Related papers: Strict $g$-convexity for generated Jacobian equati…
This is a PhD thesis about generated Jacobian equations; our purpose is twofold. First, we provide an introduction to these equations, whilst, at the same time, collating some results scattered throughout the literature. The other goal is…
We prove that two Aleksandrov solutions of a generated prescribed Jacobian equation have the same gradients at points where they are both differentiable. For the optimal transportation case where two solutions can be translated to agree at…
The study of reflector surfaces in geometric optics necessitates the analysis of certain nonlinear equations of Monge-Amp\`ere type known as generated Jacobian equations. These equations, whose general existence theory has been recently…
In this paper, we prove the existence of classical solutions to second boundary value prob- lems for generated prescribed Jacobian equations, as recently developed by the second author, thereby obtaining extensions of classical solvability…
In this paper a new class of modified-Hessian equations, closely related to the Optimal Transportation Equation, will be introduced and studied. In particular, the existence of globally smooth, classical solutions of these equations…
In this work, we study geodesic curvature of the boundary of a two dimensional Alexandrov space of curvature bounded below (CBB). We prove several comparison and globalization theorems for the geodesic curvature, generalizing the known…
We give a variational proof of the existence and uniqueness of a convex cap with the given upper boundary. The proof uses the concavity of the total scalar curvature functional on the space of generalized convex caps. As a byproduct, we…
By studying the linearization of contour dynamics equation and using implicit function theorem, we prove the existence of co-rotating and travelling global solutions for the gSQG equation, which extends the result of Hmidi and Mateu…
We prove the optimal $W^{2,\infty}$ regularity for variational problems with convex gradient constraints. We do not assume any regularity of the constraints; so the constraints can be nonsmooth, and they need not be strictly convex. When…
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…
In this paper, we establish a global regularity result for the optimal transport problem with the quadratic cost, where the domains may not be convex. This result is obtained by a perturbation argument, using a recent global regularity of…
We study the fine regularity properties of optimal potentials for the dual formulation of the Hellinger--Kantorovich problem (HK), providing sufficient conditions for the solvability of the primal Monge formulation. We also establish new…
In this paper we provide a sufficient condition, in terms of only one of the nine entries of the gradient tensor, i.e., the Jacobian matrix of the velocity vector field, for the global regularity of strong solutions to the three-dimensional…
This note concerns the relationship between conditions on cost functions and domains and the convexity properties of potentials in optimal transportation and the continuity of the associated optimal mappings. In particular, we prove that if…
We prove that outside of a closed singular set of measure zero solutions to the second boundary value problem for generated Jacobian equations are smooth.
We establish global $C^{1,\beta}$ and $W^{2, p}$ regularity for singular Monge-Amp\`ere equations of the form \[\det D^2 u \sim \text{dist}^{-\alpha}(\cdot,\partial\Omega),\quad \alpha\in (0, 1),\] under suitable conditions on the boundary…
We show a global existence result of weak solutions for a class of generalized Surface Quasi-Geostrophic equation in the inviscid case. We also prove the global regularity of such solutions for the equation with slightly supercritical…
In this paper we investigate a forced incompressible Navier-Stokes equation coupled with a parabolic type equation of Q-tensors in a domain $U\subset\R^3.$ In the case $U$ is bounded, we prove the existence of a global strong solution when…
We investigate the Cauchy problem for a two-component generalization of the Novikov equation with cubic nonlinearity -- an integrable system whose solutions may develop strong nonlinear phenomena such as gradient blow-up and interactions…
We investigate the properties of convex functions in the plane that satisfy a local inequality which generalizes the notion of sub-solution of Monge-Ampere equation for a Monge-Kantorovich problem with quadratic cost between non-absolutely…