If the cost function is not too far from the Euclidean cost, then the optimal map transporting Gaussians restricted to a ball will be regular. \ Similarly, given any cost function which is smooth in a neighborhood of two points on a manifold, there are small neighborhoods near each such that a Gaussian restricted to one is transported smoothly to a Gaussian on the other
We establish $C^{σ+α}$ interior estimates for concave nonlocal fully nonlinear equations of order $σ\in(0,2)$ with rough kernels. Namely, we prove that if $u\in C^α(\mathbb R^n)$ solves in $B_1$ a concave translation invariant equation with kernels in $\mathcal L_0(σ)$, then $u$ belongs to $C^{σ+α}(\overline{ B_{1/2}})$, with an estimate. More generally, our results allow the equation to depend on $x$ in a $C^α$ fashion. Our method of proof combines a Liouville theorem and a blow-up (compactness) procedure. Due to its flexibility, the same method can be useful in different regularity proofs fo…
Given a concave integro-differential operator $I$, we study regularity for solutions of fully nonlinear, nonlocal, parabolic, concave equations of the form $u_t-Iu=0$. The kernels are assumed to be smooth but non necessarily symmetric which accounts for a critical non-local drift. We prove a $C^{\s+\a}$ estimate in the spatial variable and a $C^{1,\a/\s}$ estimates in time assuming time regularity for the boundary data. The estimates are uniform in the order of the operator $I$, hence allowing us to extend the classical Evans-Krylov result for concave parabolic equations.
We show the existence of the strong solutions of the Monge-Ampere equation for the log-concave measures without any regularity assumption. In particular the measures with density of the form $\exp -f$, where $f$ is an $H$-convex function which may take the value $\infty$ on a set of positive measure are included.
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.
Given an open bounded subset $Ω$ of $\mathbb{R}^n$, which is convex and satisfies an interior sphere condition, we consider the pde $-Δ_{\infty} u = 1$ in $Ω$, subject to the homogeneous boundary condition $u = 0$ on $\partial Ω$. We prove that the unique solution to this Dirichlet problem is power-concave (precisely, 3/4 concave) and it is of class $C ^1(Ω)$. We then investigate the overdetermined Serrin-type problem obtained by adding the extra boundary condition $|\nabla u| = a$ on $\partial Ω$; by using a suitable $P$-function we prove that, if $Ω$ satisfies the same assumptions as above a…
For $M$ being a closed manifold or the Euclidean space we present a detailed proof of regularity properties of the composition of $H^s$-regular diffeomorphisms of $M$ for $s > 1/2\dim M + 1$.
We consider an elliptic Kolmogorov equation lambda u - Ku =f in a convex subset C of a separable Hilbert space X. We prove maximal Sobolev regularity of its weak solution, when lambda >0 and f is in L^2(C,nu), where nu is the log-concave measure associated to the system. Moreover we prove maximal estimates on the gradient of u, that allow to show that u satisfies the Neumann boundary condition in the sense of traces at the boundary of C. The general results are applied to Kolmogorov equations of reaction-diffusion stochastic PDEs and Cahn-Hilliard stochastic PDEs in convex sets of suitable Hil…
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 prove the existence and $C^{1,α}$ regularity of solutions to nonlocal fully nonlinear elliptic double obstacle problems. We also obtain boundary regularity for these problems. The obstacles are assumed to be Lipschitz semi-concave/semi-convex functions, and we do not require them to be $C^1$. Our approach is to adapt a penalization method to be applicable to the setting of nonlocal equations and their viscosity solutions.
In this note, we study the random data problem for incompressible Navier-Stokes equations in Euclidean space with arbitrary regularity, using the randomization introduced in \cite{Shen-Soffer-Wu-2021-1}.
We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of a half-equator. The proofs combine constructions of strictly convex functions and the regularity theory of quasi-linear elliptic systems. We apply these results to the spherical and Euclidean Bernstein problems for minimal hypersurfaces, obtaining new conditions under which compact minimal hypersurfaces in spheres or complete minimal hypersurfaces in Euclidean spaces are trivial.
We investigate the stability of boundary layer solutions of the two-dimensional incompressible Navier-Stokes equations. We consider shear flow solutions of Prandtl type : $$ u^ν(t,x,y) \, = \, \big (U^E(t,y) + U^{BL}(t,\frac{y}{\sqrtν})\,, \, 0 \big )\, , \quad 0<ν\ll 1\,. $$ We show that if $U^{BL}$ is monotonic and concave in $Y = y /\sqrtν$ then $u^ν$ is stable over some time interval $(0,T)$, $T$ independent of $ν$, under perturbations with Gevrey regularity in $x$ and Sobolev regularity in $y$. We improve in this way the classical stability results of Sammartino and Caflisch in analytic c…
This paper studies the regularity of solutions to the Zakharov and Klein-Gordon-Schrödinger systems at low regularity levels. The main result is that the nonlinear part of the solution flow falls in a smoother space than the initial data. This relies on a new bilinear $X^{s,b}$ estimate, which is proved using delicate dyadic and angular decompositions of the frequency domain. Such smoothing estimates have a number of implications for the long-term dynamics of the system. In this work, we give a simplified proof of the existence of global attractors for the Klein-Gordon-Schrödinger flow in the …
We consider the Dirichlet problem for positively homogeneous, degenerate elliptic, concave (or convex) Hessian equations. Under natural and necessary conditions on the geometry of the domain, with the $C^{1,1}$ boundary data, we establish the interior $C^{1,1}$-regularity of the unique (admissible) solution, which is optimal even if the boundary data is smooth. Both real and complex cases are studied by the unified (Bellman equation) approach.
