POINCARE INEQUALITIES IN WEIGHTED SOBOLEV SPACES

The weighted Poincaréinequalities in weighted Sobolev spaces are discussed, and the necessary and sufficient conditions for them to hold are given.That is,the Poincaréinequalities hold if,and only if,the ball measure of non-compactness of the natural embedding of weighted Sobolev spaces is less than 1.     

Best constants in Sobolev inequalities for ultraspherical polynomials

A family of sharp Sobolev-type inequalities for functions on the classical measure spaces associated with the ultraspherical or Gegenbauer polynomials is obtained. These estimates generalize the Sobolev inequalities for the n-sphere S ⁿ given by Beckner, and are derived from a sharp Sobolev inequality for functions on the real line. Spectral considerations allow these estimates to be expressed as multiplier inequalities for functions which have expansions in terms of Gegenbauer polynomials.     

Fractional Hardy–Lieb–Thirring and Related Inequalities for Interacting Systems

We prove analogues of the Lieb–Thirring and Hardy–Lieb–Thirring inequalities for many-body quantum systems with fractional kinetic operators and homogeneous interaction potentials, where no anti-symmetry on the wave functions is assumed. These many-body inequalities imply interesting one-body interpolation inequalities, and we show that the corresponding one- and many-body inequalities are actually equivalent in certain cases.     

On the Robin-Transmission Boundary Value Problems for the Nonlinear Darcy–Forchheimer–Brinkman and Navier–Stokes Systems

The purpose of this paper is to study a boundary value problem of Robin-transmission type for the nonlinear Darcy–Forchheimer–Brinkman and Navier–Stokes systems in two adjacent bounded Lipschitz domains in \({{\mathbb{R}}^{n} (n\in \{2,3\})}\), with linear transmission conditions on the internal Lipschitz interface and a linear Robin condition on the remaining part of the Lipschitz boundary. We also consider a Robin-transmission problem for the same nonlinear systems subject to nonlinear transmission conditions on the internal Lipschitz interface and a nonlinear Robin condition on the remaining part of the boundary. For each of these problems we exploit layer potential theoretic methods combined with fixed point theorems in order to show existence results in Sobolev spaces, when the given data are suitably small in \({L^2}\)-based Sobolev spaces or in some Besov spaces. For the first mentioned problem, which corresponds to linear Robin and transmission conditions, we also show a uniqueness result. Note that the Brinkman–Forchheimer-extended Darcy equation is a nonlinear equation that describes saturated porous media fluid flows.     

Sharp Trace Hardy–Sobolev-Maz’ya Inequalities and the Fractional Laplacian

In this work we establish trace Hardy and trace Hardy–Sobolev–Maz’ya inequalities with best Hardy constants for domains satisfying suitable geometric assumptions such as mean convexity or convexity. We then use them to produce fractional Hardy–Sobolev–Maz’ya inequalities with best Hardy constants for various fractional Laplacians. In the case where the domain is the half space, our results cover the full range of the exponent ${s \in}$ (0, 1) of the fractional Laplacians. In particular, we give a complete answer in the L ² setting to an open problem raised by Frank and Seiringer (“Sharp fractional Hardy inequalities in half-spaces,” in Around the research of Vladimir Maz’ya. International Mathematical Series, 2010).     

Weighted Sobolev Spaces for a Scalar Model of the Stationary Oseen Equations in $$\mathbb{R}^{3}$$

This paper is devoted to a scalar model of the Oseen equations, a linearized form of the Navier–Stokes equations. To control the behavior of functions at infinity, the problem is set in weighted Sobolev spaces including anisotropic weights. In a first step, some weighted Poincaré-type inequalities are obtained. In a second step, we establish existence, uniqueness and regularity results.     

On the Blow-up Criterion of 3D-NSE in Sobolev–Gevrey Spaces

In Benameur (Methods Appl 103:87–97, 2014), Benameur proved a blow-up result of the non regular solution of (NSE) in the Sobolev–Gevrey spaces. In this paper we improve this result, precisely we give an exponential type explosion in Sobolev–Gevrey spaces with less regularity on the initial condition. Fourier analysis and standard techniques are used.     

