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.     

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.     

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.     

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.     

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.     

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.     

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.     

Enclosings of decompositions of complete multigraphs in 2‐factorizations

Let k, m, n, λ, and μ be positive integers. A decomposition of into edge‐disjoint subgraphs is said to be enclosed by a decomposition of into edge‐disjoint subgraphs if and, after a suitable labeling of the vertices in both graphs, is a subgraph of and is a subgraph of for all . In this paper, we continue the study of enclosings of given decompositions by decompositions that consist of spanning subgraphs. A decomposition of a graph is a 2‐factorization if each subgraph is 2‐regular and spanning, and is Hamiltonian if each subgraph is a Hamiltonian cycle. We give necessary and sufficient conditions for the existence of a 2‐factorization of that encloses a given decomposition of whenever and . We also give necessary and sufficient conditions for the existence of a Hamiltonian decomposition of that encloses a given decomposition of whenever and either or and , or , , and .     

Some properties of infinite factorials

With the Axiom of Choice , for any infinite cardinal but, without , we cannot conclude any relationship between and for an arbitrary infinite cardinal . In this paper, we give some properties of in the absence of and compare them to those of for an infinite cardinal . Among our results, we show that “ for any infinite cardinal and any natural number n” is provable in although “ for any infinite cardinal ” is not.     

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.     

Eigenvalues of Robin Laplacians in infinite sectors

For , let denote the infinite planar sector of opening 2α, and be the Laplacian in , , with the Robin boundary condition , where stands for the outer normal derivative and . The essential spectrum of does not depend on the angle α and equals , and the discrete spectrum is non‐empty if and only if . In this case we show that the discrete spectrum is always finite and that each individual eigenvalue is a continous strictly increasing function of the angle α. In particular, there is just one discrete eigenvalue for . As α approaches 0, the number of discrete eigenvalues becomes arbitrary large and is minorated by with a suitable , and the nth eigenvalue of behaves as and admits a full asymptotic expansion in powers of α². The eigenfunctions are exponentially localized near the origin. The results are also applied to δ‐interactions on star graphs.     

Littlewood–Paley characterizations of higher‐order Sobolev spaces via averages on balls

In this article, the authors characterize higher‐order Sobolev spaces , with , and , or with , and , via the Lusin area function and the Littlewood–Paley ‐function in terms of ball averages, where denotes the maximal integer not greater than . Moreover, the authors also complement the above results in the endpoint cases of p via establishing some weak type estimates. These improve and develop the corresponding known results for Sobolev spaces with smoothness order .     

Some characterizations of BLO space

For , a real‐valued function belongs to space if In this paper, we establish a version of John–Nirenberg inequality suitable for the space with . As a corollary, it is proved that spaces are independent of the scale in sense of norm. Also, we characterize the space through weighted Lebesgue spaces and variable Lebesgue spaces, respectively.     

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.     

Ramsey Results for Cycle Spectra

Let denote the set of lengths of cycles of a graph G of order n and let denote the complement of G. We show that if , then contains all odd ? with and all even ? with , where and denote the maximum odd and the maximum even integer in , respectively. From this we deduce that the set contains at least integers, which is sharp.     

The Brezis–Nirenberg problem for fractional elliptic operators

Let be a uniformly elliptic operator in divergence form in a bounded open subset Ω of . We study the effect of the operator on the existence and nonexistence of positive solutions of the nonlocal Brezis–Nirenberg problem where denotes the fractional power of with zero Dirichlet boundary values on , , and λ is a real parameter. By assuming for all and near some point , we prove existence theorems for any , where denotes the first Dirichlet eigenvalue of . Our existence result holds true for and in the interior case () and for and in the boundary case (). Nonexistence for star‐shaped domains is obtained for any .     

On Rooted Packings,Decompositions, and Factors of Graphs

In this article, we study so‐called rooted packings of rooted graphs. This concept is a mutual generalization of the concepts of a vertex packing and an edge packing of a graph. A rooted graph is a pair , where G is a graph and . Two rooted graphs and are isomorphic if there is an isomorphism of the graphs G and H such that S is the image of T in this isomorphism. A rooted graph is a rooted subgraph of a rooted graph if H is a subgraph of G and . By a rooted ‐packing into a rooted graph we mean a collection of rooted subgraphs of isomorphic to such that the sets of edges are pairwise disjoint and the sets are pairwise disjoint. In this article, we concentrate on studying maximum ‐packings when H is a star. We give a complete classification with respect to the computational complexity status of the problems of finding a maximum ‐packing of a rooted graph when H is a star. The most interesting polynomial case is the case when H is the 2‐edge star and S contains the center of the star only. We prove a min–max theorem for ‐packings in this case.     

Many symmetrically indivisible structures

A structure in a first‐order language is indivisible if for every colouring of its universe M in two colours, there is a monochromatic such that . Additionally, we say that is symmetrically indivisible if can be chosen to be symmetrically embedded in (that is, every automorphism of can be extended to an automorphism of ). In the following paper we give a general method for constructing new symmetrically indivisible structures out of existing ones. Using this method, we construct many non‐isomorphic symmetrically indivisible countable structures in given (elementary) classes and answer negatively the following question from 6 : Let be a symmetrically indivisible structure in a language . Let . Is symmetrically indivisible?