# Complex Monge-Ampère equation

## Top PDF Complex Monge-Ampère equation:

### Stochastic Analysis for the Complex Monge-Ampère Equation. (An Introduction to Krylov's Approach.)

Monge-Ampère equation, the original required long-run estimate of the derivatives of the ow an be relaxed to a mu h more less restri tive version (and in fa t an almost be an elled) thanks to the non-degenera y assumption itself. (The argument is explained in the note.) In the ase of Monge-Ampère, the equation may degenerate, but the analysis may benet from the des ription of the boundary: if the domain is stri tly pseudo- onvex, the original required long-run estimate of the derivatives of the ow an be relaxed as well (but annot be an elled); that is, stri t pseudo- onvexity plays the role of a weak non- degenera y assumption. Finally, the analysis may also benet from the Hamilton-Ja obi- Bellman formulation, i.e. from the writing of the Monge-Ampère equation as an equation deriving from a sto hasti optimization problem: the stru ture is indeed kept invariant under some transformations of the optimization parameters. As explained below, this may also help to redu e the long-run onstraint on the deriv atives of the ow.
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### Équation de Monge-Ampère complexe, métriques kählériennes de type Poincaré et instantons gravitationnels ALF

10 Proof of Theorem 9.12 W e shall use for proving Theorem 9.12 a classical method in the study of Monge- Ampère equations: the continuity method. This method was suggested by Calabi for the resolution of the complex Monge-Ampère equation on compact Kähler manifolds. Since the successful use by Yau [Yau78] of this method, it has been adapted to diﬀerent non-compact settings; let us quote here the version by Joyce [Joy00, ch. 8] for ALE manifolds, which greatly inspired ours. We also refer the reader to Tian and Yau’s seminal work [TY90, TY91], which pioneered the research on generalizing Calabi-Yau theorem to non-compact manifolds. We shall mention a result by Hein [Hei10, Prop. 4.1] too, very similar to ours if taking Hein’s parameter β = 3, but dealing with less precise asymptotics. We hence start this part by describing the method, and follow by the analytic work (in particular, a priori estimates) it requires.
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### Flots de Monge-Ampère complexes sur les variétés hermitiennes compactes

admits a continuous solution for any bounded -pseudoconvex domain . Under natu- ral growth assumptions on Â the solution is H¨older continuous for any H¨older continous boundary data Ï. Another interesting topic is the comparison between viscosity and pluripotential theory whenever the latter can be reasonably defined. A guiding principle for us is the basic ob- servation made by Eyssidieux, Guedj and Zeriahi [ EGZ11 ] that plurisubharmonic functions correspond to viscosity subsolutions to the complex Monge-Ampère equation. We prove several analogous results for general complex nonlinear operators. It has to be stressed that the notion of a supersolution, which does not appear in pluripotential theory, is a very subtle one for nonlinear elliptic PDEs and several alternative definitions are possible. We in particular compare these and introduce a notion of supersolution that unifies the previously known approaches.
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### Equations de Monge-Ampère complexes paraboliques

then (3.1.1) admits a unique smooth solution [HL10]. We want to understand the situation when u 0 is irregular. On compact K¨ ahler manifolds, the corresponding problem was considered and solved [GZ13, DL14]. By using approximations and a priori estimates, it was shown there that the Parabolic complex Monge-Amp` ere equation admits a unique solution in a sense close to the classical solution. We expect the situation is similar on a domain Ω .

### Le problème de Dirichlet pour les équations de Monge-Ampère complexes

n. When m = 1, this equation corresponds to the Poisson equation which is classical. The case m = n corresponds to the complex Monge-Amp` ere equation. The complex Hessian equation is a natural generalisation of the complex Monge- Amp` ere equation and has some geometrical applications. For examples, this equation appears in problems related to quaternionic geometry [AV10] and in the work [STW15] for solving Gauduchon’s conjecture. Its real counterpart has been developed in the works of Trudinger, Wang and others (see for example [W09]). This all gives us a strong motivation to study the existence and regularity of weak solutions to complex Hessian equations.
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### Géométrie des équations de Monge-Ampère complexes sur des variétés kähleriennes compactes

Calabi asked in [Cal57] whether one can find a K¨ ahler form ω such that Ric(ω) = η. He showed that if the answer is positive, then the solution is unique and proposed a continuity method to prove the existence. This problem, known as the Calabi conjecture, remained open for two decades. This result was finally solved by Yau in [Yau78] and is now known as the Calabi-Yau theorem. The Calabi conjecture reduces to solving a complex Monge-Amp` ere equation as we can see here below. Fix α ∈ H 1,1 (X, R) a K¨ahler class, ω a K¨ ahler form in α and η ∈ c 1 (X) a smooth form. Since Ric(ω) also represents
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### Coupled complex Monge-Ampère equations on Fano horosymmetric manifolds

