We give a necessary and sufficient condition on the cost function so that the map solution of Monge's optimal transportation problem is continuous for arbitrary smooth positive data. This condition was first introduced by Ma, Trudinger and Wang \cite{MTW, TW} for a priori estimates of the corresponding Monge-Ampère equation. It is expressed by a so-called {\em cost-sectional curvature} being non-negative. We show that when the cost function is the squared distance of a Riemannian manifold, the cost-sectional curvature yields the sectional curvature. As a consequence, if the manifold does not have non-negative sectional curvature everywhere, the optimal transport map {\em can not be continuous} for arbitrary smooth positive data. The non-negativity of the cost-sectional curvature is shown to be equivalent to the connectedness of the contact set between any cost-convex function (the proper generalization of a convex function) and any of its supporting functions. When the cost-sectional curvature is uniformly positive, we obtain that optimal maps are continuous or Hölder continuous under quite weak assumptions on the data, compared to what is needed in the Euclidean case. This case includes the reflector antenna problem and the squared Riemannian distance on the sphere.
We provide a new proof of the known partial regularity result for the optimal transportation map (Brenier map) between two sets. Contrary to the existing regularity theory for the Monge-Amp{è}re equation, which is based on the maximum principle, our approach is purely variational. By constructing a competitor on the level of the Eulerian (Benamou-Brenier) formulation, we show that locally, the velocity is close to the gradient of a harmonic function provided the transportation cost is small. We then translate back to the Lagrangian description and perform a Campanato iteration to obtain an $ε$…
We develop an $\e$-regularity theory at the boundary for a general class of Monge-Ampère type equations arising in optimal transportation. As a corollary we deduce that optimal transport maps between Hölder densities supported on $C^2$ uniformly convex domains are $C^{1,α}$ up to the boundary, provided that the cost function is a sufficient small perturbation of the quadratic cost $-x\cdot y$.
We study the regularity of solutions to an optimal transportation problem where the dimension of the source is larger than that of the target. We demonstrate that if the target is $c$-convex, then the source has a canonical foliation whose co-dimension is equal to the dimension of the target and the problem reduces to an optimal transportation problem between spaces with equal dimensions. If the $c$-convexity condition fails, we do not expect regularity for arbitrary smooth marginals, but, in the case where the source is 2-dimensional and the target is 1 dimensional, we identify sufficient con…
In this paper, we obtain some regularities of the free boundary in optimal transportation with the quadratic cost. Our first result is about the $C^{1,α}$ regularity of the free boundary for optimal partial transport between convex domains for densities $f, g$ bounded from below and above. When $f, g \in C^α$, and $\partialΩ, \partialΩ^*\in C^{1,1}$ are far apart, by adopting our recent results on boundary regularity of Monge-Ampère equations \cite{CLW1}, our second result shows that the free boundaries are $C^{2,α}$. As an application, in the last we also obtain these regularities of the free…
This paper describes recent results obtained in collaboration with M. Huesmann and F. Otto on the regularity of optimal transport maps. The main result is a quantitative version of the well-known fact that the linearization of the Monge-Amp{è}re equation around the identity is the Poisson equation. We present two applications of this result. The first one is a variational proof of the partial regularity theorem of Figalli and Kim and the second is the rigorous validation of some predictions made by Carraciolo and al. on the structure of the optimal transport maps in matching problems.
We study a form of optimal transportation surplus functions which arise in hedonic pricing models. We derive a formula for the Ma-Trudinger-Wang curvature of these functions, yielding necessary and sufficient conditions for them to satisfy \textbf{(A3w)}. We use this to give explicit new examples of surplus functions satisfying \textbf{(A3w)}, of the form $b(x,y)=H(x+y)$ where $H$ is a convex function on $\mathbb{R}^n$. We also show that the space of equilibrium contracts in the hedonic pricing model has the maximal possible dimension, a result of potential economic interest.
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 the cost function satisfies the condition (A3), introduced in our previous work with Xinan Ma, the densities and their reciprocals are bounded and the target domain is convex with respect to the cost function, then the potential is continuously differentiable and its dual potential strictly concave with respect to the cost function. Our result extends, by d…
We survey the (old and new) regularity theory for the Monge-Ampère equation, show its connection to optimal transportation, and describe the regularity properties of a general class of Monge-Ampère type equations arising in that context.
