An Obstacle Problem and its Relation to the Complex Monge-Ampere Equation

In [Don07], S. K. Donaldson discussed the three equivalent formulations of the same PDE problem associated to a compact Riemannian manifold X, assuming suitable regularity for the solutions. One of these is the following: Given positive smooth functions ϕ 0 , ϕ 1 , with 1 − ∆ X ϕ i > 0 where ∆ X is the Riemannian Laplacian on X, define a function L on X × R by L(x, y) = max( ϕ 0 (x) − ϕ 1 (x) + y, 0). We seek a C 1 function U(x, y) on X × R with U ≥ L everywhere and satisfying the equation ∆ ϵ U : = −ϵ∂ y 2 U + ∆ X U = ∆ X ϕ 0 − 1 on the open set Ω where U > L. We use the sign convention for ∆ X so when ϵ = 1 our ∆ ϵ is the standard Laplace operator on X × R. In this paper we will cast this problem in a variational setting and use this to give a proof of existence and uniqueness of the solutions to this equation. Under the assumption that the contact set { u = L } has two disjoint components Γ 0 := { u = 0 } and Γ 1 := { u = ϕ 0 (x) − ϕ 1 (x) + y } , we prove that the solution to this equation is C loc 1,1 .