Related papers: A gradient estimate for the Hermitian Monge-Amp\`e…
Uniform bounds are obtained using the auxiliary Monge-Amp\`ere equation method for solutions of very general classes of fully non-linear partial differential equations, assuming the existence of a ${C}$-subsolution in the sense of G.…
This is mostly an exposition, aimed to be accessible to geometers, analysts, and probabilists, of a fundamental recent theorem of R. Berman with recent developments by J. Hultgren, that asserts that the second boundary value problem for the…
We study the Dirichlet problem of a class of fully nonlinear elliptic equations on Hermitian manifolds and derive a priori $C^2$ estimates which depend on the initial data on manifolds, the admissible subsolutions and the upper bound of the…
We establish several gradient estimates for second-order divergence type parabolic and elliptic systems. The coefficients and data are assumed to be H\"older or Dini continuous in the time variable and all but one spatial variables. This…
We study the exact effect of the anisotropic convexity of domains on the boundary estimate for two Monge-Amp\`ere Equations: one is singular which is from the proper affine hyperspheres with constant mean curvature; the other is degenerate…
We determine sufficient conditions for the occurrence of a pointwise gradient estimate for the evolution operators associated to nonautonomous second order parabolic operators with (possibly) unbounded coefficients. Moreover we exhibit a…
Existence and boundary regularity away from the corners are established for two-dimensional Monge-Amp\`{e}re equations on convex polytopes with Guillemin boundary conditions. An important step is to derive an expansion in terms of functions…
We study the long-time existence and convergence of general parabolic complex Monge-Ampere type equations whose second order operator is not necessarily convex or concave in the Hessian matrix of the unknown solution.
By using the Malliavin calculus and solving a control problem, Bismut type derivative formulae are established for a class of degenerate diffusion semigroups with non-linear drifts. As applications, explicit gradient estimates and Harnack…
We consider the Dirichlet problem for the complex Monge--Amp\`ere equation on strongly pseudoconvex K\"ahler manifolds when the right-hand side is decreasing in the solution. Using flow-based arguments, we establish existence of smooth…
In recent works - both experimental and theoretical - it has been shown how to use computational geometry to efficently construct approximations to the optimal transport map between two given probability measures on Euclidean space, by…
In this paper, we are interested in studying the Dirichlet problem for the complex Monge-Amp\`ere operator. We provide necessary and sufficient conditions for the problem to have H\"older continuous solutions.
In this paper, we obtain a uniform $W^{2,\varepsilon}$-estimate of solutions to the fully nonlinear uniformly elliptic equations on Riemannian manifolds with a lower bound of sectional curvature using the ABP method.
For the Monge-Amp\`{e}re equation $\det D^2 u=1$, we find new auxiliary curvature functions which attain respective maximum on the boundary. Moreover, we obtain the upper bounded estimates for the Gauss curvature and mean curvature of the…
In this paper, we consider bounded positive solutions to the Allen-Cahn equation on complete noncompact Riemannian manifolds without boundary. We derive gradient estimates for those solutions. As an application, we get a Liouville type…
In this paper, we propose a monotone mixed finite difference scheme for solving the two-dimensional Monge-Amp\`ere equation. In order to accomplish this, we convert the Monge-Amp\`ere equation to an equivalent Hamilton-Jacobi-Bellman (HJB)…
We show that a positive Borel measure of positive finite total mass, on compact Hermitian manifolds, admits a Holder continuous quasi-plurisubharmonic solution to the Monge-Ampere equation if and only if it is dominated locally by…
A new type of gradient estimate is established for diffusion semigroups on non-compact complete Riemannian manifolds. As applications, a global Harnack inequality with power and a heat kernel estimate are derived for diffusion semigroups on…
We correct the calculation of the Monge-Amp\`ere measure of a certain extremal plurisubharmonic function for the complex Euclidean ball in C^2.
We exploit the link between the transport equation and derivatives of expectations to construct efficient pathwise gradient estimators for multivariate distributions. We focus on two main threads. First, we use null solutions of the…