scholarly journals The Nirenberg problem of prescribed Gauss curvature on $S^2$

2021 ◽  
Vol 96 (2) ◽  
pp. 215-274
Author(s):  
Michael Anderson
Keyword(s):  
Gauss Curvature ◽  
Nirenberg Problem

scholarly journals Existence results for the fractional Nirenberg problem

2016 ◽  
Vol 270 (11) ◽  
pp. 4043-4086 ◽  
Author(s):  
Yan-Hong Chen ◽  
Chungen Liu ◽  
Youquan Zheng
Keyword(s):  
Existence Results ◽  
Nirenberg Problem

scholarly journals Peak solutions for the fractional Nirenberg problem

2015 ◽  
Vol 122 ◽  
pp. 100-124 ◽  
Author(s):  
Yan-Hong Chen ◽  
Youquan Zheng
Keyword(s):  
Nirenberg Problem

Regularity of almost extremal solutions of Monge–Ampère equations

1991 ◽  
Vol 117 (1-2) ◽  
pp. 21-29 ◽  
Author(s):  
John I. E. Urbas
Keyword(s):  
Large Class ◽  
Convex Domain ◽  
Gauss Curvature ◽  
Extremal Solutions ◽  
Classical Solutions ◽  
Uniformly Convex ◽  
Suitable Sense ◽  
Monge Ampere

SynopsisWe show that for a large class of Monge-Ampère equations, generalised solutions on a uniformly convex domain Ω⊂ℝn are classical solutions on any pre-assigned subdomain Ω′⋐Ω, provided the solution is almost extremal in a suitable sense. Alternatively, classical regularity holds on subdomains of Ω which are sufficiently distant from ∂Ω. We also show that classical regularity may fail to hold near ∂Ω in the nonextremal case. The main example of the class of equations considered is the equation of prescribed Gauss curvature.

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