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 continuous measures. For each measure, we introduce a discrete scale so that the measure behaves as an absolutely continuous measure up to that scale. Our main theorem then proves that such convex functions cannot exhibit any flat part at a scale larger than the corresponding discrete scales on the measures. This, in turn, implies a $C^1$ regularity result u…
We give sufficient conditions on initial and target measures supported on the sphere $§^n$ to ensure the solution to the optimal transport problem with the cost $|x-y|^2/2$ is a diffeomorphism.
In this series of lectures we introduce the Monge-Kantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x,y). Connections to geometry, inequalities, and partial differential equations will be discussed, focusing in particular on recent developments in the regularity theory for Monge-Ampere type equations. An application to microeconomics will also be described, which amounts to finding the equilibrium price distribution for a monopolist marketing a multidimensional line of products to a population of anonymou…
In this paper the regularity of optimal transportation potentials defined on round spheres is investigated. Specifically, this research generalises the calculations done by Loeper, where he showed that the strong (A3) condition of Trudinger and Wang is satisfied on the round sphere, when the cost-function is the geodesic distance squared. In order to generalise Loeper's calculation to a broader class of cost-functions, the (A3) condition is reformulated via a stereographic projection that maps charts of the sphere into Euclidean space. This reformulation subsequently allows one to verify the (…
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 weak form of our condition A3, under which we proved interior regularity in a recent paper with Xi-nan Ma. Consequently they include the quadratic cost function case of Caffarelli and Urbas as well as the various examples in the earlier work. The approach is through the derivation of global estimates for second derivatives of solutions.
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 optimal transportation in convex domains by the authors.
We formulate the half-wave maps problem with target $S^2$ and prove global regularity in sufficiently high spatial dimensions for a class of small critical data in Besov spaces.
We prove full boundary regularity for minimizing biharmonic maps with smooth Dirichlet boundary conditions. Our result, similarly as in the case of harmonic maps, is based on the nonexistence of nonconstant boundary tangent maps. With the help of recently derivated boundary monotonicity formula for minimizing biharmonic maps by Altuntas we prove compactness at the boundary following Scheven's interior argument. Then we combine those results with the conditional partial boundary regularity result for stationary biharmonic maps by Gong--Lamm--Wang.
In this paper we study the regularity of weak solutions to an elliptic-parabolic system modeling natural network formation. The system is singular and involves cubic nonlinearity. Our investigation reveals that weak solutions are Hölder continuous when the space dimension $N$ is $2$. This is achieved via an inequality associated with the Stummel-Kato class of functions and refinement of a lemma originally due to S. Campanato and C. B. Morrey (\cite{G}, p. 86).
The aim of this short note is to extend the recent variational proof of partial regularity for optimal transport maps to the case of continuous densities.
We prove that, for general cost functions on $\mathbb{R}^n$, or for the cost $d^2/2$ on a Riemannian manifold, optimal transport maps between smooth densities are always smooth outside a closed singular set of measure zero.
We prove a quantification result for harmonic maps with free boundary from arbitrary Riemannian surfaces into the unit ball of ${\mathbb R}^{n+1}$ with bounded energy. This generalizes results obtained by Da Lio on the disc.
This article addresses the regularity issue for stationary or minimizing fractional harmonic maps into spheres of order $s\in(0,1)$ in arbitrary dimensions. It is shown that such fractional harmonic maps are $C^\infty$ away from a small closed singular set. The Hausdorff dimension of the singular set is also estimated in terms of $s\in(0,1)$ and the stationarity/minimality assumption.
We prove the uniqueness of solutions to Dirichlet problem for p-harmonic maps with images in a small geodesic ball of the target manifold. As a consequence, we show that such maps have Hoelder continuous derivatives. This gives an extension of a result by S. Hildebrandt et al concerning harmonic maps.
In this paper we develop a boundary $\varepsilon$-regularity theory for optimal transport maps between bounded open sets with $C^{1,α}$-boundary. Our main result asserts sharp $C^{1,α}$-regularity of transport maps at the boundary in form of a linear estimate under certain assumptions: The main quantitative assumptions are that the local nondimensionalized transport cost is small and that the boundaries are locally almost flat in $C^{1,α}$. Our method is completely variational and builds on the recently developed interior regularity theory.
