Regularity of potential functions of the optimal transportation problem

DOI10.1007/s00205-005-0362-9zbMath1072.49035OpenAlexW2050480234MaRDI QIDQ2385976

Xu-Jia Wang, Neil S. Trudinger, Xi-Nan Ma

Publication date: 22 August 2005

Published in: Archive for Rational Mechanics and Analysis (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s00205-005-0362-9

zbMATH Keywords

mass transferoptimal transportationpotential functionMonge-Ampère type equation

Mathematics Subject Classification ID

Nonlinear boundary value problems for linear elliptic equations (35J65) Variational methods applied to PDEs (35A15) Variational problems in a geometric measure-theoretic setting (49Q20)

Related Items

Gradient estimates for Neumann boundary value problem of Monge–Ampère type equations, Existence of weak solutions to refraction problems in negative refractive index materials, On the numerical solution of the far field refractor problem, Regularity of the solution to the principal-agent problem, Second order variational heuristics for the Monge problem on compact manifolds, Blaschke's rolling ball theorem and the Trudinger-Wang monotone bending, Optimal Transport and Curvature, Monotone discretization of the Monge–Ampère equation of optimal transport, On the Degeneracy of Optimal Transportation, Sharp boundary \(\varepsilon\)-regularity of optimal transport maps, Exponential convergence of parabolic optimal transport on bounded domains, The Monge–Ampère equation and its link to optimal transportation, An iterative method for generated Jacobian equations, \(C^{1,\alpha}\)-estimates for the near field refractor, A linear finite-difference scheme for approximating randers distances on cartesian grids, On the (in)stability of the identity map in optimal transportation, First and second derivative Hölder estimates for generated Jacobian equations, Regularity of optimal mapping between hypercubes, The Space of Spaces: Curvature Bounds and Gradient Flows on the Space of Metric Measure Spaces, Strict g-Convexity for Generated Jacobian Equations with Applications to Global Regularity, Augmented Hessian equations on Riemannian manifolds: from integral to pointwise local second derivative estimates, A Perturbative Approach to the Parabolic Optimal Transport Problem, A REMARK ON THE GEOMETRIC INTERPRETATION OF THE A3W CONDITION FROM OPTIMAL TRANSPORT, Graph comparison meets Alexandrov, The boundary method for semi-discrete optimal transport partitions and Wasserstein distance computation, The multiplicity of radial \(k\)-convex solutions for an augmented Hessian equation, Adaptive mesh methods on compact manifolds via optimal transport and optimal information transport, Geometry of vectorial martingale optimal transportations and duality, On the difference between entropic cost and the optimal transport cost, Sharp convergence rates for empirical optimal transport with smooth costs, A note on second derivative estimates for Monge-Ampère-type equations, The five gradients inequality for non quadratic costs, Existence results for the \(k\)-Hessian type system with the gradients via \(\mathbb{R}_+^n\)-monotone matrices, A geometric Laplace method, Weak solutions of generated Jacobian equations, A note on optimal transport and Monge-Ampère geometry, A Geometric Approach to the Transport of Discontinuous Densities, Convergence rate of general entropic optimal transport costs, Inequalities for quermassintegrals on \(k\)-convex domains, Entire subsolutions of a kind of \(k\)-Hessian type equations with gradient terms, On estimates for augmented Hessian type parabolic equations on Riemannian manifolds, On the Dirichlet problem for Monge-Ampère type equations, A perturbation argument for a Monge-Ampère type equation arising in optimal transportation, InteriorC^2,αRegularity for Potential Functions in Optimal Transportation, From Optimal Transportation to Conformal Geometry, A note on global regularity in optimal transportion, An Aleksandrov type estimate for ${\alpha }$-convex functions, A Newton Algorithm for Semidiscrete Optimal Transport with Storage Fees, On the Second Boundary Value Problem for a Class of Modified-Hessian Equations, On Pogorelov estimates in optimal transportation and geometric optics, Gradient estimates for potentials of invertible gradient-mappings on the sphere, An Inverse Problem for the Refractive Surfaces with Parallel Lighting, An Algorithm for Optimal Transport between a Simplex Soup and a Point Cloud, An overview of mathematical modeling of geometric optics problems involving refraction, REGULARITY OF MONGE–AMPÈRE EQUATIONS IN OPTIMAL TRANSPORTATION, Ramified optimal