A Priori Estimates for High Frequency Scattering by Obstacles of Arbitrary Shape

High frequency estimates for the Dirichlet-to-Neumann and Neumann-to-Dirichlet operators are obtained for the Helmholtz equation in the exterior of bounded obstacles. These a priori estimates are used to study the scattering of plane waves by an arbitrary bounded obstacle and to prove that the total cross section of the scattered wave does not exceed four geometrical cross sections of the obstacle in the limit as the wave number k → ∞. This bound of the total cross section is sharp.     

The hydrodynamic limit for the reaction diffusion equation — An approach in terms of the GPV method

We study the hydrodynamic limit of the reaction diffusion process by means of the GPV technique (Guoet al. ⁽⁴⁾). To this end, we first derivea priori bounds on the moments of the occupation numbers using the local central limit theorem and results of stochastic analysis. The result of De Masi and Presutti⁽²⁾ for the hydrodynamic limit of the reaction diffusion process is generalized here.     

On the existence of steady flows of a Navier–Stokes liquid around a moving rigid body

We prove the existence of a strong solution to the three‐dimensional steady Navier–Stokes equations in the exterior of an obstacle undergoing a rigid motion. Unlike the classical exterior problem for the Navier–Stokes equations, that only takes into account the translational motion of the obstacle, is this case, the obstacle can also rotate. Assuming the total flux of the velocity field through the boundary to be sufficiently small, we first construct approximating solutions in bounded regions Ω_R = Ω∩ {x ∈ ?³:∣x∣< R} invading the liquid domain Ω. A set of estimates independent of R are shown to hold for the approximating solutions which allows to obtain a strong solution by taking the limit R→∞. Copyright © 2004 John Wiley & Sons, Ltd.     

Relaxation‐time limit of the three‐dimensional hydrodynamic model with boundary effects

In this paper, we study three‐dimensional (3D) unipolar and bipolar hydrodynamic models and corresponding drift‐diffusion models from semiconductor devices on bounded domain. Based on the asymptotic behavior of the solutions to the initial boundary value problems with slip boundary condition, we investigate the relation between the 3D hydrodynamic semiconductor models and the corresponding drift‐diffusion models. That is, we discuss the relation‐time limit from the 3D hydrodynamic semiconductor models to the corresponding drift‐diffusion models by comparing the large‐time behavior of these two models. These results can be showed by energy arguments. Copyrightcopyright 2011 John Wiley & Sons, Ltd.     

On the mean value property of finely harmonic and finely hyperharmonic functions

Summary It is shown that a finely superharmonic function in a planar fine domainU is greater than or equal to its lower integral with respect to harmonic measure associated with any bounded finely open setV with fine closure contained inU. Examples are given showing that this result does not extend to dimension 3 or more (unlessf is supposed to be, e.g., lower bounded onV) and also that the integral need not exist.     

First-order correction for the hydrodynamic limit of asymmetric simple exclusion processes in dimension d ≥ 3

It is well-known that the hydrodynamic limit of the asymmetric simple exclusion is governed by a viscousless Burgers equation in the Euler scale [15]. We prove that, in the same scale, the next-order correction is given by a viscous Burgers equation up to a fixed time T for dimension d ≥ 3 provided that the corresponding viscousless Burger equation has a smooth solution up to time T. The diffusion coefficient was characterized via a variation of the Green-Kubo formula by [17, 18, 6]. Within the framework of asymmetric simple exclusion, this provides a rigorous verification in a simplified setting that the correction to the Euler equation is given by the Navier-Stokes equation if the time scale is within the Euler scale. © 1997 John Wiley & Sons, Inc.     

On characterizing Vizing's edge colouring bound

We consider multigraphs G for which equality holds in Vizing's classical edge colouring bound χ′(G)≤Δ + µ, where Δ denotes the maximum degree and µ denotes the maximum edge multiplicity of G. We show that if µ is bounded below by a logarithmic function of Δ, then G attains Vizing's bound if and only if there exists an odd subset S?V(G) with |S|≥3, such that |E[S]|>((|S| ? 1)/2)(Δ + µ ? 1). The famous Goldberg–Seymour conjecture states that this should hold for all µ≥2. We also prove a similar result concerning the edge colouring bound χ′(G)≤Δ + ?µ/?g/2??, due to Steffen (here g denotes the girth of the underlying graph). Finally we give a general approximation towards the Goldberg‐Seymour conjecture in terms of Δ and µ. © 2011 Wiley Periodicals, Inc. J Graph Theory 69:160‐168, 2012     

Convergence of Weighted Empirical Measures

This article considers a system with infinitely many interacting particles, starting with a system of n interacting particles that is described by a system of n stochastic differential equations for the time-varying locations and weights. Any particle in the system interacts with others through the weighted empirical measure Uⁿ formed by the sum of weighted Dirac measures on the n particles. Weak convergence of the weighted empirical measure is studied under suitable conditions, such as bounded initial values, and linear growth of drift and diffusion coefficients. Thereafter, the limit of the weighted empirical measures is identified to be a martingale solution of the infinite interacting system.     

