Related papers: The pluripotential Cauchy-Dirichlet problem for co…
Extending a recent theory developed on compact K\"ahler manifolds by Guedj-Lu-Zeriahi and the author, we define and study pluripotential solutions to degenerate parabolic complex Monge-Amp\`ere equations on compact Hermitian manifolds.…
We study pluripotential complex Monge-Amp\`ere flows in big cohomology classes on compact K{\"a}hler manifolds. We use the Perron method, considering pluripotential subsolutions to the Cauchy problem. We prove that, under natural…
We show that the pluripotential Cauchy-Dirichlet problem for the complex Monge-Amp\`ere flow is solvable for the right-hand side of the form $dt \wedge d\mu$ where $d\mu$ is dominated by a Monge-Amp\`ere measure of a bounded…
In this paper, we study the Cauchy-Dirichlet problem for Parabolic complex Monge-Amp\`ere equations on a strongly pseudoconvex domain by the viscosity method. We extend the results in [EGZ15b] on the existence of solution and the…
We develop a parabolic pluripotential theory on compact K{\"a}hler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge-Amp{\`e}re equations. We provide a parabolic analogue of the celebrated Bedford-Taylor…
In this paper, the author studies quaternionic Monge-Amp\`ere equations and obtain the existence of the solutions to the Dirichlet problem for such equations in strictly pesudoconvex domains in quaternionic space. The stability and…
The main result of this paper is the existence and uniqueness of solution of the Dirichlet problem for quaternionic Monge-Ampere equations in quaternionic strictly pseudoconvex bounded domains in H^n. We continue the study of the theory of…
We study weak quasi-plurisubharmonic solutions to the Dirichlet problem for the complex Monge-Am\`ere equation on a general Hermitian manifold with non-empty boundary. We prove optimal subsolution theorems: for bounded and H\"older…
In this paper, we study the Cauchy-Dirichlet problem for Parabolic complex Monge-Amp\`ere equations on strongly pseudoconvex domains using the viscosity method. We prove a comparison principle for Parabolic complex Monge-Amp\`ere equations…
We prove a comparison principle for the pluripotential complex Monge-Amp\`ere flows for the right-hand side of the form $dt \wedge d\mu$ where $d\mu$ is dominated by a Monge-Amp\`ere measure of a bounded plurisubharmonic function. As a…
Studying the (long-term) behavior of the K\"ahler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Amp\'ere equations. The purpose of this article, the first of a…
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…
We show the existence of a bounded solution to the Cauchy problem for the complex Monge-Amp\`ere flow on a compact K\"ahler manifold, with the right-hand side of the form $dt \wedge d\mu$ where $d\mu$ is dominated by a Monge-Amp\`ere…
We study the Dirichlet problem for the complex Monge-Amp\`ere equation on a strictly pseudoconvex domain in Cn or a Hermitian manifold. Under the condition that the right-hand side lies in Lp function and the boundary data are H\"older…
Studying the (long-term) behavior of the K\"ahler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Amp\'ere equations. The purpose of this article, the second of a…
We survey the Dirichlet problem for the complex Homogeneous Monge-Amp\`ere Equation, both in the case of domains in $\mathbb C^n$ and the case of compact K\"ahler manifolds parametrized by a Riemann surface with boundary. We then give a…
The Monge-Amp\`ere type equations over bounded convex domains arise in a host of geometric applications. In this paper, we focus on the Dirichlet problem for a class of Monge-Amp\`ere type equations, which can be degenerate or singular near…
In this paper, we study the Dirichlet problem for Monge-Amp\`ere type equations for $p$-plurisubharmonic functions on Riemannian manifolds. The $a$ $priori$ estimates up to the second order derivatives of solutions are established. The…
We investigate the Cauchy-Dirichlet problem for linear parabolic equations in divergence form. Under mild assumptions on the source term and the domain, we prove the existence of globally H\"{o}lder continuous solutions. Notably, our…
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…