elliptic operator in nondivergence form with principal coefficients in VMO. We develop all of the basic theory of existence, uniqueness, optimal regularity, and nondegeneracy of the solutions. These results, in turn, allow us to begin the study of the regularity of the free boundary, and we show existence of blowup limits, a basic measure stability result, and a measure-theoretic version of the Caffarelli alternative proven in [C1].
We develop some of the basic theory for the obstacle problem on Riemannian Manifolds, and we use it to establish a mean value theorem. Our mean value theorem works for a very wide class of Riemannian manifolds and has no weights at all within the integral.
We show a method to eliminate a type of mixed asymptotics in certain free boundary problems, and give two examples of its application. It appears that these problems cannot be handled by the monotonicity formula of Alt et al.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.