Closing Aubry Sets I

Given a Tonelli Hamiltonian H:T*M → ? of class C^k, with k ≥ 2, we prove the following results: (1) Assume there exist a recurrent point of the projected Aubry set and a critical viscosity subsolution u such that u is a C¹ critical solution in an open neighborhood of the positive orbit of . Suppose further that u is “C² at ”. Then there exists a C^k potential V : M → ?, small in C²‐topology, for which the Aubry set of the new Hamiltonian H + V is either an equilibrium point or a periodic orbit. (2) If M is two dimensional, (1) holds replacing “C¹ critical solution + C² at ” by “CM³ critical subsolution”. These results can be considered as a first step through the attempt of proving the Mañé's conjecture in C²‐topology. In a second paper [27], we will generalize (2) to arbitrary dimension. Moreover, such an extension will need the introduction of some new techniques, which will allow us to prove in [27] the Mañé's density conjecture in C¹‐topology. Our proofs are based on a combination of techniques coming from finite‐dimensional control theory and Hamilton‐Jacobi theory, together with some of the ideas that were used to prove C¹‐closing lemmas for dynamical systems.© 2014 Wiley Periodicals, Inc.     

Partial Regularity for Singular Solutions to the Monge‐Ampère Equation

We prove that solutions to the Monge‐Ampère inequality in ?ⁿ are strictly convex away from a singular set of Hausdorff (n‐1)‐dimensional measure zero. Furthermore, we show this is optimal by constructing solutions to det D²u = 1 with singular set of Hausdorff dimension as close as we like to n‐1. As a consequence we obtain W^2,1 regularity for the Monge‐Ampère equation with bounded right‐hand side and unique continuation for the Monge‐Ampère equation with sufficiently regular right‐hand side. © 2015 Wiley Periodicals, Inc.     

Volume Estimates on the Critical Sets of Solutions to Elliptic PDEs

In this paper we study solutions to elliptic linear equations either on or a Riemannian manifold, under the assumption that the coefficient functions a^ij are Lipschitz bounded. We focus our attention on the critical set and the singular set , and more importantly on effective versions of these. Currently, with just the Lipschitz regularity of the coefficients, the strongest results in the literature say that the singular set is (n –2)–dimensional; however, at this point it has not even been shown that unless the coefficients are smooth. Fundamentally, this is due to the need of an ?‐regularity theorem that requires higher smoothness of the coefficients as the frequency increases. We introduce new techniques for estimating the critical and singular set, which avoids the need of any such ?‐regularity. Consequently, we prove that if the frequency of u is bounded by Λ, then we have the estimates and , depending on whether the equation is critical or not. More importantly, we prove corresponding estimates for the effective critical and singular sets. Even under the assumption of real analytic coefficients these results are much sharper than those currently in the literature. We also give applications of the technique to give estimates on the volume of the nodal set of solutions and estimates for the corresponding eigenvalue problem.© 2017 Wiley Periodicals, Inc.     

The Heat Equation Shrinks Ising Droplets to Points

Let be a bounded, smooth enough domain of ?². For L > 0 consider the continuous‐time, zero‐temperature heat bath stochastic dynamics for the nearest‐neighbor Ising model on (?/L)² (the square lattice with lattice spacing 1/L) with initial condition such that σ_x =?1 if x ? and σ_x = + 1 otherwise. We prove the following classical conjecture due to H. Spohn: In the diffusive limit where time is rescaled by L² and L → ∞, the boundary of the droplet of “‐” spins follows a deterministic anisotropic curve‐shortening flow such that the normal velocity is given by the local curvature times an explicit function of the local slope. Locally, in a suitable reference frame, the evolution of the droplet boundary follows the one‐dimensional heat equation. To our knowledge, this is the first proof of mean‐curvature‐type droplet shrinking for a lattice model with genuine microscopic dynamics. An important ingredient is in our forthcoming work, where the case of convex was solved. The other crucial point in the proof is obtaining precise regularity estimates on the deterministic curve‐shortening flow. This builds on geometric and analytic ideas of Grayson, Gage and Hamilton, Gage and Li, Chou and Zhu, and others.© 2015 Wiley Periodicals, Inc.     