We prove that the half-wave maps problem on $\mathbb{R}^{4+1}$ with target $S^2$ is globally well-posed for smooth initial data which are small in the critical $l^1$ based Besov space. This is a formal analogue of the result for wave maps by Tataru.
We prove that manifold constrained $p(x)$-harmonic maps are $C^{1,β}$-regular outside a set of zero $n$-dimensional Lebesgue's measure, for some $β\in (0,1)$. We also provide an estimate from above of the Hausdorff dimension of the singular set.
For minimizers in a geometrically nonlinear Cosserat model for micropolar elasticity of continua, we prove interior Hölder regularity, up to isolated singular points that may be possible if the exponent $p$ from the model is $2$ or in $(\frac{32}{15},3)$. The obstacle to full continuity turns out to be the existence of certain minimizing homogeneous $p$-harmonic maps to $S^3$. For those, we slightly improve existing regularity theorems in order to achieve our result on the Cosserat model.
In dimensions greater than or equal to 3, we prove that the Schroedinger map initial-value problem is globally well-posed for small data in the critical Besov space.
We prove that, in the optimal transportation problem with general costs and positive continuous densities, the potential function is always of class $W^{2,p}_{loc}$ for any $p \geq 1$ outside of a closed singular set of measure zero. We also establish global $W^{2,p}$ estimates when the cost is a small perturbation of the quadratic cost. The latter result is new even when the cost is exactly the quadratic cost.
We introduce n/p-harmonic maps as critical points of E(v) the Lp-Norm of the alpha-laplacian of v, where pointwise v maps Rn into a sphere, and alpha = n/p. This energy combines the non-local behaviour of the fractional harmonic maps introduced by Riviere and the first author with the degenerate arguments of the n-laplacian. In this setting, we will prove Holder continuity.
We prove the regularity of weak 1/2-harmonic maps from the real line into a sphere. The key point in our result is first a formulation of the 1/2-harmonic map equation in the form of a non-local linear Schrödinger type equation with a 3-terms commutators in the right-hand-side . We then establish a sharp estimate for these 3-commutators.
We establish that solving an optimal transportation problem in which the source and target densities are defined on manifolds with different dimensions, is equivalent to solving a new nonlocal analog of the Monge-Ampère equation, introduced here for the first time. Under suitable topological conditions, we also establish that solutions are smooth if and only if a local variant of the same equation admits a smooth and uniformly elliptic solution. We show that this local equation is elliptic, and $C^{2,α}$ solutions can therefore be bootstrapped to obtain higher regularity results, assuming smoo…
Using the harmonic map heat flow and the function spaces of Tataru and the author, we establish a large data local well-posedness result in the energy class for wave maps from two-dimensional Minkowski space $\R^{1+2}$ to hyperbolic spaces $\H^m$. This is one of the five claims required in an earlier paper in this series to prove global regularity for such wave maps.
We prove that the solutions to the initial-value problem for 2-dimensional Schrödinger maps are unique in $C_tL_x^{\infty} \cap L_t^{\infty} (\dot{H}^1_x\cap \dot{H}^2_x)$. For the proof, we follow McGahagan's argument with improving its technical part, combining Yudovich's argument.
We consider the polar factorization of vector valued mappings introduced by Y. Brenier. In the case of a family of mappings depending on a parameter. We investigate the regularity with respect to this parameter of the terms of the polar factorization by constructing some a priori bounds. To do so, we consider the linearization of the associated Monge-Ampere equation which we view as a conservation law.
We prove continuity on domains up to the boundary for n/2-polyharmonic maps into manifolds. Technically, we show how to adapt Helein's direct approach to the fractional setting. This extends a remark by the author that this is possible in the setting of Riviere's famous regularity result for critical points of conformally invariant variational functionals. Moreover, pointwise behavior for the involved three-commutators is established. Continuity up to the boundary is then obtained via an adaption of Hildebrandt and Kaul's technique to the non-local setting.
Using the harmonic map heat flow, we construct an energy class for wave maps $ϕ$ from two-dimensional Minkowski space $\R^{1+2}$ to hyperbolic spaces $\H^m$, and then show (conditionally on a large data well-posedness claim for such wave maps) that no stationary, travelling, self-similar, or degenerate wave maps exist in this energy class. These results form three of the five claims required in our earlier paper (arXiv:0805.4666) to prove global regularity for such wave maps. (The conditional claim of large data well-posedness is one of the remaining claims required in that paper.)
