| الملخص: | <p>In this thesis we obtain new estimates and regularity results for some fully nonlinear elliptic equations arising in conformal geometry.</p>
<p>Our first set of new results concern local pointwise second derivative estimates for elliptic solutions to the σ<sub>k</sub>-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.</p>
<p>In Chapter 4 we study the existence of conformal metrics satisfying g<sup>-1</sup> A<sub>g</sub><sup>τ</sup> ∈ Γ<sub>2</sub><sup>+</sup>, where A<sub>g</sub><sup>τ</sup> is the trace-modified Schouten tensor. When τ = 1, this is an important question in the context of the σ<sub>2</sub>-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.</p>
<p>In Chapter 5 we obtain integral estimates for a fourth order perturbation of the (trace-modified) σ<sub>2</sub>-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. </p>
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