transportation in geodesic metric spaces, A variational formulation of the BDF2 method for metric gradient flows, Continuity of optimal transport maps and convexity of injectivity domains on small deformations of 𝕊², Variational Heuristics for Optimal Transportation Maps on Compact Manifolds, Optimal transport with discrete long-range mean-field interactions, A Jacobian Inequality for Gradient Maps on the Sphere and Its Application to Directional Statistics, Boundary $C^{1,\alpha}$ regularity of an optimal transport problem with cost close to $-x\cdot y$, On the Convexity of Injectivity Domains on Nonfocal Manifolds, Optimal transportation, topology and uniqueness, Pointwise Estimates and Regularity in Geometric Optics and Other Generated Jacobian Equations, Partial regularity for optimal transport maps, Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular), Second order analysis on (𝒫₂(ℳ),𝒲₂), Variational approach to regularity of optimal transport maps: general cost functions, On Neumann problem for the degenerate Monge-Ampère type equations, Radial solutions for fully nonlinear elliptic equations of Monge-Ampère type, On the numerical solution of the near field refractor problem, \(C^{1}\) regularity of solutions of the Monge-Ampère equation for optimal transport in dimension two, On strict convexity and continuous differentiability of potential functions in optimal transportation, The refractor problem in reshaping light beams, \(C^{1, \alpha}\) estimates for the parallel refractor, Stability results on the smoothness of optimal transport maps with general costs, A characterization for solutions of the Monge-Kantorovich mass transport problem, Optimal partition of a large labor force into working pairs, Hedonic price equilibria, stable matching, and optimal transport: Equivalence, topology, and uniqueness, When optimal transport meets information geometry, Pseudo-Riemannian geometry encodes information geometry in optimal transport, Synthetic theory of Ricci curvature bounds, Dynamics of optimal partial transport, Oblique boundary value problems for augmented Hessian equations. I, On Monge-Ampère type equations arising in optimal transportation problems, On the local theory of prescribed Jacobian equations revisited, The radial solution for an eigenvalue problem of singular augmented Hessian equation, A glimpse into the differential topology and geometry of optimal transport, Multi-marginal optimal transport and multi-agent matching problems: uniqueness and structure of solutions, On the local theory of prescribed Jacobian equations, The exchange value embedded in a transport system, On the Ma-Trudinger-Wang curvature on surfaces, Hölder continuity and injectivity of optimal maps, On the Dirichlet problem for general augmented Hessian equations, Representation of Markov chains by random maps: existence and regularity conditions, Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds, Strict convexity and \(C^{1,\alpha}\) regularity of potential functions in optimal transportation under condition A3w, Existence and regularity of the reflector surfaces in \(\mathbb R^{n+1}\), On the regularity of solutions of optimal transportation problems, An iterative scheme for solving the optimal transportation problem, On the regularity of optimal transportation potentials on round spheres, The Sinkhorn algorithm, parabolic optimal transport and geometric Monge-Ampère equations, The Kähler geometry of certain optimal transport problems, Structural results on convexity relative to cost functions, Light reflection is nonlinear optimization, Optimal transportation between unequal dimensions, On Aleksandrov-Fenchel inequalities for \(k\)-convex domains, Continuity for the Monge mass transfer problem in two dimensions, A McLean theorem for the moduli space of Lie solutions to mass transport equations, Positivity of Ma-Trudinger-Wang curvature on Riemannian surfaces, Regularity in Monge's mass transfer problem, Regularity for an obstacle problem of hessian equations on Riemannian manifolds, Bipolar comparison, The Neumann problem for parabolic Hessian quotient equations, On the local geometry of maps with c-convex potentials, Convergence rates for discretized Monge-Ampère equations and quantitative stability of optimal transport, Regularity for the near field parallel refractor and reflector problems, Free discontinuities in optimal transport, On the classical solvability of near field reflector problems, Strict convexity and \(C^1\) regularity of solutions to generated Jacobian equations in dimension two, Wasserstein barycenters over Riemannian manifolds, Optimal martingale