We derive a boundary monotonicity formula for a class of biharmonic maps with Dirichlet boundary conditions. A monotonicity formula is crucial in the theory of partial regularity in super-critical dimensions. As a consequence of such a boundary monotonicity formula, one is able to show partial regularity for variationally biharmonic maps and full boundary regularity for minimizing biharmonic maps.
In this paper we continue the investigation of the regularity of the so-called weak $\frac{n}{p}$-harmonic maps in the critical case. These are critical points of the following nonlocal energy \[ {\mathcal{L}}_s(u)=\int_{\mathbb{R}^n}| ( {-Δ})^{\frac{s}{2}} u(x)|^p dx\,, \] where $u\in \dot{H}^{s,p}(\mathbb{R}^n,\mathcal{N})$ and ${\mathcal{N}}\subset\mathbb{R}^N$ is a closed $k$ dimensional smooth manifold and $s=\frac{n}{p}$. We prove Hölder continuity for such critical points for $p \leq 2$. For $p > 2$ we obtain the same under an additional Lorentz-space assumption. The regularity theory i…
We shall prove new contraction properties of general transportation costs along nonnegative measure-valued solutions to Fokker-Planck equations in $R^d$, when the drift is a monotone (or $λ$-monotone) operator. A new duality approach to contraction estimates has been developed: it relies on the Kantorovich dual formulation of optimal transportation problems and on a variable-doubling technique. The latter is used to derive a new comparison property of solutions of the backward Kolmogorov (or dual) equation. The advantage of this technique is twofold: it directly applies to distributional solut…
In this paper, we study the relaxed energy for biharmonic maps from a $m$-dimensional domain into spheres. By an approximation method, we prove the existence of a minimizer of the relaxed energy of the Hessian energy, and that the minimizer is biharmonic and smooth outside a singular set $Σ$ of finite $(m-4)$-dimensional Hausdorff measure. Moreover, when $m=5$, we prove that the singular set $Σ$ is 1-rectifiable.
We study the regularity of weak solutions for two elliptic systems involving the $n$-Laplacian and a critical nonlinearity in the right hand side: $H$-systems and $n$-harmonic maps into compact Riemannian manifolds. Under the assumptions that the solutions belong to $W^{n/2,2}$ in an even dimension $n$, we prove their continuty. The tools used in the proof involve Hardy spaces and BMO, and the Rivière--Uhlenbeck decomposition (with estimates in Morrey spaces). A prominent role is played by the Coifman--Rochberg--Weiss commutator theorem.
We show that the wave map equation in $\R^{1+1}$ is in general ill-posed in the critical space $\dot H^{1/2}$, and the Besov space $\dot B^{1/2,1}_2$. The problem is attributed to the bad behaviour of the one-dimensional bilinear expression $D^{-1}(f Dg)$ in these spaces.
We consider the Dirichlet problem for stationary biharmonic maps $u$ from a bounded, smooth domain $Ω\subset\mathbb R^n$ ($n\ge 5$) to a compact, smooth Riemannian manifold $N\subset\mathbb R^l$ without boundary. For any smooth boundary data, we show that if, in addition, $u$ satisfies a certain boundary monotonicity inequality, then there exists a closed subset $Σ\subset\barΩ$, with $H^{n-4}(Σ)=0$, such that $u\in C^\infty(\barΩ\setminusΣ, N)$.
We extend the variational approach to regularity for optimal transport maps initiated by Goldman and the first author to the case of general cost functions. Our main result is an $ε$-regularity result for optimal transport maps between Hölder continuous densities slightly more quantitative than the result by De Philippis-Figalli. One of the new contributions is the use of almost-minimality: if the cost is quantitatively close to the Euclidean cost function, a minimizer for the optimal transport problem with general cost is an almost-minimizer for the one with quadratic cost. This further highl…
In this paper, we will study the partial regularity theorem for stationary harmonic maps from a Riemannian manifold into a Lorentzian manifold. For a weakly stationary harmonic map $(u,v)$ from a smooth bounded open domain $Ω\subset\R^m$ to a Lorentzian manifold with Dirichlet boundary condition, we prove that it is smooth outside a closed set whose $(m-2)$-dimension Hausdorff measure is zero. Moreover, if the target manifold $N$ does not admit any harmonic sphere $S^l$, $l=2,...,m-1$, we will show $(u,v)$ is smooth.