Free Boundary Regularity in the Parabolic Fractional Obstacle Problem

The parabolic obstacle problem for the fractional Laplacian naturally arises in American option models when the asset prices are driven by pure‐jump Lévy processes. In this paper we study the regularity of the free boundary. Our main result establishes that, when , the free boundary is a C^1,α graph in x and t near any regular free boundary point . Furthermore, we also prove that solutions u are C^{1 + s} in x and t near such points, with a precise expansion of the form (1) with , and . © 2018 Wiley Periodicals, Inc.     

A Classification of Isolated Singularities of Elliptic Monge‐Ampére Equations in Dimension Two

Let ?₁ denote the space of solutions z(x,y) to an elliptic, real analytic Monge‐Ampére equation whose graphs have a non‐removable isolated singularity at the origin. We prove that ?₁ is in one‐to‐one correspondence with ?₂ × ?₂, where ?₂ is a suitable subset of the class of regular, real analytic, strictly convex Jordan curves in ?₂. We also describe the asymptotic behavior of solutions of the Monge‐Ampére equation in the C^k‐smooth case, and a general existence theorem for isolated singularities of analytic solutions of the more general equation .© 2015 Wiley Periodicals, Inc.     

Cycle Factors and Renewal Theory

For which values of k does a uniformly chosen 3‐regular graph G on n vertices typically contain n/k vertex‐disjoint k ‐cycles (a k ‐cycle factor)? To date, this has been answered for k = n and for k ? log n ; the former, the Hamiltonicity problem, was finally answered in the affirmative by Robinson and Wormald in 1992, while the answer in the latter case is negative since with high probability (w.h.p.) most vertices do not lie on k ‐cycles. A major role in our study of this problem is played by renewal processes without replacement, where one wishes to estimate the probability that in a uniform permutation of a given set of positive integers, the partial sums hit a designated target integer. Using sharp tail estimates for these renewal processes, which may be of independent interest, we settle the cycle factor problem completely: the “threshold” for a k ‐cycle factor in G as above is κ ₀ log₂ n with . To be precise, G contains a k ‐cycle factor w.h.p. if and w.h.p. does not contain one if . Thus, for most values of n the threshold concentrates on the single integer K ₀(n ). As a byproduct, we confirm the “comb conjecture,” an old problem concerning the embedding of certain spanning trees in the random graph (n,p ).© 2015 Wiley Periodicals, Inc.     

Strong Asymptotics of the Orthogonal Polynomials with Respect to a Measure Supported on the Plane

We consider the orthogonal polynomials with respect to the measure over the whole complex plane. We obtain the strong asymptotic of the orthogonal polynomials in the complex plane and the location of their zeros in a scaling limit where n grows to infinity with N . The asymptotics are described in terms of three (probability) measures associated with the problem. The first measure is the limit of the counting measure of zeros of the polynomials, which is captured by the g‐function much in the spirit of ordinary orthogonal polynomials on the real line. The second measure is the equilibrium measure that minimizes a certain logarithmic potential energy, supported on a region K of the complex plane. The third measure is the harmonic measure of K ^c with a pole at ∞ . This appears as the limit of the probability measure given (up to the normalization constant) by the squared modulus of the n^th orthogonal polynomial times the orthogonality measure, i.e., The compact region K that is the support of the second measure undergoes a topological transition under the variation of the parameter in a double scaling limit near the critical point given by we observe the Hastings‐McLeod solution to Painlevé II in the asymptotics of the orthogonal polynomials. © 2014 Wiley Periodicals, Inc.     

Infinitely many solutions for p‐Kirchhoff equation with concave–convex nonlinearities in

In this paper, we study the existence of infinitely many solutions to p‐Kirchhoff‐type equation (0.1) where f(x,u) = λh₁(x)|u|^{m ? 2}u + h₂(x)|u|^{q ? 2}u,a≥0,μ > 0,τ > 0,λ≥0 and . The potential function verifies , and h₁(x),h₂(x) satisfy suitable conditions. Using variational methods and some special techniques, we prove that there exists λ₀>0 such that problem 0.1 admits infinitely many nonnegative high‐energy solutions provided that λ∈[0,λ₀) and . Also, we prove that problem 0.1 has at least a nontrivial solution under the assumption f(x,u) = h₂|u|^{q ? 2}u,p < q< min{p*,p(τ + 1)} and has infinitely many nonnegative solutions for f(x,u) = h₁|u|^{m ? 2}u,1 < m < p. Copyright © 2015 John Wiley & Sons, Ltd.     

