# The obstacle problem for Monge Ampere equation

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We study the parabolic complex Monge-AmpÃ¨re type equations on closed Hermitian manifolds. We derive uniform $$C^\infty $$ a priori estimates for normalized solutions, and then prove the $$C^\infty $$ convergence. The result also yields a way to carry out the method of continuity for elliptic...

- A Priori Lâˆž-Estimates for Degenerate Complex Mongeâ€“AmpÃ¨re Equations. Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed // IMRN: International Mathematics Research Notices;Jan2008, Vol. 2008, p1
We study families of complex Mongeâ€“AmpÃ¨re equations, focusing on the case where the cohomology classes degenerate to a nonbig class. We establish uniform a priori Lâˆž-estimates for the normalized solutions, generalizing the recent work of S. KoÅ‚odziej and G. Tian. This has...

- Large solutions to complex Monge-AmpÃ¨re equations: Existence, uniqueness and asymptotics. Xiang, Ni; Yang, Xiaoping // Chinese Annals of Mathematics;Jul2011, Vol. 32 Issue 4, p569
The authors consider the complex Monge-AmpÃ¨re equation det $\left( {u_{i\bar j} } \right)$ = Ïˆ( z, u, âˆ‡ u) in bounded strictly pseudoconvex domains Î©, subject to the singular boundary condition u = âˆž on âˆ‚Î©. Under suitable conditions on Ïˆ, the existence, uniqueness...

- The Monge problem in $ {\mathbb{R}^d} $: Variations on a theme. Champion, T.; Pascale, L. // Journal of Mathematical Sciences;Mar2012, Vol. 181 Issue 6, p856
In a recent paper, the authors proved that, under natural assumptions on the first marginal, the Monge problem in $ {\mathbb{R}^d} $ for the cost given by a general norm admits a solution. Although the basic idea of the proof is simple, it involves some complex technical results. Here we will...

- On the regularity of the polar factorization for time dependent maps. Loeper, G. // Calculus of Variations & Partial Differential Equations;Mar2005, Vol. 22 Issue 3, p343
We consider the polar factorization of vector valued mappings, introduced in [3], in the case of a family of mappings depending on a parameter. We investigate the regularity with respect to this parameter of the terms of the polar factorization by constructing some a priori bounds. To do so, we...

- MULTI-VALUED SOLUTIONS TO A CLASS OF PARABOLIC MONGE-AMPERE EQUATIONS. LIMEI DAI // Communications on Pure & Applied Analysis;May2014, Vol. 13 Issue 3, p1061
In this paper, we investigate the multi-valued solutions of a class of parabolic Monge-AmpÃ¨re equation -ut det(DÂ²u) = f. Using the Perron method, we obtain the existence of finitely valued and infinitely valued solutions to the parabolic Monge-AmpÃ¨re equations. We generalize the results...

- ERROR ANALYSIS OF A MIXED FINITE ELEMENT METHOD FOR THE MONGE-AMPÃˆRE EQUATION. AWANOU, GERARD; HENGGUANG LI // International Journal of Numerical Analysis & Modeling;2014, Vol. 11 Issue 4, p745
We analyze the convergence of a mixed finite element method for the elliptic Monge- Amp`ere equation in dimensions 2 and 3. The unknowns in the formulation, the scalar variable and a discrete Hessian, are approximated by Lagrange finite element spaces. The method originally proposed by Lakkis...

- Quadratic mixed finite element approximations of the Monge-AmpÃ¨re equation in 2D. Awanou, Gerard // Calcolo;Dec2015, Vol. 52 Issue 4, p503
We give error estimates for a mixed finite element approximation of the two-dimensional elliptic Monge-AmpÃ¨re equation with the unknowns approximated by Lagrange finite elements of degree two. The variables in the formulation are the scalar variable and the Hessian matrix.

- Characterization of n-rectifiability in terms of Jones' square function: part I. Tolsa, Xavier // Calculus of Variations & Partial Differential Equations;Dec2015, Vol. 54 Issue 4, p3643
In this paper it is shown that if $$\mu $$ is a finite Radon measure in $${\mathbb R}^d$$ which is n-rectifiable and $$1\le p\le 2$$ , then where with the infimum taken over all the n-planes $$L\subset {\mathbb R}^d$$ . The $$\beta _{\mu ,p}^n$$ coefficients are the same as the ones considered...