arXiv research

Paper finds periodic orbits for convex Lagrangian systems on noncompact manifolds.

We construct singular solutions to special Lagrangian equa- tions with subcritical phases and minimal surface systems. A priori estimate breaking families of smooth solutions are also produced cor- respondingly. A priori estimates for special Lagrangian equations with certain convexity are largely known by now.

We prove that for a uniformly convex Lagrangian system L on a compact manifold M, almost all energy levels contain a periodic orbit. We also prove that below Ma ne's critical value of the lift of the Lagrangian to the universal cover, almost all energy levels have conjugate points. We prove that if the energy level [E=…

Generic potential primes have no self-intersections or intersections.

We make use of a symmetry reduction technique called Routh reduction to show that the solutions of the Euler-Lagrange equations of a strongly convex autonomous Lagrangian which lie on a specific energy level can be thought of as geodesics of an associated Finsler function.

Proves flows of two-convex Lagrangians are regular, global, and converge.

Equivalence of convex optimization, saddle-point problems, and variational inequalities is a well-established concept. The variational inequality (VI) is a static problem which is studied under dynamical settings using a framework called the projected dynamical system, whose stationary points coincide with the static s…

Ancient Lagrangian flows get limited convex solutions.

New findings on convexity of special Lagrangian geodesics.

Convex solutions to a specific equation are smooth when the phase is smooth enough.

Estimates for special Lagrangian curvature equations in critical and convex cases.

Recent results in Compressive Sensing have shown that, under certain conditions, the solution to an underdetermined system of linear equations with sparsity-based regularization can be accurately recovered by solving convex relaxations of the original problem. In this work, we present a novel primal-dual analysis on a …

Proves smoothness and estimates for special Lagrangian solutions with semi-convexity.

Solves a specific Dirichlet problem for Lagrangian mean curvature equations.

The paper proves optimal smoothness for certain Lagrangian graphs with specific Hölder continuity.

Real analytic solutions found for special Lagrangian equation.

Classifies regularity for Lagrangian mean curvature type equations.

This article studies the mean curvature flow of Lagrangian submanifolds. In particular, we prove the following global existence and convergence theorem: if the potential function of a Lagrangian graph in T^{2n} is convex, then the flow exists for all time and converges smoothly to a flat Lagrangian submanifold.

We derive a Bernstein type result for the special Lagrangian equation, namely, any global convex solution must be quadratic. In terms of minimal surfaces, the result says that any global minimal Lagrangian graph with convex potential must be a hyper-plane.

The paper proves a Liouville theorem for special Lagrangian equations with convexity conditions.

Variational reduction simplifies Lagrangian systems with scaling symmetries.

We find all extremal Lagrangian tori in symplectic unit balls and some toric domains.

Paper develops reduction theory for controlled Lagrangian systems with symmetry and momentum map.

Ricci curvature links volume convexity and minimal submanifolds.

We discuss a recently proposed variational principle for deriving the variational equations associated to any Lagrangian system. The principle gives simultaneously the Lagrange and the variational equations of the system. We define a new Lagrangian in an extended configuration space ---which we call D'Alambert's--- com…

We prove the existence of non-smooth solutions to Special Lagrangian Equations in the non-convex case.

Bayesian framework discovers interpretable Lagrangian from data.

Visible Lagrangians in Hitchin systems are studied for pillowcase covers.

New curvature measure for optimal transport with specific cost function.

New proof for convex solutions of Monge-Ampère equation.

New curvature-dimension condition for Lagrangians on manifolds.

The paper finds global Darboux coordinates for a new family of symplectic forms on the deformation space of $\mathbb{R}\mathbb{P}^2$-structures.

Automates discovery of interpretable Lagrangians from data.

Alternative approach to regularize time-dependent singular Lagrangian systems.

We derive a Liouville type result for special Lagrangian equations with certain "convexity" and restricted linear growth assumptions on the solutions.

We study non-conservative like SODEs admitting explicit Lagrangian descriptions. Such systems are equivalent to the system of Lagrange equations of some Lagrangian $L$, including a covariant force field which represents non-conservative forces. We find necessary and sufficient conditions for the existence of a differen…

The paper studies bifurcations in Lagrangian systems and geodesics.

We discuss the local differential geometry of convex affine spheres in $\re^3$ and of minimal Lagrangian surfaces in Hermitian symmetric spaces. In each case, there is a natural metric and cubic differential holomorphic with respect to the induced conformal structure: these data come from the Blaschke metric and Pick f…

New method fills cluster seeds with exact Lagrangian structures.

In recent years, constrained optimization has become increasingly relevant to the machine learning community, with applications including Neyman-Pearson classification, robust optimization, and fair machine learning. A natural approach to constrained optimization is to optimize the Lagrangian, but this is not guarantee…

We discuss the characterization of relative equilibria of Lagrangian systems with symmetry.

We consider a {\em Hamiltonian setup} $\sextuple$, where $(\mathcal M,ω)$ is a symplectic manifold, $\mathfrak L$ is a distribution of Lagrangian subspaces in $\mathcal M$, $\mathcal P$ a Lagrangian submanifold of $\mathcal M$, $H$ is a smooth time dependent Hamiltonian function on $\mathcal M$ and $Γ:[a,b]\to\mathcal…

A solution for the Weinstein's Problem in the general framework of generalized Lie algebroids is the target of this paper. We present the mechanical systems called by use, mechanical (?; ?)-systems, Lagrange mechanical (?; ?)-systems or Finsler mechanical (?; ?)-systems and we develop their geometries. We obtain the ca…

Survey of Lagrangian reduction for discrete mechanical systems.

Geodesic extensions for systems with nonholonomic constraints.

The purpose of this paper is describe Lagrangian Mechanics for constrained systems on Lie algebroids, a natural framework which covers a wide range of situations (systems on Lie groups, quotients by the action of a Lie group, standard tangent bundles...). In particular, we are interested in two cases: singular Lagrangi…

We discuss two generalizations of the inverse problem of the calculus of variations, one in which a given mechanical system can be brought into the form of Lagrangian equations with non-conservative forces of a generalized Rayleigh dissipation type, the other leading to Lagrangian equations with so-called gyroscopic fo…

The classes of Monge-Ampère systems, decomposable and bi-decomposable Monge-Ampère systems, including equations for improper affine spheres and hypersurfaces of constant Gauss-Kronecker curvature are introduced. They are studied by the clear geometric setting of Lagrangian contact structures, based on the existence of …