In this paper we study a general system of Monge-Ampère equations which in- corporates the three generalizations of Kähler-Einstein metrics mentioned above as well as coupled Kähler-Einstein metrics (see Equation 1 below). Our main theorem (Theorem 1.2 below) is a necessary and sufficient condition for existence of solu- tions to this equation on horosymmetric manifolds, a class of manifolds introduced by the first named author in [Del18]. This class is strictly included in the class of spherical manifolds and strictly includes the class of group compactifications. We deduce this theorem as a consequence of two independent results. The first of these is that Yau’s higher order estimates along the continuity method for Monge-Ampère equations, as well as local solvability, apply to a suitable continuity path containing Equation 1 (see Theorem 1.1 below). The second of these is that the C 0 -estimates
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### Monotone and Consistent discretization of the Monge-Ampere operator

1 Introduction We introduce a new discretization of the Monge-Ampere operator, on two dimensional cartesian grids, which is consistent and preserves at the discrete level a fundamental property of the continuous operator: degenerate ellipticity. Discrete degenerate ellipticity [Obe06] implies strong guarantees for the numerical scheme: a comparison principle, convergence of discrete solutions towards the continuous one in the setting of viscosity solutions, and convergence of Euler iterative solvers for the discrete system [Obe06]. Some Degenerate Elliptic (DE) schemes for the Monge- Ampere (MA) Partial Differential Equation (PDE) already exist [FO11, Obe06], but they suffer from several flaws: they are strongly non-local, and only approximately consistent. Consistent non DE schemes such as [LR05, BN12] offer better accuracy, but require the PDE solution to be sufficiently smooth and the discrete numerical solver to be well initialized. Filtered schemes [FO13] nonlinearly combine several existing schemes, in order to cumulate their advantages (here degenerate ellipticity and consistency), or mitigate their defects. Their definition and their analysis are however complex, and their application requires to adjust several parameters. For a recent overview of the numerical approaches to solving the Monge-Ampère equation, see Glowinski, Feng and Neilan [FGN13].
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### A generalized dual maximizer for the Monge--Kantorovich transport problem

µ, p Y (π) ≤ ν} where M + X×Y denotes the non-negative Borel measures π on X × Y with norm kπk = π(X × Y ). By p X (π) ≤ µ (resp. p Y (π) ≤ ν) we mean that the projection of π onto X (resp. onto Y ) is dominated by µ (resp. ν). We denote P ε := inf {hc, πi : π ∈ Π ε (µ, ν)} . This partial transport problem has recently been studied by Caffarelli and McCann [CM06] as well as Figalli [Fig09]. In their work the emphasis is on a finer analysis of the Monge problem for the squared Euclidean distance on R n , and pertains to a fixed ε > 0. In the

### A general duality theorem for the Monge-Kantorovich transport problem

A GENERAL DUALITY THEOREM FOR THE MONGE–KANTOROVICH TRANSPORT PROBLEM MATHIAS BEIGLB ¨ OCK, CHRISTIAN LEONARD, WALTER SCHACHERMAYER Abstract. The duality theory of the Monge–Kantorovich transport problem is analyzed in a general setting. The spaces X, Y are assumed to be polish and equipped with Borel probability measures µ and ν. The transport cost function c : X × Y → [0, ∞] is assumed to be Borel. Our main result states that in this setting there is no duality gap, provided the optimal transport problem is formulated in a suitably relaxed way. The relaxed transport problem is defined as the limiting cost of the partial transport of masses 1 − ε from (X, µ) to (Y, ν), as ε > 0 tends to zero.
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### Construction of a blow-up solution for the Complex Ginzburg-Landau equation in some critical case

For our purpose, we consider CGL independently from any particular physical context and investigate it as a mathematical model in partial differential equations with p > 1. We note also that the interest on the study of singular solutions in CGL comes also from the analogies with the three-dimensional Navier-Stokes. The two equations have the same scaling properties and the same energy identity (for more details see the work of Plech´ aˇc and ˇ Sver´ ak [Pˇ S01]; the authors in this work give some evidence for the existence of a radial solution which blow up in a selfsimilar way). Their argument is based on matching a numerical solution in an inner region with an analytical solution in an outer region. In the same direction we can also cite the work of Rottsch¨ afer [Rot08] and [Rot13]. The Cauchy problem for equation (1) can be solved in a variety of spaces using the semi-group theory as in the case of the heat equation (see [Caz03, GV96, GV97]).
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### A DOUBLE LARGE DEVIATION PRINCIPLE FOR MONGE-AMPERE GRAVITATION

So we have achieved, at least at a formal level, the main goal of our paper which was, through a double application of the large deviation principle, the derivation of the (discrete) Monge-Amp`ere gravitational model from one of the simplest think- able model of particles: N independent Brownian trajectories, with an intriguing interplay between their indistinguishability (for the first application of the LDP) and their individuality (for the second application of the LDP).

