Related papers: Monge-Amp\`ere Systems with Lagrangian Pairs
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…
The complex Monge-Amp\`ere operator has been defined for locally bounded plurisubharmonic functions by Bedford-Taylor in the 80's. This definition has been extended to compact complex manifolds, and to various classes of mildly unbounded…
We study continuity properties of generalized Monge-Amp\`ere operators for plurisubharmonic functions with analytic singularities. In particular, we prove continuity for a natural class of decreasing approximating sequences. We also prove a…
We investigate finite-dimensional constrained structured optimization problems, featuring composite objective functions and set-membership constraints. Offering an expressive yet simple language, this problem class provides a modeling…
This article is concerned with the question: For which pairs of hyperbolic Euler-Lagrange systems in the plane does there exist a rank-$1$ B\"acklund transformation relating them? We express some obstructions to such existence in terms of…
We present a complete theory of higher-order autonomous contact mechanics, which allows us to describe higher-order dynamical systems with dissipation. The essential tools for the theory are the extended higher-order tangent bundles, ${\rm…
By constructing explicit supersolutions, we obtain the optimal global H\"older regularity for several singular Monge-Amp\`ere equations on general bounded open convex domains including those related to complete affine hyperbolic spheres,…
We generalize and strenghten Ko{\l}odziej's stability theorem. In particular we give sharp stability exponent and treat the case with more singular right hand side of the Monge-Amp\`ere equation.
In the tangent bundle of $(M,g)$, it is well-known that the Monge-Amp\`ere equation $(\partial\bar\partial \sqrt\rho)^n=0$ has the asymptotic expansion $ \rho(x+iy)=\sum_{ij} g_{ij} (x) y_{i} y_{j} + O(y^4)$ near $M$. Those 4th order terms…
We use geometric methods to calculate a formula for the complex Monge-Amp\`ere measure $(dd^cV_K)^n$, for $K \Subset \RR^n \subset \CC^n$ a convex body and $V_K$ its Siciak-Zaharjuta extremal function. Bedford and Taylor had computed this…
This study examines geodesics and plurisubharmonic envelopes within the Cegrell classes on bounded hyperconvex domains in $\mathbb{C}^n$. We establish that solutions possessing comparable singularities to the complex Monge-Amp\`ere equation…
Lagrangian duality in mixed integer optimization is a useful framework for problems decomposition and for producing tight lower bounds to the optimal objective, but in contrast to the convex counterpart, it is generally unable to produce…
In this paper, by the method of moving planes, we establish the monotonicity and symmetry properties of convex solutions for Monge-Ampere systems on bounded smooth planar domains.
We study the complex Monge-Amp\`ere operator in bounded hyperconvex domains of $\C^n$. We introduce a scale of classes of weakly singular plurisubharmonic functions : these are functions of finite weighted Monge-Amp\`ere energy. They…
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 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…
In this paper we study an obstacle problem for Monge-Amp\`ere type functionals, whose Euler-Lagrange equations are a class of fourth order equations, including the affine maximal surface equations and Abreu's equation.
Let (M, \om) be a symplectic manifold. A Lagrangian fiber bundle \pi : M -> B determines a completely integrable system on M. First integrals of this system are the pull-backs of functions on the base of the bundle. We show that for each…
We describe geometrically contact Lagrangian systems under impulsive forces and constraints, as well as instantaneous nonholonomic constraints which are not uniform along the configuration space. In both situations, the vector field…
This paper concerns with the convergence analysis of a fourth order singular perturbation of the Dirichlet Monge-Amp\`ere problem in the $n$-dimensional radial symmetric case. A detailed study of the fourth order problem is presented. In…