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Michael Goldman

Publications and source records attributed to Michael Goldman.

At least 37 records · Page 2Linked to original sources
arXiv2023

Non-convex functionals penalizing simultaneous oscillations along two independent directions: structure of the defect measure

We continue the analysis of a family of energies penalizing oscillations in oblique directions: they apply to functions $u(x_1,x_2)$ with $x_l\in\mathbb{R}^{n_l}$ and vanish when $u(x)$ is of the form $u_1(x_1)$ or $u_2(x_2)$. We mainly study the rectifiability properties of the defect measure $\nabla_1\nabla_2u$ of functions with finite energy. The energies depend on a parameter $θ\in(0,1]$ and the set of functions with finite energy grows with $θ$. For $θ<1$ we prove that the defect measure is $(n_1-1,n_2-1)$-tensor rectifiable in $Ω_1\timesΩ_2$. We first get the result for $n_1=n_2=1$ and deduce the general case through slicing using White's rectifiability criterion. When $θ=1$ the situation is less clear as measures of arbitrary dimensions from zero to $n_1+n_2-1$ are possible. We show however, in the case $n_1=n_2=1$ and for Lipschitz continuous functions, that the defect measures are $1\,$-rectifiable. This case bears strong analogies with the study of entropic solutions of the eikonal equation.

math.AP↗
arXiv2023

Optimal transport methods for combinatorial optimization over two random point sets

We investigate the minimum cost of a wide class of combinatorial optimization problems over random bipartite geometric graphs in $\mathbb{R}^d$ where the edge cost between two points is given by a $p$-th power of their Euclidean distance. This includes e.g.\ the travelling salesperson problem and the bounded degree minimum spanning tree. We establish in particular almost sure convergence, as $n$ grows, of a suitable renormalization of the random minimum cost, if the points are uniformly distributed and $d \ge 3$, $1\le p<d$. Previous results were limited to the range $p<d/2$. Our proofs are based on subadditivity methods and build upon new bounds for random instances of the Euclidean bipartite matching problem, obtained through its optimal transport relaxation and functional analytic techniques.

math.PR↗
arXiv2023

On the concave one-dimensional random assignment problem and Young integration theory

We investigate the one-dimensional random assignment problem in the concave case, i.e., the assignment cost is a concave power function, with exponent $0<p<1$, of the distance between $n$ source and $n$ target points, that are i.i.d. random variables with a common law on an interval. We prove that the limit of a suitable renormalization of the costs exists if the exponent $p$ is different than $1/2$. Our proof in the case $1/2<p<1$ makes use of a novel version of the Kantorovich optimal transport problem based on Young integration theory, where the difference between two measures is replaced by the weak derivative of a function with finite $q$-variation, which may be of independent interest. We also prove a similar result for the random bipartite Traveling Salesperson Problem.

math.PR↗
arXiv2023

From energy bounds to dimensional estimates in a branched transport model for type-I superconductors

We consider a branched transport type problem which describes the magnetic flux through type-I superconductors in a regime of very weak applied fields. At the boundary of the sample, deviation of the magnetization from being uniform is penalized through a negative Sobolev norm. It was conjectured by S. Conti, F. Otto and S. Serfaty that as a result, the trace of the magnetization on the boundary should be a measure of Hausdorff dimension $8/5$. We prove that this conjecture is equivalent to the proof of local energy bounds with an optimal exponent. We then obtain local bounds which are however not optimal. These yield improved lower bounds on the dimension of the irrigated measure but unfortunately does not improve on the trivial upper bound. In order to illustrate the dependence of this dimension on the choice of penalization, we consider in the last part of the paper a toy model where the boundary energy is given by a Wasserstein distance to Lebesgue. In this case minimizers are finite graphs and thus the trace is atomic.

math.AP↗
arXiv2022

A deformation theorem for tensor flat chains and applications (complement to ''Tensor rectifiable G-flat chains'')

