相关论文: On the integrability of symplectic Monge-Amp\'ere …

We study the quaternionic Monge-Amp\`ere equation on HKT manifolds admitting an HKT foliation having corank 4. We show that in this setting the quaternionic Monge-Amp\`ere equation has always a unique solution for every basic datum. This…

This paper is concerned with developing and analyzing convergent semi-Lagrangian methods for the fully nonlinear elliptic Monge-Amp\`ere equation on general triangular grids. This is done by establishing an equivalent (in the viscosity…

We consider a general class of elliptic equations on hypercomplex manifolds which includes the quaternionic Monge-Amp\`ere equation, the quaternionic Hessian equation and the Monge-Amp\`ere equation for quaternionic $(n-1)$-plurisubharmonic…

In this paper, we prove second derivative estimates together with classical solvability for the Dirichlet problem of certain Monge-Ampere type equations under sharp hypotheses. In particular we assume that the matrix function in the…

In this paper, we obtain gradient estimates and Laplacian estimates for the solution to the singular complex Monge-Amp\`ere equation by applying the integral method.

Let $\phi: \Bbb R^n \mapsto \Bbb R$ be a strictly convex and smooth function, and $\mu= \text{det}\,D^2 \phi$ be the Monge-Amp\`ere measure generated by $\phi.$ For $x\in \Bbb R^n$ and $t>0$, let $S(x,t):=\{y\in \Bbb R^n:…

We characterize non-degenerate Lagrangians of the form $ \int f(u_x, u_y, u_t) dx dy dt $ such that the corresponding Euler-Lagrange equations $ (f_{u_x})_x+ (f_{u_y})_y+ (f_{u_t})_t=0 $ are integrable by the method of hydrodynamic…

We study the complex Monge-Amp\`ere equation $(dd^c u)^n=\mu$ in a strictly pseudoconvex domain $\Omega$ with the boundary condition $u=\varphi$, where $\varphi\in C(\partial\Omega)$. We provide a non-trivial sufficient condition for…

We study four particular 3-dimensional natural Hamiltonian systems defined in conformally Euclidean spaces. We prove their superintegrability and we obtain, in the four cases, the maximal number of functionally independent integrals of…

Let $(u_j)$ be a deaceasing sequence of psh functions in the domain of definition $\cal D$ of the Monge-Amp\`ere operator on a domain $\Omega$ of $\mathbb{C}^n$ such that $u=\inf_j u_j$ is plurisubharmonic on $\Omega$. In this paper we are…

We investigate non-degenerate Lagrangians of the form $$ \int f(u_x, u_y, u_t) dx dy dt $$ such that the corresponding Euler-Lagrange equations $ (f_{u_x})_x+ (f_{u_y})_y+ (f_{u_t})_t=0 $ are integrable by the method of hydrodynamic…

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…

The partial Legendre transform of a non-linear elliptic differential equation is shown to be another non-linear elliptic differential equation. In particular, the partial Legendre transform of the Monge-Amp\`ere equation is another equation…

The Liouville type theorem on the parabolic Monge--Amp\`ere equation $-u_t\det D^2u=1$ states that any entire parabolically convex classical solution must be of form $-t+|x|^2/2$ up to a re-scaling and transformation, under additional…

This paper addresses several isotopy problems on $4$-manifolds. First, we classify the isotopy classes of embeddings of $\Sigma$ in $\Sigma\times S^2$ that are geometrically dual to $\{\mbox{pt}\}\times S^2$, where $\Sigma$ is a closed…

The Monge-Amp\`{e}re equation arises in the theory of optimal transport. When more complicated cost functions are involved in the optimal transportation problem, which are motivated e.g. from economics, the corresponding equation for the…

This paper develops a symplectic bifurcation theory for integrable systems in dimension four. We prove that if an integrable system has no hyperbolic singularities and its bifurcation diagram has no vertical tangencies, then the fibers of…

Let M be a connected, symplectic 4-manifold. A semitoric integrable system on M essentially consists of a pair of independent, real-valued, smooth functions J and H on the manifold M, for which J generates a Hamiltonian circle action under…

Let $\Omega\subset \R^n$ be a bounded convex domain and $\phi\in C(\bar\Omega)$ be a convex function such that $\phi$ is sufficiently smooth on $\partial\Omega$ and the Monge--Amp\`ere measure $\det D^2\phi$ is bounded away from zero and…

We prove that the image of an isometric embedding into ${\mathbb R}^3$ of a two dimensionnal complete Riemannian manifold $(\Sigma, g)$ without boundary is a convex surface provided both the embedding and the metric $g$ enjoy a…