Global bounded weak solutions to a degenerate quasilinear chemotaxis system with rotation

This paper deals with the quasilinear Keller–Segel system with rotation where is a bounded domain with smooth boundary, D(u) is supposed to be sufficiently smooth and satisfies D(u)≥D₀u^{m ? 1}(m≥1) and D(u)≤D₁(u + 1)^{K ? m}u^{m ? 1}(K≥1) for all u≥0 with some positive constants D₀ and D₁, and f(u) is assumed to be smooth enough and non‐negative for all u≥0 and f(0) = 0, while S(u,v,x) = (s_ij)_{n × n} is a matrix with and with l≥2, where is nondecreasing on [0,∞). It is proved that when , the system possesses at least one global and bounded weak solution for any sufficiently smooth non‐negative initial data. Copyright © 2015 John Wiley & Sons, Ltd.     

ShapeFit: Exact Location Recovery from Corrupted Pairwise Directions

Let t₁,…,t_n ∊ ℝ^d for d ≥ 2 and consider the location recovery problem: given a subset of pairwise direction observations , where a constant fraction of these observations are arbitrarily corrupted, find up to a global translation and scale. We propose a novel algorithm for the location recovery problem, which consists of a simple convex program over $dn$ real variables. We prove that this program recovers a set of n i.i.d. Gaussian locations exactly and with high probability if the observations are given by an Erdős‐Rényi graph, with d large enough, and provided that at most a constant fraction of observations involving any particular location are adversarially corrupted. We also prove that the program exactly recovers Gaussian locations for d = 3 if the fraction of corrupted observations at each location is, up to poly‐logarithmic factors, at most a constant. Both of these recovery theorems are based on a set of deterministic conditions that we prove are sufficient for exact recovery. © 2017 Wiley Periodicals, Inc.     

What Is Variable Bandwidth?

We propose a new notion of variable bandwidth that is based on the spectral subspaces of an elliptic operator where p > 0 is a strictly positive function. Denote by the orthogonal projection of A_p corresponding to the spectrum of A_p in ; the range of this projection is the space of functions of variable bandwidth with spectral set in Λ. We will develop the basic theory of these function spaces. First, we derive (nonuniform) sampling theorems; second, we prove necessary density conditions in the style of Landau. Roughly, for a spectrum the main results say that, in a neighborhood of , a function of variable bandwidth behaves like a band‐limited function with local bandwidth . Although the formulation of the results is deceptively similar to the corresponding results for classical band‐limited functions, the methods of proof are much more involved. On the one hand, we use the oscillation method from sampling theory and frame‐theoretic methods; on the other hand, we need the precise spectral theory of Sturm‐Liouville operators and the scattering theory of one‐dimensional Schrödinger operators. © 2017 Wiley Periodicals, Inc.     

Multi‐to One‐Dimensional Optimal Transport

We consider the Monge‐Kantorovich problem of transporting a probability density on to another on the line, so as to optimize a given cost function. We introduce a nestedness criterion relating the cost to the densities, under which it becomes possible to solve this problem uniquely by constructing an optimal map one level set at a time. This map is continuous if the target density has connected support. We use level‐set dynamics to develop and quantify a local regularity theory for this map and the Kantorovich potentials solving the dual linear program. We identify obstructions to global regularity through examples. More specifically, fix probability densities f and g on open sets and with . Consider transporting f onto g so as to minimize the cost . We give a nondegeneracy condition on that ensures the set of x paired with [g‐a.e.] y ∈ Y lie in a codimension‐n submanifold of X. Specializing to the case m > n = 1, we discover a nestedness criterion relating s to (f,g) that allows us to construct a unique optimal solution in the form of a map . When and g and f are bounded, the Kantorovich dual potentials (u,υ) satisfy , and the normal velocity V of with respect to changes in y is given by . Positivity of V locally implies a Lipschitz bound on f; moreover, if intersects transversally. On subsets where this nondegeneracy, positivity, and transversality can be quantified, for each integer the norms of and are controlled by these bounds, , and the smallness of . We give examples showing regularity extends from $X to part of , but not from Y to . We also show that when s remains nested for all (f,g), the problem in reduces to a supermodular problem in . © 2017 Wiley Periodicals, Inc.     

Regularity for Shape Optimizers: The Nondegenerate Case

We consider minimizers of (1) where F is a function strictly increasing in each parameter, and is the k^th Dirichlet eigenvalue of Ω. Our main result is that the reduced boundary of the minimizer is composed of C^1,α graphs and exhausts the topological boundary except for a set of Hausdorff dimension at most n – 3. We also obtain a new regularity result for vector‐valued Bernoulli‐type free boundary problems.© 2018 Wiley Periodicals, Inc.     