In this note we extend White's deformation theorem for G-flat chains to the setting of G-flat tensor chains. As a corollary we obtain that the groups of normal tensor chains identify with some subgroups of normal chains. Moreover the corresponding natural group isomorphisms are isometric with respect to norms based on the coordinate slicing mass. The coordinate slicing mass of a k-chain is the integral of the mass of its 0-slices along all coordinate-planes of codimension k. The fact that this quantity is equivalent to the usual mass is not straightforward. To prove it, we use the deformation theorem and a partial extension of the restriction operator defined for all chains (not only of finite mass). On the contrary, except in some limit or degenerate cases, the whole groups of tensor chains and of finite mass tensor chains do not identify naturally with subgroups of chains.

math.AP↗
arXiv2022

Tensor rectifiable G-flat chains

A rigidity result for normal rectifiable $k$-chains in $\mathbb{R}^n$ with coefficients in an Abelian normed group is established. Given some decompositions $k=k_1+k_2$, $n=n_1+n_2$ and some rectifiable $k$-chain $A$ in $\mathbb{R}^n$, we consider the properties:(1) The tangent planes to $μ_A$ split as $T_xμ_A=L^1(x)\times L^2(x)$ for some $k_1$-plane $L^1(x)\subset\mathbb{R}^{n_1}$ and some $k_2$-plane $L^2(x)\subset\mathbb{R}^{n_2}$.(2) $A=A_{\vertΣ^1\timesΣ^2}$ for some sets $Σ^1\subset\mathbb{R}^{n_1}$, $Σ^2\subset\mathbb{R}^{n_2}$ such that $Σ^1$ is $k_1$-rectifiable and $Σ^2$ is $k_2$-rectifiable (we say that $A$ is $(k_1,k_2)$-rectifiable).The main result is that for normal chains, (1) implies (2), the converse is immediate. In the proof we introduce the new groups of tensor flat chains (or $(k_1,k_2)$-chains) in $\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}$ which generalize Fleming's $G$-flat chains. The other main tool is White's rectifiable slices theorem. We show that on the one hand any normal rectifiable chain satisfying~(1) identifies with a normal rectifiable $(k_1,k_2)$-chain and that on the other hand any normal rectifiable $(k_1,k_2)$-chain is $(k_1,k_2)$-rectifiable.

math.AP↗
arXiv2022

Rigidity of the ball for an isoperimetric problem with strong capacitary repulsion

We consider a variational problem involving competition between surface tension and charge repulsion. We show that, as opposed to the case of weak (short-range) interactions where we proved ill-posedness of the problem in a previous paper, when the repulsion is stronger the perimeter dominates the capacitary term at small scales. In particular we prove existence of minimizers for small charges as well as their regularity. Combining this with the stability of the ball under small $C^{1,γ}$ perturbations, this ultimately leads to the minimality of the ball for small charges. We cover in particular the borderline case of the $1-$capacity where both terms in the energy are of the same order.

math.AP↗
arXiv2021

Compactness and structure of zero-states for unoriented Aviles-Giga functionals

Motivated by some models of pattern formation involving an unoriented director field in the plane, we study a family of unoriented counterparts to the Aviles-Giga functional. We introduce a nonlinear curl operator for such unoriented vector fields as well as a family of even entropies which we call "trigonometric entropies". Using these tools we show two main theorems which parallel some results in the literature on the classical Aviles-Giga energy. The first is a compactness result for sequences of configurations with uniformly bounded energies. The second is a complete characterization of zero-states, that is, the limit configurations when the energies go to 0. These are Lipschitz continuous away from a locally finite set of points, near which they form either a vortex pattern or a disclination with degree 1/2. The proof is based on a combination of regularity theory together with techniques coming from the study of the Ginzburg-Landau energy. Our methods provide alternative proofs in the classical Aviles-Giga context.

math.AP↗
arXiv2021

On the quadratic random matching problem in two-dimensional domains

We investigate the average minimum cost of a bipartite matching, with respect to the squared Euclidean distance, between two samples of n i.i.d. random points on a bounded Lipschitz domain in the Euclidean plane, whose common law is absolutely continuous with strictly positive H{ö}lder continuous density. We confirm in particular the validity of a conjecture by D. Benedetto and E. Caglioti stating that the asymptotic cost as n grows is given by the logarithm of n multiplied by an explicit constant times the volume of the domain. Our proof relies on a reduction to the optimal transport problem between the associated empirical measures and a Whitney-type decomposition of the domain, together with suitable upper and lower bounds for local and global contributions, both ultimately based on PDE tools. We further show how to extend our results to more general settings, including Riemannian manifolds, and also give an application to the asymptotic cost of the random quadratic bipartite travelling salesperson problem.