Centro-sectional measures for log-concave functions
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences（匈牙利科学院阿尔弗雷德·雷尼数学研究所）；Institute of Mathematics, Eötvös University（厄特沃什大学数学研究所）；Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien（维也纳工业大学离散数学与几何研究所）；School of Mathematics and statistics, Henan University（河南大学数学与统计学院）
In this paper we analyze the optimal value function $v$ associated to a general parametric optimization problems via the theory of viscosity solutions. The novelty is that we obtain regularity properties of $v$ by showing that it is a viscosity solution to a set of first-order equations. As a consequence, in Banach spaces, we provide sufficient conditions for local and global Lipschitz properties of $v$. We also derive, in finite dimensions, conditions for optimality through a comparison principle. Finally, we study the relationship between viscosity and Clarke generalized solutions to get fur…
In this paper, we study the regularity of optimal mappings in Monge's mass transfer problem. Using the approximation to Monge's cost function given by the Euclidean distance c(x,y)=dist(x,y) through the costs c_\eps(x,y)=(\eps^2+dist(x,y)^2)^{1/2}, we consider the optimal mappings T_\eps for these costs, and we prove that the eigenvalues of the Jacobian matrix DT_\eps, which are all positive, are locally uniformly bounded. By an example we prove that T_\eps is in general not uniformly Lipschitz continuous as \eps-0, even if the mass distributions are positive and smooth, and the domains are c-…
On a bounded domain $Ω$ in euclidean space $\mathbb{R}^n$, we study the homogeneous Dirichlet problem for the eikonal equation associated with a system of smooth vector fields, which satisfies Hörmander's bracket generating condition. We prove that the solution is smooth in the complement of a closed set of Lebesgue measure zero.
In this paper we propose new insights and ideas to set up quantitative boundary estimates for solutions to Dirichlet problem of a class of fully non-linear elliptic equations on compact Hermitian manifolds with real analytic Levi flat boundary. With the quantitative boundary estimates at hand, we can establish the gradient estimate and give a unified approach to investigate the existence and regularity of solutions of Dirichlet problem with sufficiently smooth boundary data, which include the geodesic equation in the space of Kähler metrics as a special case. Our method can also be applied to …
We construct low regularity solutions of the vacuum Einstein constraint equations. In particular, on 3-manifolds we obtain solutions with metrics in $H^s\loc$ with $s>{3\over 2}$. The theory of maximal asymptotically Euclidean solutions of the constraint equations descends completely the low regularity setting. Moreover, every rough, maximal, asymptotically Euclidean solution can be approximated in an appropriate topology by smooth solutions. These results have application in an existence theorem for rough solutions of the Einstein evolution equations.
This paper studies the Sobolev-Lorentz capacity and its regularity in the Euclidean setting for $n \ge 1$ integer. We extend here our previous results on the Sobolev-Lorentz capacity obtained for $n \ge 2.$ Moreover, for $n \ge 2$ integer we obtain a few new results concerning the $n,1$ relative and global capacities. We obtain sharp estimates for the $n,1$ relative capacity of the concentric condensers $(\overline{B}(0,r), B(0,1))$ for all $r$ in $[0,1).$ As a consequence we obtain the exact value of the $n,1$ capacity of a point relative to all its bounded open neighborhoods from ${\mathbf{R…
We investigate the logarithmic and power-type convexity of the length of the level curves for $a$-harmonic functions on smooth surfaces and related isoperimetric inequalities. In particular, our analysis covers the $p$-harmonic and the minimal surface equations. As an auxiliary result, we obtain higher Sobolev regularity properties of the solutions, including the $W^{2,2}$ regularity. The results are complemented by a number of estimates for the derivatives $L'$ and $L''$ of the length of the level curve function $L$, as well as by examples illustrating the presentation. Our work generalizes r…
We show that directed minimal cones in (n+1)-dimensional Euclidean space which have at most one singularity are - besides the trivial cases: empty set, whole space - half spaces. Using blow-up techniques, this result can be used to get C^{1,lambda}-regularity for the measure-theoretic boundary of almost minimal Caccioppoli sets which are representable as subgraphs in R^n, n<=8. This provides a different method to obtain a result due to De Giorgi. We also prove a splitting theorem for general directed minimal cones. Such a cone is the Cartesian product of R^k and C, where C is an undirected min…
We prove local higher-order asymptotics for extreme water waves with vorticity near stagnation points. We obtain that the behaviour of solutions and their regularity depend substantially on the vorticity. In particular, we show that extreme waves with a negative vorticity distribution have concave profiles near the crest. Our approach is based on new regularity results and asymptotic analysis of the corresponding nonlinear problem in a half-strip. Our main result is local and therefore is valid for a broad range of problems, such as for waves with a piecewise constant vorticity, stratified wav…
In the Engel group with its Carnot group structure we study subsets of locally finite subRiemannian perimeter and possessing constant subRiemannian normal. We prove the rectifiability of such sets: more precisely we show that, in some specific coordinates, they are upper-graphs of entire Lipschitz functions (with respect to the Euclidean distance). However we find that, when they are written as intrinsic upper-graphs with respect to the direction of the normal, then the function defining the set might even fail to be continuous. Nevertheless, we can prove that one can always find other horizon…
In this paper, Monge-Kantorovich problem is considered in the infinite dimension on an abstract Wiener space $(W, H,μ)$, where $H$ is Cameron-Martin space and $μ$ is the Gaussian measure. We study the regularity of optimal transport maps with a quadratic cost function assuming that both initial and target measures have a strictly positive Radon-Nikodym density with respect to $μ$. Under conditions on the density functions, the forward and backward transport maps can be written in terms of Sobolev derivative of so-called Monge-Brenier maps, or Monge potentials. We show Sobolev regularity of the…