On the Location of Maxima of Solutions of Schrödinger's Equation

We prove an inequality with applications to solutions of the Schrödinger equation. There is a universal constant c > 0 such that if is simply connected, vanishes on the boundary ∂Ω, and |u| assumes a maximum in , then (1) It was conjectured by Pólya and Szegő (and proven, independently, by Makai and Hayman) that a membrane vibrating at frequency λ contains a disk of size . Our inequality implies a refined result: the point on the membrane that achieves the maximal amplitude is at distance from the boundary. We also give an extension to higher dimensions (generalizing results of Lieb and of Georgiev and Mukherjee): if u solves on with Dirichlet boundary conditions, then the ball B with radius centered at the point in which |u| assumes a maximum is almost fully contained in Ω in the sense that © 2018 Wiley Periodicals, Inc.     

Global existence and optimal decay rates of solutions to conservation laws with diffusion‐type terms

We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion‐type source term. Based on a low‐frequency and high‐frequency decomposition, Green's function method and the classical energy method, we not only obtain L² time‐decay estimates but also establish the global existence of solutions to Cauchy problem when the initial data u₀(x) satisfies the smallness condition on , but not on . Furthermore, by taking a time‐frequency decomposition, we obtain the optimal decay estimates of solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Convergence of the Two‐Dimensional Dynamic Ising‐Kac Model to

The Ising‐Kac model is a variant of the ferromagnetic Ising model in which each spin variable interacts with all spins in a neighborhood of radius γ ^{? 1} for around its base point. We study the Glauber dynamics for this model on a discrete two‐dimensional torus for a system size and for an inverse temperature close to the critical value of the mean field model. We show that the suitably rescaled coarse‐grained spin field converges in distribution to the solution of a nonlinear stochastic partial differential equation. This equation is the dynamic version of the quantum field theory, which is formally given by a reaction‐diffusion equation driven by an additive space‐time white noise. It is well‐known that in two spatial dimensions such equations are distribution valued and a Wick renormalization has to be performed in order to define the nonlinear term. Formally, this renormalization corresponds to adding an infinite mass term to the equation. We show that this need for renormalization for the limiting equation is reflected in the discrete system by a shift of the critical temperature away from its mean field value.© 2016 by the authors. Communications on Pure and Applied Mathematics is published by Wiley Periodicals, Inc., on behalf of the Courant Institute of Mathematics.     

Dissipative Euler Flows with Onsager‐Critical Spatial Regularity

For any ? > 0 we show the existence of continuous periodic weak solutions v of the Euler equations that do not conserve the kinetic energy and belong to the space ; namely, x ? v (x,t) is ??ε‐Hölder continuous in space at a.e. time t and the integral is finite. A well‐known open conjecture of L. Onsager claims that such solutions exist even in the class .© 2016 Wiley Periodicals, Inc.     

The existence of ‐bounded solution operators of the thermoelastic plate equation with Dirichlet boundary conditions

We consider the linearized thermoelastic plate equation with the Dirichlet boundary condition in a general domain Ω, given by with the initial condition u|_(t=0)=u₀, u_t|_(t=0)=u₁, and θ|_(t=0)=θ₀ in Ω and the boundary condition u=∂_νu=θ=0 on Γ, where u=u(x,t) denotes a vertical displacement at time t at the point x=(x₁,⋯,x_n)∈Ω, while θ=θ(x,t) describes the temperature. This work extends the result obtained by Naito and Shibata that studied the problem in the half‐space case. We prove the existence of ‐bounded solution operators of the corresponding resolvent problem. Then, the generation of C₀ analytic semigroup and the maximal L_p‐L_q‐regularity of time‐dependent problem are derived.     

Global Existence of Weak Solutions of the Nematic Liquid Crystal Flow in Dimension Three

For any bounded smooth domain , we establish the global existence of a weak solution of the initial boundary value (or the Cauchy) problem of the simplified Ericksen‐Leslie system LLF modeling the hydrodynamic flow of nematic liquid crystals for any initial and boundary (or Cauchy) data , with (the upper hemisphere). Furthermore, (u,d) satisfies the global energy inequality.© 2016 Wiley Periodicals, Inc.