Design beam shapers with double freeform surfaces to form a desired wavefront with prescribed illumination pattern by solving a Monge-Ampère type equation

2016 ◽  
Vol 18 (12) ◽  
pp. 125602 ◽  
Author(s):  
Shengqian Chang ◽  
Rengmao Wu ◽  
Li An ◽  
Zhenrong Zheng
Keyword(s):  
Type Equation ◽  
Freeform Surfaces ◽  
Monge Ampere

Existence and Hölder continuity to solutions of the complex Monge–Ampère-type equations with measures vanishing on pluripolar subsets

2021 ◽  
Author(s):  
Le Mau Hai ◽  
Vu Van Quan
Keyword(s):  
Pseudoconvex Domain ◽  
Hölder Continuity ◽  
Type Equation ◽  
Holder Continuity ◽  
Continuous Solutions ◽  
Hölder Continuous ◽  
Strictly Pseudoconvex Domain ◽  
Monge Ampere

In this paper, we establish existence of Hölder continuous solutions to the complex Monge–Ampère-type equation with measures vanishing on pluripolar subsets of a bounded strictly pseudoconvex domain [Formula: see text] in [Formula: see text].

On a time-depending Monge–Ampère type equation

2012 ◽  
Vol 75 (10) ◽  
pp. 4006-4013 ◽  
Author(s):  
B. Brandolini
Keyword(s):  
Type Equation ◽  
Monge Ampere

Designing two freeform surfaces with Monge–Ampère equation method for point-like sources

2018 ◽  
Author(s):  
Rengmao Wu ◽  
Zhenrong Zheng ◽  
Shengqian Chang ◽  
Zhanghao Ding
Keyword(s):  
Equation Method ◽  
Freeform Surfaces ◽  
Monge Ampere

scholarly journals A Parabolic Monge–Ampère Type Equation of Gauduchon Metrics

2017 ◽  
Vol 2019 (17) ◽  
pp. 5497-5538 ◽  
Author(s):  
Tao Zheng
Keyword(s):  
Smooth Function ◽  
Existence And Uniqueness ◽  
Type Equation ◽  
Uniqueness Of Solution ◽  
Long Time Existence ◽  
Hermitian Manifolds ◽  
Long Time ◽  
Monge Ampere ◽  
Time Existence

Abstract We prove the long time existence and uniqueness of solution to a parabolic Monge–Ampère type equation on compact Hermitian manifolds. We also show that the normalization of the solution converges to a smooth function in the smooth topology as $t$ approaches infinity which, up to scaling, is the solution to a Monge–Ampère type equation. This gives a parabolic proof of the Gauduchon conjecture based on the solution of Székelyhidi, Tosatti, and Weinkove to this conjecture.

Anisotropic Gauss curvature flows and their associated Dual Orlicz-Minkowski problems

2021 ◽  
pp. 1-15
Author(s):  
Li Chen
Keyword(s):  
Type Equation ◽  
Gauss Curvature ◽  
Curvature Flow ◽  
Strictly Convex ◽  
Convex Solution ◽  
Orlicz Minkowski Problem ◽  
Closed Hypersurfaces ◽  
Smooth Measures ◽  
Gauss Curvature Flow ◽  
Monge Ampere

In this paper we study a normalized anisotropic Gauss curvature flow of strictly convex, closed hypersurfaces in the Euclidean space. We prove that the flow exists for all time and converges smoothly to the unique, strictly convex solution of a Monge-Ampère type equation and we obtain a new existence result of solutions to the Dual Orlicz-Minkowski problem for smooth measures, especially for even smooth measures.

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