Multigrid methods for H(div) in three dimensions with nonoverlapping domain decomposition smoothers

We design and analyze V‐cycle multigrid methods for an H(div) problem discretized by the lowest‐order Raviart–Thomas hexahedral element. The smoothers in the multigrid methods involve nonoverlapping domain decomposition preconditioners that are based on substructuring. We prove uniform convergence of the V‐cycle methods on bounded convex hexahedral domains (rectangular boxes). Numerical experiments that support the theory are also presented.     

Exponential estimate for reaction diffusion models

Summary. We consider the superposition of a speeded up symmetric simple exclusion process with a Glauber dynamics, which leads to a reaction diffusion equation. Using a method introduced in [Y] based on the study of the time evolution of the H ₋₁ norm, we prove that the mean density of particles on microscopic boxes of size N ^α , for any 12/13<α<1, converges to the solution of the hydrodynamic equation for times up to exponential order in N, provided the initial state is in the basin of attraction of some stable equilibrium of the reaction–diffusion equation. From this result we obtain a lower bound for the escape time of a domain in the basin of attraction of the stable equilibrium point. Received: 3 March 1995 / In revised form: 2 February 1996     

Bounds on cost effective domination numbers

A vertex υ in a set S is said to be cost effective if it is adjacent to at least as many vertices in V\S as it is in S and is very cost effective if it is adjacent to more vertices in V\S than to vertices in S. A dominating set S is (very) cost effective if every vertex in S is (very) cost effective. The minimum cardinality of a (very) cost effective dominating set of G is the (very) cost effective domination number of G. Our main results include a quadratic upper bound on the very cost effective domination number of a graph in terms of its domination number. The proof of this result gives a linear upper bound for hereditarily sparse graphs which include trees. We show that no such linear bound exists for graphs in general, even when restricted to bipartite graphs. Further, we characterize the extremal trees attaining the bound. Noting that the very cost effective domination number is bounded below by the domination number, we show that every value of the very cost effective domination number between these lower and upper bounds for trees is realizable. Similar results are given for the cost effective domination number.     

The Jacobian and the Ginzburg-Landau energy

We study the Ginzburg-Landau functional for , where U is a bounded, open subset of . We show that if a sequence of functions satisfies , then their Jacobians are precompact in the dual of for every . Moreover, any limiting measure is a sum of point masses. We also characterize the -limit of the functionals , in terms of the function space B2V introduced by the authors in [16,17]: we show that I(u) is finite if and only if , and for is equal to the total variation of the Jacobian measure Ju. When the domain U has dimension greater than two, we prove if then the Jacobians are again precompact in for all , and moreover we show that any limiting measure must be integer multiplicity rectifiable. We also show that the total variation of the Jacobian measure is a lower bound for the limit of the Ginzburg-Landau functional. Received: 15 December 2000 / Accepted: 23 January 2001 / Published online: 25 June 2001     

ASYMPTOTIC LIMITS FOR QUANTUM TRAJECTORY MODELS

ABSTRACT

The semi-classical and the inviscid limit in quantum trajectory models given by a one-dimensional steady-state hydrodynamic system for quantum fluids are rigorously performed. The model consists of the momentum equation for the particle density in a bounded domain, with prescribed current density, and the Poisson equation for the electrostatic potential. The momentum equation can be written as a dispersive third-order differential equation which may include viscous terms. It is shown that the semi-classical and inviscid limit commute for sufficiently small data (i.e. current density) corresponding to subsonic states, where the inviscid non-dispersive solution is regular. In addition, we show that these limits do not commute in general. The proofs are based on a reformulation of the problem as a singular second-order elliptic system and on elliptic and W ^1,1 estimates.     

