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.     

Scattering and Localization Properties of Highly Oscillatory Potentials

We investigate scattering, localization, and dispersive time decay properties for the one‐dimensional Schrödinger equation with a rapidly oscillating and spatially localized potential , where is periodic and mean zero with respect to y. Such potentials model a microstructured medium. Homogenization theory fails to capture the correct low‐energy (k small) behavior of scattering quantities, e.g., the transmission coefficient as ? tends to zero. We derive an effective potential well such that is small, uniformly for as well as in any bounded subset of a suitable complex strip. Within such a bounded subset, the scaled limit of the transmission coefficient has a universal form, depending on a single parameter, which is computable from the effective potential. A consequence is that if ?, the scale of oscillation of the microstructure potential, is sufficiently small, then there is a pole of the transmission coefficient (and hence of the resolvent) in the upper half‐plane on the imaginary axis at a distance of order from . It follows that the Schrödinger operator has an bound state with negative energy situated a distance from the edge of the continuous spectrum. Finally, we use this detailed information to prove the local energy time decay estimate: where denotes the projection onto the continuous spectral part of . © 2013 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.     

Infinite Speed of Propagation and Regularity of Solutions to the Fractional Porous Medium Equation in General Domains

We study the positivity and regularity of solutions to the fractional porous medium equations in for m > 1 and s ∈ (0,1), with Dirichlet boundary data u = 0 in and nonnegative initial condition . Our first result is a quantitative lower bound for solutions that holds for all positive times t > 0. As a consequence, we find a global Harnack principle stating that for any t > 0 solutions are comparable to d^s/m , where d is the distance to ?Ω. This is in sharp contrast with the local case s = 1, where the equation has finite speed of propagation. After this, we study the regularity of solutions. We prove that solutions are classical in the interior (C ^∞ in x and C ^1,α in t ) and establish a sharp regularity estimate up to the boundary. Our methods are quite general and can be applied to wider classes of nonlocal parabolic equations of the form in Ω, both in bounded and unbounded domains.© 2016 Wiley Periodicals, Inc.     

Uniqueness of Radial Solutions for the Fractional Laplacian

We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian (?Δ)^s with s ? (0,1) for any space dimensions N ≥ 1. By extending a monotonicity formula found by Cabré and Sire , we show that the linear equation has at most one radial and bounded solution vanishing at infinity, provided that the potential V is radial and nondecreasing. In particular, this result implies that all radial eigenvalues of the corresponding fractional Schrödinger operator H = (?Δ)^s + V are simple. Furthermore, by combining these findings on linear equations with topological bounds for a related problem on the upper half‐space , we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation for arbitrary space dimensions N ≥ 1 and all admissible exponents α > 0. This generalizes the nondegeneracy and uniqueness result for dimension N = 1 recently obtained by the first two authors and, in particular, the uniqueness result for solitary waves of the Benjamin‐Ono equation found by Amick and Toland .© 2016 Wiley Periodicals, Inc.     

Entanglement Thresholds for Random Induced States

For a random quantum state on obtained by partial tracing a random pure state on , we consider the question whether it is typically separable or typically entangled. For this problem, we show the existence of a sharp threshold of order roughly . More precisely, for any and for d large enough, such a random state is entangled with very large probability when , and separable with very large probability when . One consequence of this result is as follows: for a system of N identical particles in a random pure state, there is a threshold such that two subsystems of k particles each typically share entanglement if k > k₀, and typically do not share entanglement if k < k₀. Our methods also work for multipartite systems and for “unbalanced” systems such as , . The arguments rely on random matrices, classical convexity, high‐dimensional probability, and geometry of Banach spaces; some of the auxiliary results may be of reference value. © 2013 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.     

Boundary Value Problems for Second‐Order Elliptic Operators Satisfying a Carleson Condition

Let Ω be a Lipschitz domain in , and be a second‐order elliptic operator in divergence form. We establish the solvability of the Dirichlet regularity problem with boundary data in and of the Neumann problem with data for the operator L on Lipschitz domains with small Lipschitz constant. We allow the coefficients of the operator L to be rough, obeying a certain Carleson condition with small norm. These results complete the results of Dindo?, Petermichl, and Pipher (2007), where the Dirichlet problem was considered under the same assumptions, and Dindo? and Rule (2010), where the regularity and Neumann problems were considered on two‐dimensional domains.© 2016 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.     

A Rigidity Result for Overdetermined Elliptic Problems in the Plane

Let be a (locally) Lipschitz function and a domain whose boundary is unbounded and connected. If there exists a positive bounded solution to the overdetermined elliptic problem we prove that Ω is a half‐plane. In particular, we obtain a partial answer to a question raised by H. Berestycki, L. Caffarelli, and L. Nirenberg in 1997.© 2017 Wiley Periodicals, Inc.     

A Liouville Theorem for the Complex Monge-Ampère Equation on Product Manifolds

Let Y be a closed Calabi-Yau manifold. Let ω be the Kähler form of a Ricci-flat Kähler metric on . We prove that if ω is uniformly bounded above and below by constant multiples of , where is the standard flat Kähler form on and ω_Y is any Kähler form on Y, then ω is a product Kähler form up to a certain automorphism of . © 2018 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.     

Unique Continuation at the Boundary for Harmonic Functions in C1 Domains and Lipschitz Domains with Small Constant

Let be a domain, or more generally, a Lipschitz domain with small local Lipschitz constant. In this paper it is shown that if u is a function harmonic in and continuous in , which vanishes in a relatively open subset ; moreover, the normal derivative vanishes in a subset of with positive surface measure; then u is identically zero. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

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.     

Robustness of Liouville Measure under a Family of Stable Diffusions

Consider a C^∞ closed connected Riemannian manifold (M, g) with negative sectional curvature. The unit tangent bundle SM is foliated by the (weak) stable foliation of the geodesic flow. Let Δ^s be the leafwise Laplacian for and let X be the geodesic spray, i.e., the vector field that generates the geodesic flow. For each ρ , the operator generates a diffusion for . We show that, as ρ → − ∞ , the unique stationary probability measure for the leafwise diffusion of converge to the normalized Liouville measure on SM . © 2020 Wiley Periodicals LLC     

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.     

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.     

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.     

On Heavy-Tail Phenomena in Some Large-Deviations Problems

We revisit the proof of the large-deviations principle of Wiener chaoses partially given by Borell and then by Ledoux in its full form. We show that some heavy-tail phenomena observed in large deviations can be explained by the same mechanism as for the Wiener chaoses, meaning that the deviations are created, in a sense, by translations. More precisely, we prove a general large-deviations principle for a certain class of functionals , where is some metric space, under the n -fold probability measure , where α ∈ (0, 2] , for which the large deviations are due to translations. We retrieve, as an application, the large-deviations principles known for the Wigner matrices without Gaussian tails, in works by Bordenave and Caputo on one hand, and the author on the other hand, of the empirical spectral measure, the largest eigenvalue, and traces of polynomials. We also apply our large-deviations result to the last-passage time, which yields a large-deviations principle when the weights follow the law , with α ∈ (0, 1) . © 2020 Wiley Periodicals LLC