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.     

Stability of elastic bodies with nonmonotone multivalued boundary conditions of the quasidifferential type

Necessary conditions for the stability of elastic bodies subjected to nonmonotone multivalued boundary conditions are derived. These conditions are assumed to be derived from nonconvex and nonsmooth, quasidifferentiable energy functions. A difference convex approximation of the potential energy function is written based on an appropriate quasidifferential formulation. Under appropriate assumptions for the convex and the concave parts we prove the existence of at least one nontrivial solution to the nonlinear eigenvalue problem.     

接触问题的互补变分原理及非线性互补模型

给出了接触问题的互补虚功原理,互补能量原理及变分不等式等变分提法,并讨论了三类提法的关系。其此建立了有限维非线性互补模型并给出解的存在唯一性定理。文中还阐明了有,无摩擦接触问题之间以及二维和三维有摩擦接触问题之间的本质区别。     

SOME GENERALIZED GRONWALL-LIKE INEQUALITIES WITH DELAY ON TIME SCALES

In this paper, we establish some new Gronwall-like inequalities which can be used as tools in the theory of integral equations with delay on time scales.     

Variational Problem,Generalized Convexity,and Application to a Competitive Equilibrium Problem

The main purpose of this article is to present a new formulation of a competitive equilibrium in terms of a suitable quasivariational inequality involving multivalued maps. More precisely, a pure exchange economy is considered where the consumer's preferences are represented by utility functions that we assume to be generalized concave and non-differentiable. In the concave context, we have characterized the equilibrium by means of a variational problem involving the subdifferential. Now, by relaxing concavity and differentiability assumptions on utility functions, the subdifferential operator of the utility function is replaced by a suitable multimap involving a new concept, recently introduced in [1 D. Aussel and N. Hadjisavvas ( 2005 ). Adjusted sublevel sets, normal operator and quasiconvex programming . SIAM J. Optim. 16 : 358 – 367 .[Crossref], [Web of Science ®] , [Google Scholar]]: the normal operator to the adjusted sublevel sets. Thanks to this variational formulation we are able to achieve the existence of equilibrium points by using arguments of the set-valued analysis. Finally, we provide some example of utility functions which verify our assumptions.     

Poincaré type inequalities for group measure spaces and related transportation cost inequalities

Let G be a countable discrete group with an orthogonal representation α on a real Hilbert space H  . We prove _Lp$L_p$ Poincaré inequalities for the group measure space _L∞(_ΩH,γ)?G$L_{\infty }({\Omega }_H,\gamma )?G$, where both the group action and the Gaussian measure space (_ΩH,γ)$({\Omega }_H,\gamma )$ are associated with the representation α  . The idea of proof comes from Pisier?s method on the boundedness of Riesz transform and Lust-Piquard?s work on spin systems. Then we deduce a transportation type inequality from the _Lp$L_p$ Poincaré inequalities in the general noncommutative setting. This inequality is sharp up to a constant (in the Gaussian setting). Several applications are given, including Wiener/Rademacher chaos estimation and new examples of Rieffel?s compact quantum metric spaces.     

Inequalities among eigenvalues of Sturm–Liouville problems with distribution potentials

This paper deals with the eigenvalue problems for the Sturm–Liouville operators generated by the differential expression

L(y)=−(p(x)y^′)^′+q(x)y$L\left(y\right)=-{\left(p\left(x\right)y^{\prime }\right)}^{\prime }+q\left(x\right)y$

with singular coefficients q(x)$q\left(x\right)$ in the sense of distributions. We obtain the inequalities among the eigenvalues corresponding to different self-adjoint boundary conditions. The continuity region, the differentiability and the monotonicity of the n$n$th eigenvalue corresponding to the separated boundary conditions are given. Oscillation properties of the eigenfunctions of all the coupled Sturm–Liouville problems are characterized. The main results of this paper can also be applied to solve a class of transmission problems.     

Explicit estimates on the torus for the sup-norm and the dissipative length scale of solutions of the Swift–Hohenberg Equation in one and two space dimensions

In this work we have obtained explicit and accurate estimates of the sup-norm for solutions of the Swift–Hohenberg Equation (SHE) in one and two space dimensions. By using the best (so far) available estimates of the embedding constants which appear in the classical functional interpolation inequalities used in the study of solutions of dissipative partial differential equations, we have evaluated in an explicit manner the values of the sup-norm of the solutions of the SHE. In addition we have calculated the so-called time-averaged dissipative length scale associated to the above solutions.