XU-DISSERTATION-2022.pdf (438.07 kB)

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

thesis

posted on 2022-08-01, 00:00 authored by Jinhua XuIn [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 .

## History

## Advisor

Ross, Julius## Chair

Ross, Julius## Department

Mathematics , Statistics, and Computer Science## Degree Grantor

University of Illinois at Chicago## Degree Level

- Doctoral

## Degree name

PhD, Doctor of Philosophy## Committee Member

Sparber, Christof Dumas, David Van Limbeek, Wouter Keller, Julien## Submitted date

August 2022## Thesis type

application/pdf## Language

- en