Abstract
We prove local pointwise second derivative estimates for positive W^{2,p} solutions to the sigma _k-Yamabe equation on Euclidean domains, addressing both the positive and negative cases. Generalisations for augmented Hessian equations are also considered.
Highlights
When k = 2, a similar divergence structure holds for general H and we obtain the following: Theorem 1.8 Let Ω be a domain in Rn (n ≥ 3), f ∈ Cl1o,c1(Ω × R × Rn) a positive function and
This prompts us to consider the divergence structure of the linearised operator, which we address in Sect. 3, and motivate the estimates established from Sect. 4 onwards
4 Main estimates we prove our main estimates, which will be used in the proof of our main results in Sect
Summary
We obtain local pointwise second derivative estimates for positive W 2,p solutions to the equations σk1/k( Au(x)) = f (x, u(x), ∇u(x)) > 0, λ( Au(x)) ∈ Γk+ for a.e. x ∈ Ω (1.1+). As far as the authors are aware, Theorem 1.2 currently provides the only available local second derivative estimate for solutions to the σk-Yamabe equation in the negative case. When k = 2, a similar divergence structure holds for general H and we obtain the following: Theorem 1.8 Let Ω be a domain in Rn (n ≥ 3), f ∈ Cl1o,c1(Ω × R × Rn) a positive function and. Our proofs of Theorems 1.5 and 1.8 use an integrability improvement argument, from which the Cl1o,c1 estimate is obtained by the Moser iteration technique. The estimate (3.12) follows from the definition of V [u] j
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