Degenerate Monge-Ampere Equations Research Articles

Monge-Ampère equations on compact Hessian manifolds

We consider degenerate Monge-Ampere equations on compact Hessian manifolds. We establish compactness properties of the set of normalized quasi-convex functions and show local and global comparison principles for twisted Monge-Ampere operators. We then use the Perron method to solve Monge-Ampere equations whose RHS involves an arbitrary probability measure, generalizing works of Cheng-Yau, Delanoe, Caffarelli-Viaclovsky and Hultgren-Onnheim. The intrinsic approach we develop should be useful in deriving similar results on mildly singular Hessian varieties, in line with the Strominger-Yau-Zaslow conjecture.

Uniqueness of Nontrivial Solutions for Degenerate Monge–Ampere Equations

Uniqueness of Nontrivial Solutions for Degenerate Monge–Ampere Equations

The degenerate Monge-Ampère equations with the Neumann condition

In this paper, we study a priori derivative estimates (up to the second order) of solutions for the Monge-Ampere equation \begin{document}$ \det D^{2}u = f(x) $\end{document} with the Neumann boundary value condition, which are independent of \begin{document}$ \inf f $\end{document} . Based on these uniform estimates, the existence and uniqueness of the global \begin{document}$ C^{1,1} $\end{document} solution to the Neumann problem of the degenerate Monge-Ampere equation are established under the assumption \begin{document}$ f^{\frac{1}{n-1}}\in C^{1,1}(\bar{\Omega}) $\end{document} .

The Dirichlet Problem on Almost Hermitian Manifolds

We prove second-order a priori estimate on the boundary for the Dirichlet problem of a class of fully nonlinear equations on compact almost Hermitian manifolds with smooth boundary. As applications, we solve the Dirichlet problem of the Monge–Ampere type equation and of the degenerate Monge–Ampere equation.

Schauder estimates for degenerate Monge–Ampère equations and smoothness of the eigenfunctions

We obtain \(C^{2,\beta }\) estimates up to the boundary for solutions to degenerate Monge–Ampere equations of the type $$\begin{aligned} \det D^2 u = f~~\text {in}~\Omega , \quad ~f\sim \text {dist}^{\alpha }(\cdot , \partial \Omega )~\text {near}~\partial \Omega ,~\alpha >0. \end{aligned}$$ As a consequence we obtain global \(C^\infty \) estimates up to the boundary for the eigenfunctions of the Monge–Ampere operator \((\det D^2 u)^{1/n}\) on smooth, bounded, uniformly convex domains in \(\mathbb R^n\).

Sobolev spaces of isometric immersions of arbitrary dimension and co-dimension

We prove the $$C^{1}_{\mathrm{{loc}}}$$ regularity and developability of $$W^{2,p}_{\mathrm{{loc}}}$$ isometric immersions of n-dimensional flat domains into $${\mathbb {R}}^{n+k}$$ where $$p\ge \min \{2k, n\}$$ . We also prove similar rigidity and regularity results for scalar functions of n variables for which the rank of the Hessian matrix is a.e. bounded by some $$k<n$$ , again assuming $$W^{2,p}_{\mathrm{{loc}}}$$ regularity for $$p\ge \min \{2k,n\}$$ . In particular, this includes results about the degenerate Monge–Ampere equation, $$\mathrm{{det}} D^2 u = 0$$ , corresponding to the case $$k=n-1$$ .

A localization theorem and boundary regularity for a class of degenerate Monge–Ampere equations

We consider degenerate Monge–Ampere equations of the typedetD2u=fin Ω,f∼d∂Ωαnear ∂Ω, where d∂Ω represents the distance to the boundary of the domain Ω and α>0 is a positive power. We obtain C2 estimates at the boundary under natural conditions on the boundary data and the right-hand side. Similar estimates in two dimensions were obtained by J.X. Hong, G. Huang and W. Wang in [3].

On degenerate Monge-Ampere equations over closed Kahler manifolds

In this short paper, we study complex Monge-Ampère equations over closed Kähler manifolds. The generalization from classic case is that the cohomology class can be no longer Kähler. The center of the argument is to derive a priori L∞ estimate for the solution. With some help from a classic result in several complex variables, we can prove the continuity of the solution for the most interesting case. Our proof makes use of pluripotential theory and is a quite direct generalization of the works of S. Kolodziej.

On the dirichlet problem for degenerate Monge-Ampère equations

We prove global second derivative estimates for the Dirichlet problem for degenerate Monge-Ampere equations which yield corresponding existence and regularity results. Our conditions are essentially optimal and our techniques, while drawing on previous investigations, are substantially new.