相关论文: Variational models for prestrained plates with Mon…
A variational model to simultaneously treat Stress-Driven Rearrangement Instabilities, such as boundary discontinuities, internal cracks, external filaments, edge delamination, wetting, and brittle fractures, is introduced. The model is…
The aim of the paper is to investigate the structure of plurifinely open sets. As an application, we will prove an equality on complex Monge-Amp\`ere measures in plurifinely open sets.
We prove that every $\mathcal{C}^1(\bar\omega)$-regular subsolution of the Monge-Amp\`ere system posed on a $2$-dimensional domain $\omega$ and with target codimension $2$, can be uniformly approximated by its exact solutions with…
We provide a formal derivation of a reduced model for twisted bilayer graphene (TBG) from Density Functional Theory. Our derivation is based on a variational approximation of the TBG Kohn-Sham Hamiltonian and asymptotic limit techniques. In…
We present an iterative approach to approximate the solution to the Dirichlet complex Monge-Amp\`ere eigenvalue problem on a bounded strictly pseudoconvex domain in $\C^n$. This approach is inspired by a similar approach initiated by F.…
In this paper, we give several new approaches to study interior estimates for a class of fourth order equations of Monge-Amp\`ere type. First, we prove interior estimates for the homogeneous equation in dimension two by using the partial…
We study degenerate complex Monge-Amp\`ere equations on a compact K\"ahler manifold $(X,\omega)$. We show that the complex Monge-Amp\`ere operator $(\omega + dd^c \cdot)^n$ is well-defined on the class ${\mathcal E}(X,\omega)$ of…
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 correspondance is established between the sigma models, the minimal surfaces and the Monge-Ampere equation. The Lax -Pairs of the minimality condition of the minimal surfaces and the Monge-Ampere equations are given. Existance of…
By developing an integral approach, we present a new method for the interior regularity of strictly convex solution of the Monge-Amp\`{e}re equation $\det D^2 u = 1$.
It has been found in numerical experiments that when one removes a sector from an elastic sheet and glues the edges of the sector back together, the resulting configuration is radially symmetric and nearly conical. We make a rigorous…
Let $X$ be a compact K\"ahler manifold and let $\mu$ be a non-pluripolar measure on $X$. We give a necessary and sufficient condition for $\mu$ so that the complex Monge-Amp\`ere equation (in a K\"ahler class in $X$) having $\mu$ as the…
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…
We classify global solutions of the Monge-Amp\`ere equation $\det D^2 u=1 $ on the first quadrant in the plane with quadratic boundary data. As an application, we obtain global $C^{2,\alpha}$ estimates for the non-degenerate Monge-Amp\`ere…
One of the central objects in the theory of optimal transport is the Brenier map: the unique monotone transformation which pushes forward an absolutely continuous probability law onto any other given law. A line of recent work has analyzed…
We study second-order PDEs in 4D for which the conformal structure defined by the characteristic variety of the equation is half-flat (self-dual or anti-self-dual) on every solution. We prove that this requirement implies the Monge-Ampere…
We construct convex functions on $\mathbb{R}^3$ and $\mathbb{R}^4$ that are smooth solutions to the Monge-Amp\`{e}re equation $\det D^2u = 1$ away from compact one-dimensional singular sets, which can be Y-shaped or form the edges of a…
In this paper, we showed that for some given suitable density and pressure, there exist infinitely many compactly supported solutions with prescribed energy profile. The proof is mainly based on the convex integration scheme. We construct…
We show a general existence theorem to the complex Monge-Amp\`ere type equation on compact K\"ahler manifolds.
Determining the electronic properties of nanoscopic, low-dimensional materials free of external influences is key to discovery and understanding of new physical phenomena. An example is the suspension of graphene, which has allowed access…