scholarly journals The eigenvalue problem for the Monge-Ampère operator on general bounded convex domains

2018 ◽  
pp. 1519-1559
Author(s):  
Nam Q. Le
Keyword(s):  
Eigenvalue Problem ◽  
Convex Domains ◽  
Monge Ampere ◽  
Bounded Convex

scholarly journals Optimal boundary regularity for some singular Monge-Ampère equations on bounded convex domains

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Nam Q. Le
Keyword(s):  
Boundary Data ◽  
Boundary Regularity ◽  
Hölder Regularity ◽  
Convex Domains ◽  
Open Convex ◽  
Optimal Boundary ◽  
Monge Ampere ◽  
Holder Regularity ◽  
Bounded Convex

<p style='text-indent:20px;'>By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as <inline-formula><tex-math id="M1">\begin{document}$ \det D^2 u = |u|^{-n-2-k} (x\cdot Du -u)^{-k} $\end{document}</tex-math></inline-formula> with zero boundary data, have unexpected degenerate nature.</p>

scholarly journals The Lindelöf principle and angular derivatives in convex domains of finite type

2002 ◽  
Vol 73 (2) ◽  
pp. 221-250 ◽  
Author(s):  
Marco Abate ◽  
Roberto Tauraso
Keyword(s):  
Finite Type ◽  
Boundary Behaviour ◽  
Convex Domains ◽  
Kobayashi Distance ◽  
Angular Derivatives ◽  
Circular Domains ◽  
Lindelöf Principle ◽  
Real Analytic ◽  
Bounded Convex ◽  
Carathéodory Theorem

AbstractWe describe a generalization of the classical Julia-Wolff-Carathéodory theorem to a large class of bounded convex domains of finite type, including convex circular domains and convex domains with real analytic boundary. The main tools used in the proofs are several explicit estimates on the boundary behaviour of Kobayashi distance and metric, and a new Lindelöf principle.

Nontrivial solutions for Monge-Ampère type operators in convex domains

1993 ◽  
Vol 79 (1) ◽  
pp. 13-26
Author(s):  
Franchi Bruno ◽  
Kutev Nicolai ◽  
Polidoro Sergio
Keyword(s):  
Nontrivial Solutions ◽  
Convex Domains ◽  
Monge Ampere
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