Estimates and regularity for some fully nonlinear PDEs in conformal geometry

Thesis

In this thesis we obtain new estimates and regularity results for some fully nonlinear elliptic equations arising in conformal geometry.

Our first set of new results concern local pointwise second derivative estimates for elliptic solutions to the σ_k-Yamabe equation. In Chapter 2 we obtain such estimates for W2,p-strong solutions on Euclidean domains, addressing both the so-called positive and negative cases. We explore two methods for obtaining these estimates: an integrability improvement argument coupled with Moser iteration, and a method using the Alexandrov-Bakelman-Pucci estimate. Our estimates are obtained in the more general context of augmented Hessian equations. In Chapter 3 we obtain similar estimates for smooth solutions on manifolds when k = 2. Our work here contributes to a growing literature on the regularity theory for the σk-Yamabe equation and, from a broader perspective, the regularity theory for fully nonlinear, non-uniformly elliptic equations.

In Chapter 4 we study the existence of conformal metrics satisfying g^-1 A_g^τ ∈ Γ₂⁺, where A_g^τ is the trace-modified Schouten tensor. When τ = 1, this is an important question in the context of the σ₂-Yamabe problem, and it is also of independent geometric and topological interest when τ ≤ 1. Our focus will be on three dimensions; we prove a new existence result when τ < 1, and obtain a new proof of a result of Ge, Lin & Wang [GLW10] when τ = 1, with an eye towards tackling some related problems.

In Chapter 5 we obtain integral estimates for a fourth order perturbation of the (trace-modified) σ₂-Yamabe equation in three dimensions, in the spirit of Chang, Gursky & Yang [CGY02b]. Our study of this equation is partly motivated by the existence problems considered in Chapter 4, but also from the analytic viewpoint of using fourth order regularisations to study non-uniformly elliptic PDEs of second order.

Authors: Duncan, JAJ

• DOI: 10.5287/ora-5jm02y8q0
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: second derivative estimates; conformal; augmented Hessian equations; regularity; Yamabe; fully nonlinear
• Subjects: Geometric analysis; Differential equations, Partial; Conformal geometry