scholarly journals Applications of the duality between the Homogeneous Complex Monge–Ampère Equation and the Hele-Shaw flow

2019 ◽  
Vol 69 (1) ◽  
pp. 1-30
Author(s):  
Julius Ross ◽  
David Witt Nyström
Keyword(s):  
Homogeneous Complex ◽  
Monge Ampere

scholarly journals Pluricomplex Green's functions and Fano manifolds

2019 ◽  
Vol Volume 3 ◽  
Author(s):  
Nicholas McCleerey ◽  
Valentino Tosatti
Keyword(s):  
Green's Functions ◽  
Einstein Metrics ◽  
Green’S Functions ◽  
Fano Manifold ◽  
Fano Manifolds ◽  
Continuity Method ◽  
Homogeneous Complex ◽  
Analytic Singularities ◽  
Kähler Einstein Metrics ◽  
Monge Ampere

We show that if a Fano manifold does not admit Kahler-Einstein metrics then the Kahler potentials along the continuity method subconverge to a function with analytic singularities along a subvariety which solves the homogeneous complex Monge-Ampere equation on its complement, confirming an expectation of Tian-Yau. Comment: EpiGA Volume 3 (2019), Article Nr. 9

scholarly journals Local regularity for concave homogeneous complex degenerate elliptic equations dominating the Monge–Ampère equation

2021 ◽  
Author(s):  
Soufian Abja ◽  
Guillaume Olive
Keyword(s):  
Elliptic Equations ◽  
Regularity Result ◽  
Nonlinear Elliptic Equations ◽  
Local Regularity ◽  
Degenerate Elliptic Equations ◽  
Nonlinear Elliptic ◽  
Homogeneous Complex ◽  
Degenerate Elliptic ◽  
Monge Ampere

AbstractIn this paper, we establish a local regularity result for $$W^{2,p}_{{\mathrm {loc}}}$$ W loc 2 , p solutions to complex degenerate nonlinear elliptic equations $$F(D^2_{\mathbb {C}}u)=f$$ F ( D C 2 u ) = f when they dominate the Monge–Ampère equation. Notably, we apply our result to the so-called k-Monge–Ampère equation.

scholarly journals Symplectic structures and symmetries of solutions of the complex Monge-Ampére equation

1998 ◽  
Vol 150 ◽  
pp. 63-83
Author(s):  
Stanley M. Einstein-Matthews
Keyword(s):  
Dirichlet Problem ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Symplectic Structures ◽  
Absolutely Continuous ◽  
Nonlinear Dirichlet Problem ◽  
Homogeneous Complex ◽  
Regularity Results ◽  
Monge Ampere

Abstract.The graphs that arise from the gradients of solutions u of the homogeneous complex Monge-Ampère equation are characterized in terms of the natural symplectic structure on the cotangent bundle. This characterization is invariant under symplectic biholomorphisms. Using the symplectic structures we construct symmetries (to be called Lempert transformations) for real valued functions u which are absolutely continuous on lines. We then use these symmetries to generate interesting solutions to the homogeneous complex Monge-Ampère equation and to transform the Poincaré-Lelong equation and the ∂-equation. An example of Lempert transform is given and the main theorem is applied to prove regularity results for exterior nonlinear Dirichlet problem for the homogeneous complex Monge-Ampère equation.

scholarly journals The Cauchy problem for the homogeneous Monge–Ampère equation, III. Lifespan

2017 ◽  
Vol 2017 (724) ◽  
Author(s):  
Yanir A. Rubinstein ◽  
Steve Zelditch
Keyword(s):  
Cauchy Problem ◽  
A Priori ◽  
Complex Domain ◽  
Regularity Conditions ◽  
Uniqueness Of Solutions ◽  
The Real ◽  
Homogeneous Complex ◽  
The Cauchy Problem ◽  
Complex Settings ◽  
Monge Ampere

AbstractWe prove several results on the lifespan, regularity, and uniqueness of solutions of the Cauchy problem for the homogeneous complex and real Monge–Ampère equations (HCMA/HRMA) under various a priori regularity conditions. We use methods of characteristics in both the real and complex settings to bound the lifespan of solutions with prescribed regularity. In the complex domain, we characterize the

scholarly journals The exterior Dirichlet problems of Monge–Ampère equations in dimension two

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Limei Dai
Keyword(s):  
Asymptotic Behavior ◽  
Unit Ball ◽  
Bounded Domain ◽  
Viscosity Solutions ◽  
Sufficient Conditions ◽  
Radial Solutions ◽  
Dirichlet Problems ◽  
Necessary And Sufficient ◽  
Monge Ampere ◽  
Prescribed Asymptotic Behavior

AbstractIn this paper, we study the Monge–Ampère equations $\det D^{2}u=f$ det D 2 u = f in dimension two with f being a perturbation of $f_{0}$ f 0 at infinity. First, we obtain the necessary and sufficient conditions for the existence of radial solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a unit ball. Then, using the Perron method, we get the existence of viscosity solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a bounded domain.

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