transport between radially symmetric marginals in general dimensions, On optimal partitions, individual values and cooperative games: does a wiser agent always produce a higher value?, Degenerate conformally invariant fully nonlinear elliptic equations, Refractor surfaces determined by near-field data, An iterated projection approach to variational problems under generalized convexity constraints, Partial \(W^{2,p}\) regularity for optimal transport maps, Regularity of optimal transport in curved geometry: the nonfocal case, Boundary \(\varepsilon\)-regularity in optimal transportation, On the Dirichlet problem for a class of augmented Hessian equations, Tangent cut loci on surfaces, The Ma-Trudinger-Wang curvature for natural mechanical actions, When is multidimensional screening a convex program?, Convexity and multi-dimensional screening for spaces with different dimensions, On the regularity of the free boundary in the optimal partial transport problem for general cost functions, On asymptotic behaviour and \(W^{2, p}\) regularity of potentials in optimal transportation, Global regularity of optimal mappings in non-convex domains, Free boundaries in optimal transport and Monge-Ampère obstacle problems, On local well-posedness of the thin-film equation via the Wasserstein gradient flow, Continuity and injectivity of optimal maps, Continuity, curvature, and the general covariance of optimal transportation, Regularity of optimal maps on the sphere: the quadratic cost and the reflector antenna, On the second boundary value problem for Monge-Ampère type equations and geometric optics, Existence and nonexistence of subsolutions for augmented Hessian equations, Gradient estimates for Neumann boundary value problem of Monge-Ampère type equations on Riemannian manifolds, Stability of a 4th-order curvature condition arising in optimal transport theory, Local Hölder regularity of solutions to generated Jacobian equations, Computational semi-discrete optimal transport with general storage fees, A solution to the Monge transport problem for Brownian martingales, \({\mathcal{W}}_\infty \)-transport with discrete target as a combinatorial matching problem, Regularity of optimal transport maps on locally nearly spherical manifolds, Local pointwise second derivative estimates for strong solutions to the \(\sigma_k\)-Yamabe equation on Euclidean domains, Hölder regularity of optimal mappings in optimal transportation, Global regularity for the Monge-Ampère equation with natural boundary condition, Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds, Convergence of a Newton algorithm for semi-discrete optimal transport, Ricci curvature and measures, On the Dirichlet problem for degenerate Monge-Ampère type equations, Elliptic solutions to nonsymmetric Monge-Ampère type equations. II. A priori estimates and the Dirichlet problem, Constructing a relativistic heat flow by transport time steps, On open flat sets in spaces with bipolar comparison, Entropic optimal transport: geometry and large deviations, Necessary and sufficient conditions of entire subsolutions to Monge-Ampère type equations, On semi-discrete Monge-Kantorovich and generalized partitions, Full characterization of optimal transport plans for concave costs, Boundary \(C^{2, \alpha}\) estimates for Monge-Ampère type equations, Recent developments on the MTW tensor

Cites Work

• Boundary regularity of maps with convex potentials. II
• Classical solvability in dimension two of the second boundary-value problem associated with the Monge-Ampère operator
• Dirichlet problems for general Monge-Ampère equations
• Regularity theory for quasilinear elliptic systems and Monge-Ampère equations in two dimensions
• On a Monge-Ampère equation arising in geometric optics
• On the Dirichlet problem for Hessian equations
• The geometry of optimal transportation
• Oblique boundary value problems for equations of Monge-Ampère type
• On the design of a reflector antenna. II
• Existence and uniqueness of monotone measure-preserving maps
• Minkowski problems for complete noncompact convex hypersurfaces
• Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs
• Polar factorization and monotone rearrangement of vector‐valued functions
• The Regularity of Mappings with a Convex Potential
• On the regularity of the monge-ampère equation det (∂2 u/∂xi ∂xj) = f(x, u)
• Differential equations methods for the Monge-Kantorovich mass transfer problem
• On the second boundary value problem for equations of Monge-Ampère type.
• On the design of a reflector antenna
• On the Monge mass transfer problem
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item