相关论文: A Green's function approach to linearized Monge-Am…
In this note, we obtain sharp bounds for the Green's function of the linearized Monge-Amp\`ere operators associated to convex functions with either Hessian determinant bounded away from zero and infinity or Monge-Amp\`ere measure satisfying…
We study the solvability of singular Abreu equations which arise in the approximation of convex functionals subject to a convexity constraint. Previous works established the solvability of their second boundary value problems either in two…
We establish global H\"older estimates for solutions to inhomogeneous linearized Monge-Amp\`ere equations in two dimensions with the right hand side being the divergence of a bounded vector field. These equations arise in the…
We study the solvability of the second boundary value problem for a class of highly singular fourth order equations of Monge-Amp\`ere type. They arise in the approximation of convex functionals subject to a convexity constraint using Abreu…
In this paper, we obtain boundary Harnack estimates and comparison theorem for nonnegative solutions to the linearized Monge-Amp\`ere equations under natural assumptions on the domain, Monge-Amp\`ere measures and boundary data. Our results…
In this paper, we establish local potential estimates and H\"older estimates for solutions of linearized Monge-Amp\`ere equations with the right-hand side being a signed measure, under suitable assumptions on the data. In particular, the…
The convexity of solutions to boundary value problems for fully nonlinear elliptic partial differential equations (such as real or complex $k$-Hessian equations) is a challenging topic. In this paper, we establish the power convexity of…
In this paper, we study interior estimates for solutions to linearized Monge-Amp\`ere equations in divergence form with drift terms and the right-hand side containing the divergence of a bounded vector field. Equations of this type appear…
By constructing appropriate smooth, possibly non-convex supersolutions, we establish sharp lower bounds near the boundary for the modulus of nontrivial solutions to singular and degenerate Monge-Amp\`ere equations of the form $\det D^2 u…
In this paper, we establish boundary H\"older gradient estimates for solutions to the linearized Monge-Amp\`ere equations with $L^{p}$ ($n<p\leq\infty$) right hand side and $C^{1,\gamma}$ boundary values under natural assumptions on the…
We consider a Monge-Amp\`ere functional and its corresponding second boundary value problem, a nonlinear fourth order PDE with two Dirichlet boundary conditions. This problem was solved by Trudinger-Wang and Le under the assumption that the…
In this paper, we investigate the interior H\"older regularity of solutions to the linearized Monge-Amp\`ere equation. In particular, we focus on the cases with singular right-hand side, which arise from the study of the semigeostrophic…
We study the non-Archimedean Monge-Amp\`ere equation on a smooth projective variety over a discretely or trivially valued field. First, we give an example of a Green's function, associated to a divisorial valuation, which is not Q-PL (i.e.…
In this paper, we introduce the pluricomplex Green function of the Monge-Amp\`{e}re equation for $(n-1)$-plurisubharmonic functions by solving the Dirichlet problem for the form type Monge-Amp\`{e}re and Hessian equations on a punctured…
We obtain boundary Holder gradient estimates and regularity for solutions to the linearized Monge-Ampere equations under natural assumptions on the domain, Monge-Ampere measures and boundary data. Our results are affine invariant analogues…
These lecture notes are concerned with the solvability of the second boundary value problem of the prescribed affine mean curvature equation and related regularity theory of the Monge-Amp\`ere and linearized Monge-Amp\`ere equations. The…
We give a new proof for the interior regularity of strictly convex solutions of the Monge-Amp\`ere equation. Our approach uses a doubling inequality for the Hessian in terms of the extrinsic distance function on the maximal Lagrangian…
Uniform $L^1$ and lower bounds are obtained for the Green's function on compact K\"ahler manifolds. Unlike in the classic theorem of Cheng-Li for Riemannian manifolds, the lower bounds do not depend directly on the Ricci curvature, but only…
We establish a Schauder-type boundary regularity result for a two-dimensional singular Monge-Amp\'ere equation on convex polytopes with Guillemin boundary conditions. This extends the previous work of Rubin and Huang to the case where the…
Let $u$ be a smooth, strictly $k$-plurisubharmonic function on a bounded domain $\Omega\in\mathbb C^n$ with $2\leq k\leq n$. The purpose of this paper is to study the regularity of solution to the linearized complex Monge-Amp\`ere and…