The $\mathcal C^{2,\alpha}$ estimate of complex Monge-Ampere equation

Dinew, Sławomir; Zhang, Xi; Zhang, Xiangwen

doi:10.1512/iumj.2011.60.4444

Cited by 26 publications

(28 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recently, Tian [20] extended his method to the conic case. In [8], Dinew-Zhang-Zhang use a new method to establish the C 2,α estimate depending on the Hölder bound of ψ and the bound for the real Hessian of u. And their estimate is optimal according to the Hölder exponent.…”

Section: Introductionmentioning

confidence: 99%

$$C^{2,\alpha }$$ C 2 , α regularities and estimates for nonlinear elliptic and parabolic equations in geometry

Chu

2016

Calc. Var.

View full text Add to dashboard Cite

We give sharp C 2,α estimates for solutions of some fully nonlinear elliptic and parabolic equations in complex geometry and almost complex geometry, assuming a bound on the Laplacian of the solution. We also prove the analogous results to complex Monge-Ampère equations with conical singularities. As an application, we obtain a local estimate for Calabi-Yau equation in almost complex geometry. We also improve the C 2,α regularities and estimates for viscosity solutions to some uniformly elliptic and parabolic equations. All our results are optimal regarding the Hölder exponent.

show abstract

Section: Introductionmentioning

confidence: 99%

$$C^{2,\alpha }$$ C 2 , α regularities and estimates for nonlinear elliptic and parabolic equations in geometry

Chu

2016

Calc. Var.

View full text Add to dashboard Cite

show abstract

“…Hence this implies that V 2 C˛. Together with the uniform ellipticity, this implies that 2 C 2;˛, by the theory of complex Monge-Ampere equations [25,50]. Then rather standard Schauder theory and a bootstrapping argument imply that is smooth.…”

Section: Weak Solutions Of the Scalar Curvature Equationmentioning

confidence: 89%

Constant Scalar Curvature Equation and Regularity of Its Weak Solution

Zeng

2018

Comm Pure Appl Math

View full text Add to dashboard Cite

In this paper we study the constant scalar curvature equation (CSCK), a nonlinear fourth‐order elliptic equation, and its weak solutions on Kähler manifolds. We first define the notion of a weak solution of CSCK for an L∞ Kähler metric. The main result is to show that such a weak solution (with uniform L∞ bound) is smooth. As an application, this answers in part a conjecture of Chen regarding the regularity of K‐energy minimizers. The new technical ingredient is a W2, 2 regularity result for the Laplacian equation Δgu=f on Kähler manifolds, where the metric has only L∞ coefficients. It is well‐known that such a W2, 2 regularity (W2, p regularity for any p > 1) fails in general (except for dimension 2) for uniform elliptic equations of the form aij∂ij2u=f for aij ∊ L∞ without certain smallness assumptions on the local oscillation of aij. We observe that the Kähler condition plays an essential role in obtaining a W2, 2 regularity for elliptic equations with only L∞ elliptic coefficients on compact manifolds. © 2017 Wiley Periodicals, Inc.

show abstract

“…In the non homogeneous case, it seems that the only genuine interior Schauder estimates for (1.5), with constant depending only on the diameter of the compact subset K ⊂ Ω s and not on Ω s is to rely on the corresponding result for the complex Monge-Ampère equation in [21,Theorem 4]. See also the corresponding A.M.S.…”

Section: Appendixmentioning

confidence: 99%

Standard finite elements for the numerical resolution of the elliptic Monge–Ampère equation: Aleksandrov solutions

Awanou

2017

ESAIM: M2AN

View full text Add to dashboard Cite

Abstract. We prove a convergence result for a natural discretization of the Dirichlet problem of the elliptic Monge-Ampère equation using finite dimensional spaces of piecewise polynomial C 0 or C 1 functions. Standard discretizations of the type considered in this paper have been previous analyzed in the case the equation has a smooth solution and numerous numerical evidence of convergence were given in the case of non smooth solutions. Our convergence result is valid for non smooth solutions, is given in the setting of Aleksandrov solutions, and consists in discretizing the equation in a subdomain with the boundary data used as an approximation of the solution in the remaining part of the domain. Our result gives a theoretical validation for the use of a non monotone finite element method for the Monge-Ampère equation.

show abstract

The $\mathcal C^{2,\alpha}$ estimate of complex Monge-Ampere equation

Abstract: Abstract. In this paper, we prove a C 2,α -estimate for the solution to the complex MongeAmpère equation det(u ij ) = f with 0 < f ∈ C α , under the assumption that u ∈ C 1,β for some β < 1 which depends on n and α.

Cited by 26 publications

References 30 publications

$$C^{2,\alpha }$$ C 2 , α regularities and estimates for nonlinear elliptic and parabolic equations in geometry

$$C^{2,\alpha }$$ C 2 , α regularities and estimates for nonlinear elliptic and parabolic equations in geometry

Constant Scalar Curvature Equation and Regularity of Its Weak Solution

Standard finite elements for the numerical resolution of the elliptic Monge–Ampère equation: Aleksandrov solutions

Contact Info

Product

Resources

About