Related papers: The obstacle problem for a fractional Monge--Amp\`…
We develop an existence and regularity theory for a class of degenerate one-phase free boundary problems. In this way we unify the basic theories in free boundary problems like the classical one-phase problem, the obstacle problem, or more…
In this paper, we deal with an elliptic problem with the Dirichlet boundary condition. We operate in Sobolev spaces and the main analytic tool we use is the Lax-Milgram lemma. First, we present the variational approach of the problem which…
We obtain sharp interior Schauder estimates for solutions to nonlocal Poisson problems driven by fractional powers of nondivergence form elliptic operators $(-a^{ij}(x) \partial_{ij})^s$, for $0<s<1$, in bounded domains under minimal…
In this paper we prove an existence and uniqueness result for a Monge-Amp\`{e}re type equation in the Cegrell class $\mathcal{E}_{\chi}$.
We solve the Dirichlet Problem of Monge-Amp\`ere equation near an isolate Klt singularity, which generalizes the result of Eyssidieux-Guedj-Zeriahi \cite{EGZ}, where the Monge-Amp\`ere equation is solved on singular varieties without…
Recently, the $L_p$ dual Minkowski problem for unbounded closed convex sets in a pointed closed convex cone was proposed and a weak solution to this problem was provided. In smooth setting, this problem is equivalent to solving the…
We obtain necessary optimality conditions for variational problems with a Lagrangian depending on a Caputo fractional derivative, a fractional and an indefinite integral. Main results give fractional Euler-Lagrange type equations and…
The Monge-Amp\`ere equation is a fully nonlinear partial differential equation (PDE) of fundamental importance in analysis, geometry and in the applied sciences. In this paper we solve the Dirichlet problem associated with the…
We prove optimal regularity and a detailed analysis of the free boundary of the solutions to the thin obstacle problem for nonparametric minimal surfaces with flat obstacles.
We consider degenerate Monge-Ampere equations of the type $$\det D^2 u= f \quad \{in $\Om$}, \quad \quad f \sim \, d_{\p \Om}^\alpha \quad \{near $\p \Om$,}$$ where $d_{\p \Om}$ represents the distance to the boundary of the domain $\Om$…
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…
The Guillemin boundary condition naturally appears in the study of K\"ahler geometry of toric manifolds. In the present paper, the following Guillemin boundary value problem is investigated \begin{align} \label{eq1} &\det D^2…
In this paper, we solve the Dirichlet problem for Monge-Amp\`ere type equations for $(n-1)$-plurisubharmonic functions on Hermitian manifolds.
We study the solvability and uniqueness for several degenerate Monge--Amp\`ere equations including the Monge--Amp\`ere eigenvalue problem in real Euclidean spaces that involve singular Borel measures. Our approach systematically analyzes…
In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Amp\`{e}re operator $v\rightarrow \det \mathbf{D}^2 v$. The methodology we employ relies on the…
We prove the existence and uniqueness of the solutions of some very general type of degenerate complex Monge-Amp\`ere equations. This type of equations is precisely what is needed in order to construct K\"ahler-Einstein metrics over…
We study the Parabolic complex Monge-Amp\'ere equation in a bounded strictly pseudoconvex domain in \mathbb{C}^n, with the boundary condition u=\varphi and the initial condition u=u_0. In this paper, we consider the case where \varphi is…
We introduce a novel monotone discretization method for addressing obstacle problems involving the integral fractional Laplacian with homogeneous Dirichlet boundary conditions over bounded Lipschitz domains. This problem is prevalent in…
This paper is concerned with the existence of globally smooth solutions for the second boundary value problem for Monge-Ampere equations and the application to regularity of potentials in optimal transportation. The cost functions satisfy a…
The objective of our paper is to investigate fractional elliptic equations of the form $(-\Delta)^s u=\frac{\lambda }{(a-u)^2}$ within a bounded domain $\Omega$, subject to zero Dirichlet boundary conditions. Here, $s\in(0,1)$, $\lambda>0$,…