Convex Sobolev Inequalities Derived from Entropy Dissipation

We study families of convex Sobolev inequalities, which arise as entropy–dissipation relations for certain linear Fokker–Planck equations. Extending the ideas recently developed by the first two authors, a refinement of the Bakry–émery method is established, which allows us to prove non-trivial inequalities even in situations where the classical Bakry–émery criterion fails. The main application of our theory concerns the linearized fast diffusion equation in dimensions d ≧ 1, which admits a Poincaré, but no logarithmic Sobolev inequality. We calculate bounds on the constants in the interpolating convex Sobolev inequalities, and prove that these bounds are sharp on a specified range. In dimension d = 1, our estimates improve the corresponding results that can be obtained by the measure-theoretic techniques of Barthe and Roberto. As a by-product, we give a short and elementary alternative proof of the sharp spectral gap inequality first obtained by Denzler and McCann. In further applications of our method, we prove convex Sobolev inequalities for a mean field model for the redistribution of wealth in a simple market economy, and the Lasota model for blood cell production.     

Applications of the division theorem inH s

We prove a version of the division theorem in Sobolev spaces with an estimate of the constant ass tends to infinity. We then apply it to derive spatial decay estimates for time-periodic solutions of linear wave equations in one space dimension and to prove that the space of decaying solutions is finite-dimensional. The main point is to show that some of the arguments used to analyze embedded eigenvalues of Schrödinger operators can be extended to cases where positivity arguments are not available. This has implications for nonlinear Klein-Gordon equations. A different approach, based on the proof of the stable manifold theorem, is also worked out, under slightly different assumptions.     

Elliptic Equations in Divergence Form with Partially BMO Coefficients

The solvability in Sobolev spaces is proved for divergence form second order elliptic equations in the whole space, a half space, and a bounded Lipschitz domain. For equations in the whole space or a half space, the leading coefficients a ^ij are assumed to be only measurable in one direction and have locally small BMO semi-norms in the other directions. For equations in a bounded domain, additionally we assume that a ^ij have small BMO semi-norms in a neighborhood of the boundary of the domain. We give a unified approach of both the Dirichlet boundary problem and the conormal derivative problem. We also investigate elliptic equations in Sobolev spaces with mixed norms under the same assumptions on the coefficients.     

New Conservation Laws of Energy and C-D Inequalities in Continua Without Microstructure

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A consistent nonlocal scheme based on filters for the homogenization of heterogeneous linear materials with non-separated scales

In this work, the question of homogenizing linear elastic, heterogeneous materials with periodic microstructures in the case of non-separated scales is addressed. A framework if proposed, where the notion of mesoscopic strain and stress fields are defined by appropriate integral operators which act as low-pass filters on the fine scale fluctuations. The present theory extends the classical linear homogenization by substituting averaging operators by integral operators, and localization tensors by nonlocal operators involving appropriate Green functions. As a result, the obtained constitutive relationship at the mesoscale appears to be nonlocal. Compared to nonlocal elastic models introduced from a phenomenological point of view, the nonlocal behavior has been fully derived from the study of the microstructure. A discrete version of the theory is presented, where the mesoscopic strain field is approximated as a linear combination of basis functions. It allows computing the mesoscopic nonlocal operator by means of a finite number of transformation tensors, which can be computed numerically on the unit cell.     

How to Approximate the Heat Equation with Neumann Boundary Conditions by Nonlocal Diffusion Problems

We present a model for nonlocal diffusion with Neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary. We study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero. We prove that the solutions of this family of problems converge to a solution of the heat equation with Neumann boundary conditions.     

Criteria for the Existence and Lower Bounds of Principal Eigenvalues of Time Periodic Nonlocal Dispersal Operators and Applications

The current paper is concerned with the spectral theory, in particular, the principal eigenvalue theory, of nonlocal dispersal operators with time periodic dependence, and its applications. Nonlocal and random dispersal operators are widely used to model diffusion systems in applied sciences and share many properties. There are also some essential differences between nonlocal and random dispersal operators, for example, a smooth random dispersal operator always has a principal eigenvalue, but a smooth nonlocal dispersal operator may not have a principal eigenvalue. In this paper, we first establish criteria for the existence of principal eigenvalues of time periodic nonlocal dispersal operators with Dirichlet type, Neumann type, or periodic type boundary conditions. It is shown that a time periodic nonlocal dispersal operator possesses a principal eigenvalue provided that the nonlocal dispersal distance is sufficiently small, or the time average of the underlying media satisfies some vanishing condition with respect to the space variable at a maximum point or is nearly globally homogeneous with respect to the space variable. Next we obtain lower bounds of the principal spectrum points of time periodic nonlocal dispersal operators in terms of the corresponding time averaged problems. Finally we discuss the applications of the established principal eigenvalue theory to time periodic Fisher or KPP type equations with nonlocal dispersal and prove that such equations are of monostable feature, that is, if the trivial solution is linearly unstable, then there is a unique time periodic positive solution which is globally asymptotically stable.     