math.AP↗
arXiv2021

Existence and stability results for an isoperimetric problem with a non-local interaction of Wasserstein type

The aim of this paper is to prove the existence of minimizers for a variational problem involving the minimization under volume constraint of the sum of the perimeter and a non-local energy of Wasserstein type. This extends previous partial results to the full range of parameters. We also show that in the regime where the perimeter is dominant, the energy is uniquely minimized by balls.

math.AP↗
arXiv2021

A fluctuation result for the displacement in the optimal matching problem

The aim of this paper is to justify in dimensions two and three the ansatz of Caracciolo et al. stating that the displacement in the optimal matching problem is essentially given by the solution to the linearized equation i.e. the Poisson equation. Moreover, we prove that at all mesoscopic scales, this displacement is close in suitable negative Sobolev spaces to a curl-free Gaussian free field. For this we combine a quantitative estimate on the difference between the displacement and the linearized object, which is based on the large-scale regularity theory recently developed in collaboration with F. Otto, together with a qualitative convergence result for the linearized problem.

math.PR↗
arXiv2021

Quantitative linearization results for the Monge-Ampère equation

This paper is about quantitative linearization results for the Monge-Ampère equation with rough data. We develop a large-scale regularity theory and prove that if a measure $μ$ is close to the Lebesgue measure in Wasserstein distance at all scales, then the displacement of the macroscopic optimal coupling is quantitatively close at all scales to the gradient of the solution of the corresponding Poisson equation. The main ingredient we use is a harmonic approximation result for the optimal transport plan between arbitrary measures. This is used in a Campanato iteration which transfers the information through the scales.

math.AP↗
arXiv2020

Convergence of asymptotic costs for random Euclidean matching problems

We investigate the average minimum cost of a bipartite matching between two samples of n independent random points uniformly distributed on a unit cube in d $\ge$ 3 dimensions, where the matching cost between two points is given by any power p $\ge$ 1 of their Euclidean distance. As n grows, we prove convergence, after a suitable renormalization, towards a finite and positive constant. We also consider the analogous problem of optimal transport between n points and the uniform measure. The proofs combine sub-additivity inequalities with a PDE ansatz similar to the one proposed in the context of the matching problem in two dimensions and later extended to obtain upper bounds in higher dimensions.

math.AP↗
arXiv2019

A variational approach to regularity theory in optimal transportation

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.

math.AP↗
arXiv2019

Non-convex functionals penalizing simultaneous oscillations along independent directions: rigidity estimates

We study a family of non-convex functionals $\{\mathcal{E}\}$ on the space of measurable functions$u: Ω_1\timesΩ_2 \subset \mathbb{R}^{n_1}\times\mathbb{R}^{n_2} \to \mathbb{R}$. These functionals vanish on the non-convex subset $S(Ω_1\timesΩ_2)$ formed by functions of the form $u(x_1,x_2)=u_1(x_1)$ or $u(x_1,x_2)=u_2(x_2)$. We investigate under which conditions the converse implication "$\mathcal{E}(u) = 0 \Rightarrow u \in S(Ω_1\timesΩ_2)$" holds. In particular, we show that the answer depends strongly on the smoothness of u. We also obtain quantitative versions of this implication by proving that (at least for some parameters) $\mathcal{E}(u)$ controls in a strong sense the distance of $u$ to $S(Ω_1\timesΩ_2)$.

math.AP↗
arXiv2018

A two-point function approach to connectedness of drops in convex potentials

We establish connectedness of volume constrained minimisers of energies involving surface tensions and convex potentials. By a previous result of McCann, this implies that minimisers are convex in dimension two. This positively answers an old question of Almgren. We also prove convexity of minimisers when the volume constraint is dropped. Our proof is based on the introduction of a new "two-point function" which measures the lack of convexity and which gives rise to a negative second variation of the energy.

math.AP↗