### SECOND ORDER VARIATIONAL HEURISTICS FOR THE MONGE PROBLEM ON COMPACT MANIFOLDS

satisfied µ-almost everywhere yields C(φ) > C(ϕ) by using (1), which shows that ϕ solves, indeed, the Monge problem  Remark 1.1 If, in Proposition 1.2, the manifold M has no boundary and we strengthen (3b) by assuming that the map m ∈ M → c(m, p) is differ- entiable for \$-almost all p ∈ P , then either C has no stationary point or Monge’s problem is trivial. Indeed, if so, letting ϕ ∈ Diff µ,\$ (Ω) be station-

### From the Schrödinger problem to the Monge-Kantorovich problem

MONGE-KANTOROVICH PROBLEM CHRISTIAN L´ EONARD Abstract. The aim of this article is to show that the Monge-Kantorovich problem is the limit of a sequence of entropy minimization problems when a fluctuation parameter tends down to zero. We prove the convergence of the entropic values to the optimal transport cost as the fluctuations decrease to zero, and we also show that the limit points of the entropic minimizers are optimal transport plans. We investigate the dynamic versions of these problems by considering random paths and describe the connections between the dynamic and static problems. The proofs are essentially based on convex and functional analysis. We also need specific properties of Γ-convergence which we didn’t find in the literature. Hence we prove these Γ-convergence results which are interesting in their own right.
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### An Hilbertian Framework for the Time-Continuous Monge-Kantorovich Problem

Unit´e de recherche INRIA Lorraine, Technopˆole de Nancy-Brabois, Campus scientifique, ` NANCY 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LES Unit´e de recherche INRIA Rennes, Ir[r]

### ARTheque - STEF - ENS Cachan | Le rôle de l'image dans l'interprétation et la prédiction des combinaisons chimiques chez Ampère

Les particules ultimes de matière se répartissent les unes par rapport aux autres sous l'effet de forces attractives et répulsives jusqu'à ce qu'un équilibre s'établisse entre toutes les[r]

### Numerical solution of the Monge-Kantorovich problem by Picard iterations

Résolution numérique du problème de Monge-Kantorovich par des itérations de Picard Résumé : Nous présentons une méthode itérative pour résoudre numériquement le problème L 2 de Monge-Kantorovich. La méthode est basée sur des itérations de Picard du problème linéarisé. Des exemples relatifs aux transports de densités bidimensionnelles montrent que cette méthode réduit considérablement le temps de calcul par rapport aux méthodes existantes, en particulier lorsque la distance de Wasserstein entre les densités est petite.

### Quantitative linearization results for the Monge-Ampère equation

1.1 Strategy of the proofs In this section, we will explain the key ideas for the proofs of our two main theorems. The rough picture is as follows: The first step in the proof of Theorem 1.2 is the harmonic approximation result Theorem 1.4 itself. As in [26], this allows to run a Campanato-style iteration scheme which transfers the information (1.8), namely that the displacement is controlled at the macroscopic scale (here R) down to the microscopic scale (here 1), provided the data are well-behaved in the sense of (1.9). Due to the iterative application of Theorem 1.4 this results into a cumulative shift. The last ingredient for the proof of Theorem 1.2 is the fact that to leading order, this shift is equal to the flux averaged over a microscopic region of the solution of the linearized equation i.e. the Poisson equation, at the macroscopic scale.
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### SUR L'EQUATION DE SCHRÖDINGER ET L'EQUATION DE DIRAC-WEYL-FOCK.

SUR L’ ´ EQUATION DE SCHR ¨ ODINGER ET L’ ´ EQUATION DE DIRAC-WEYL-FOCK. JONOT JEAN LOUIS Abstract. We provide some results for the notion of spacetime which allows travel through parts of the universe where a chronological field exists. This foliation leads to a generalisation of the Schrodinger equation and the Dirac- Weyl-Fock equation.

### Minimal convex extensions and finite difference discretisation of the quadratic Monge–Kantorovich problem

theory of Barles and Souganidis [BS91]. This is a powerful framework which can be applied, in particular, to general degenerate elliptic non-linear second order equations. Our problem however has two specificities. First, the Monge-Ampère operator is degenerate elliptic only on the cone of convex function and this constraint must somehow be satisfied by the discretization and preserved in the convergence process. Second, the theory requires a uniqueness principle stating that viscosity sub-solutions are below super-solutions. The first problem or more generally the approximation of convex functions on a grid, has attracted a lot of attention, see [Mir16a] and the references therein. The Lattice Basis Reduction (LBR) technique in particular was applied to the MA Dirichlet problem in [BCM16]. The second issue (uniquess principle) is more delicate and, even though the BV2 reformulation in [BFO14] clearly belongs to the family of oblique boundary conditions (see [CIL92] for references) there is, to the best of our knowledge, no treatment of the specific (MA-BV2) and convexity constraint in the viscosity theory literature.
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