On the First Eigenvalues of Free Vibrating Membranes in Conformal Regular Domains

In 1961 G. Polya published a paper about the eigenvalues of vibrating membranes. The “free vibrating membrane” corresponds to the Neumann–Laplace operator in bounded plane domains. In this paper we obtain estimates for the firstnon-trivial eigenvalue of this operator in a large class of domains that we call conformal regular domains. This class includes convex domains, John domains etc. On the basis of our estimates we conjecture that the eigenvalues of the Neumann–Laplace operator depend on the hyperbolic metrics of plane domains. We propose a new method for the estimates which is based on weighted Poincaré–Sobolev inequalities, obtained by the authors recently.     

Fractional-order relaxation laws in non-linear viscoelasticity

Viscoelastic constitutive equations are constructed by assuming that the stress is a nonlinear function of the current strain and of a set of internal variables satisfying relaxation equations of fractional order. The dependence of the relaxation equations on the strain can also be nonlinear. The resulting constitutive equations are examined as mapping between appropriate Sobolev spaces. The proposed formulation is easier to implement numerically than history-based formulations.      

Generalized heat-transport equations: parabolic and hyperbolic models

We derive two different generalized heat-transport equations: the most general one, of the first order in time and second order in space, encompasses some well-known heat equations and describes the hyperbolic regime in the absence of nonlocal effects. Another, less general, of the second order in time and fourth order in space, is able to describe hyperbolic heat conduction also in the presence of nonlocal effects. We investigate the thermodynamic compatibility of both models by applying some generalizations of the classical Liu and Coleman–Noll procedures. In both cases, constitutive equations for the entropy and for the entropy flux are obtained. For the second model, we consider a heat-transport equation which includes nonlocal terms and study the resulting set of balance laws, proving that the corresponding thermal perturbations propagate with finite speed.     

On the <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Solvability of Higher Order Parabolic and Elliptic Systems with BMO Coefficients

We prove the solvability in Sobolev spaces for both divergence and non-divergence form higher order parabolic and elliptic systems in the whole space, on a half space, and on a bounded domain. The leading coefficients are assumed to be merely measurable only in the time variable and have small mean oscillations with respect to the spatial variables in small balls or cylinders. For the proof, we develop a set of new techniques to produce mean oscillation estimates for systems on a half space.     

A Whitham–Boussinesq long-wave model for variable topography

We study the propagation of water waves in a channel of variable depth using the long-wave asymptotic regime. We use the Hamiltonian formulation of the problem in which the non-local Dirichlet–Neumann operator appears explicitly in the Hamiltonian, and propose a Hamiltonian model for bidirectional wave propagation in shallow water that involves pseudo-differential operators that simplify the variable-depth Dirichlet–Neumann operator. The model generalizes the Boussinesq system, as it includes the exact dispersion relation in the case of constant depth. Analogous models were proposed by Whitham for unidirectional wave propagation. We first present results for the normal modes and eigenfrequencies of the linearized problem. We see that variable depth introduces effects such as a steepening of the normal modes with the increase in depth variation, and a modulation of the normal mode amplitude. Numerical integration also suggests that the constant depth nonlocal Boussinesq model can capture qualitative features of the evolution obtained with higher order approximations of the Dirichlet–Neumann operator. In the case of variable depth we observe that wave-crests have variable speeds that depend on the depth. We also study the evolutions of Stokes waves initial conditions and observe certain oscillations in width of the crest and also some interesting textures and details in the evolution of wave-crests during the passage over obstacles.     

基于能量等效原理的应变局部化分析:Ⅰ.一维解析解

基于热力学第一定律和非局部塑性理论,提出了一种求解应变局部化问题的非局部方法.对材料的每一点定义了局部和非局部两种状态空间,局部状态空间的内变量通过非局部权函数映射到非局部空间,成为非局部内变量.在应变软化过程中,局部状态空间中的塑性变形服从正交流动法则,材料的软化律在非局部状态空间中被引入.通过两个状态空间的塑性应变能耗散率的等效,得到了应变软化过程中明确定义的局部化区域以及其中的塑性应变分布.应用本方法导出了一维应变局部化问题的解析解.解析解表明,应变局部化区域的尺寸只与材料内尺度有关;对于高斯型非局部权函数,局部化区域的尺寸大约是材料内尺度的6倍.一维算例表明,局部化区域的塑性应变分布以及载荷-位移曲线仅与材料参数和结构几何尺寸有关,变形局部化区域的尺寸随着材料内尺度的减小而减小,同时塑性应变也随着材料内尺度的减小变得更加集中.当内尺度趋近于零时,应用本文方法得到的解与采用传统的局部塑性理论得到的解相同.