Precoloring extension for K4‐minor‐free graphs

Let G=(V, E) be a graph where every vertex v∈V is assigned a list of available colors L(v). We say that G is list colorable for a given list assignment if we can color every vertex using its list such that adjacent vertices get different colors. If L(v)={1, …, k} for all v∈V then a corresponding list coloring is nothing other than an ordinary k‐coloring of G. Assume that W?V is a subset of V such that G[W] is bipartite and each component of G[W] is precolored with two colors taken from a set of four. The minimum distance between the components of G[W] is denoted by d(W). We will show that if G is K₄‐minor‐free and d(W)≥7, then such a precoloring of W can be extended to a 4‐coloring of all of V. This result clarifies a question posed in 10. Moreover, we will show that such a precoloring is extendable to a list coloring of G for outerplanar graphs, provided that |L(v)|=4 for all v∈V\W and d(W)≥7. In both cases the bound for d(W) is best possible. © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 284‐294, 2009     

Variational problem with an obstacle in ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript> for a class of quadratic functionals

  A variational problem with an obstacle for a certain class of quadratic functionals is considered. Admissible vector-valued functions are assumed to satisfy the Dirichlet boundary condition, and the obstacle is a given smooth (N − 1)-dimensional surface S in ℝ ^N . The surface S is not necessarily bounded. It is proved that any minimizer u of such an obstacle problem is a partially smooth function up to the boundary of a prescribed domain. It is shown that the (n − 2)-Hausdorff measure of the set of singular points is zero. Moreover, u is a weak solution of a quasilinear system with two kinds of quadratic nonlinearities in the gradient. This is proved by a local penalty method. Bibliography: 25 titles. Dedicated to V. A. Solonnikov on the occasion of his jubilee Published in Zapiski Nauchnykh. Seminarov POMI, Vol. 362, 2008, pp. 15–47.     

An iterative method for backward time‐fractional diffusion problem

The aim of this work is to solve the backward problem for a time‐fractional diffusion equation with variable coefficients in a general bounded domain. The problem is ill‐posed in L ² norm sense. An iteration scheme is proposed to obtain a regularized solution. Two kinds of convergence rates are obtained using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Numerical examples in one‐dimensional and two‐dimensional cases are provided to show the effectiveness of the proposed methods. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 2029–2041, 2014     

Direct algorithm for the solution of two‐sided obstacle problems with M‐matrix

A direct algorithm for the solution to the affine two‐sided obstacle problem with an M‐matrix is presented. The algorithm has the polynomial bounded computational complexity O(n³) and is more efficient than those in (Numer. Linear Algebra Appl. 2006; 13 :543–551). Copyright © 2010 John Wiley & Sons, Ltd.     

Approximating Euclidean distances by small degree graphs

Given an undirected edge-weighted graphG=(V,E), a subgraphG′=(V,E′) is at-spanner ofG if, for everyu, v∈V, the weighted distance betweenu andv inG′ is at mostt times the weighted distance betweenu andv inG. We consider the problem of approximating the distances among points of a Euclidean metric space: given a finite setV of points in ℝd, we want to construct a sparset-spanner of the complete weighted graph induced byV. The weight of an edge in these graphs is the Euclidean distance between the endpoints of the edge. We show by a simple greedy argument that, for anyt>1 and anyV ⊂ ℝd, at-spannerG ofV exists such thatG has degree bounded by a function ofd andt. The analysis of our bounded degree spanners improves over previously known upper bounds on the minimum number of edges of Euclideant-spanners, even compared with spanners of boundedaverage degree. Our results answer two open problems, one proposed by Vaidya and the other by Keil and Gutwin. The main result of the paper concerns the case of dimensiond=2. It is fairly easy to see that, for somet (t≥7.6),t-spanners of maximum degree 6 exist for any set of points in the Euclidean plane, but it was not known that degree 5 would suffice. We prove that for some (fixed)t, t-spanners of degree 5 exist for any set of points in the plane. We do not know if 5 is the best possible upper bound on the degree. This research was supported by Conselho Nacional de Desenvolvimento Cientifico e Tecnológico, Proc 203039/87.4 (Brazil).     

Quantitative Derivation of the Gross‐Pitaevskii Equation

Starting from first‐principle many‐body quantum dynamics, we show that the dynamics of Bose‐Einstein condensates can be approximated by the time‐dependent nonlinear Gross‐Pitaevskii equation, giving a bound on the rate of the convergence. Initial data are constructed on the bosonic Fock space applying an appropriate Bogoliubov transformation on a coherent state with expected number of particles N. The Bogoliubov transformation plays a crucial role; it produces the correct microscopic correlations among the particles. Our analysis shows that, on the level of the one‐particle reduced density, the form of the initial data is preserved by the many‐body evolution, up to a small error that vanishes as N^?1/2 in the limit of large N.© 2015 Wiley Periodicals, Inc.     

Extremal graphs having no matching cuts

A graph G = (V, E) is matching immune if there is no matching cut in G. We show that for any matching immune graph G, |E|≥?3(|V|?1)/2?. This bound is tight, as we define operations that construct, from a given vertex, exactly the class of matching immune graphs that attain the bound. © 2011 Wiley Periodicals, Inc. J Graph Theory 69:206‐222, 2012 .