Estimates on the Green function and existence of positive solutions for some polyharmonic nonlinear equations in the half space

Let G_m,n be the Green function of with Dirichlet boundary conditions We establish some estimates on G_m,n, including a 3G-Theorem. Next, we introduce a Kato class of functions and we exploit properties of these functions to study the existence of positive solutions of some m-polyharmonic nonlinear elliptic problems.Mathematics Subject Classification (2000): 34B27, 35J40     

Green function for X-elliptic operators

We study the Green function associated to second order X-elliptic operators with non-regular coefficients. An existence and uniqueness theorem is given, along with some local estimations and a representation formula for the solution of the Dirichlet problem.Mathematics Subject Classification (2000): 34B27, 35H99, 35J70, 35C15Revised version: 28 June 2004     

Symmetry and singularity properties of second-order ordinary differential equations of Lie's type III

We extend the work of Abraham-Shrauner [B. Abraham-Shrauner, Hidden symmetries and linearization of the modified Painlevé-Ince equation, J. Math. Phys. 34 (1993) 4809-4816] on the linearization of the modified Painlevé-Ince equation to a wider class of nonlinear second-order ordinary differential equations invariant under the symmetries of time translation and self-similarity. In the process we demonstrate a remarkable connection with the parameters obtained in the singularity analysis of this class of equations.     

A Class of Operator Equations in Nonlinear Mechanics

A nonlinear evolution equation in Hilbert space is introduced and is shown to govern a broad class of problems of nonlinear solid mechanics. The simplest oscillations of this equation are described by a “generalized Liénard equation.”

AMS(MOS) Classification Nos: 73D35, 73F15, 58D25, 35L35, 34G20, 34C15.     

Nonlinear Schrödinger equations with symmetric multi-polar potentials

This paper deals with a class of nonlinear elliptic equations involving a critical power-nonlinearity as well as a potential featuring multiple inverse square singularities. When the poles form a symmetric structure, it is natural we wonder how the symmetry affects such mutual interaction. The present paper means to study this aspect from the point of view of the existence of solutions inheriting the same symmetry properties as the set of singularities. Mathematics Subject Classification (2000) 35J60, 35J20, 35B33     

Positive solutions of linear elliptic equations with critical growth in the Neumann boundary condition

We study the existence of positive solutions of a linear elliptic equation with critical Sobolev exponent in a nonlinear Neumann boundary condition. We prove a result which is similar to a classical result of Brezis and Nirenberg who considered a corresponding problem with nonlinearity in the equation. Our proof of the fact that the dimension three is critical uses a new Pohoaev-type identity.AMS Subject Classification: Primary: 35J65; Secondary: 35B33.     

Limit behavior of thin heterogeneous domain with rapidly oscillating boundary

Abstract We give a generalization of the work presented in [6] where the asymptotic behaviour, as ε→0, of a monotone nonlinear problem in a bounded multidomain of R^N depending on ε was addressed. We extend the previous results to the case where the nonlinear operator depends both on the slow and rapid variable and we prove that, due to the presence of the rapid variable, the algebraic equation contained in the limit problem obtained in [6] must be replaced by a partial differential equation with respect to the microscopic variable y′. Keywords: Homogenization, Dimension reduction, Multidomain, Rapid variable, Limit problem Mathematics Subject Classification (2000): 35B27, 35J60     

Relation between Different Types of Global Attractors of Set-Valued Nonautonomous Dynamical Systems

The article is devoted to the study of the relation between forward and pullback attractors of set-valued nonautonomous dynamical systems (cocycles). Here it is proved that every compact global forward attractor is also a pullback attractor of the set-valued nonautonomous dynamical system. The inverse statement, generally speaking, is not true, but we prove that every global pullback attractor of an α-condensing set-valued cocycle is always a local forward attractor. The obtained general results are applied while studying periodic and homogeneous systems. We give also a new criterion of the absolute asymptotic stability of nonstationary discrete linear inclusions. Dedicated to our friend Professor Enrico Primo Tomasini on the occasion of his 55th birthdayMathematics Subject Classifications (2000) Primary: 34C35, 34D20, 34D40, 34D45, 58F10,58F12, 58F39; secondary: 35B35, 35B40.     

Reply to the note on article “The evidential reasoning approach for multiple attribute decision analysis using interval belief degrees”

In a very recent note by Gao and Ni [B. Gao, M.F. Ni, A note on article “The evidential reasoning approach for multiple attribute decision analysis using interval belief degrees”, European Journal of Operational Research, in press, doi:10.1016/j.ejor.2007.10.0381], they argued that Yen’s combination rule [J. Yen, Generalizing the Dempster–Shafer theory to fuzzy sets, IEEE Transactions on Systems, Man and Cybernetics 20 (1990) 559–570], which normalizes the combination of multiple pieces of evidence at the end of the combination process, was incorrect. If this were the case, the nonlinear programming models we proposed in [Y.M. Wang, J.B. Yang, D.L. Xu, K.S. Chin, The evidential reasoning approach for multiple attribute decision analysis using interval belief degrees, European Journal of Operational Research 175 (2006) 35–66] would also be incorrect. In this reply to Gao and Ni, we re-examine their numerical illustrations and reconsider their analysis of Yen’s combination rule. We conclude that Yen’s combination rule is correct and our nonlinear programming models are valid.     

A note on article “The evidential reasoning approach for multiple attribute decision analysis using interval belief degrees”

Wang et al. use an evidential reasoning approach for solving multiple attribute decision analysis (MADA) problems under interval belief degrees [Y.M. Wang, J.B. Yang, D.L. Xu, K.S. Chin, The evidential reasoning approach for multiple attribute decision analysis using interval belief degrees, European Journal of Operational Research 175 (2006) 35–66]. In this note it is shown some nonlinear optimization models in that paper are incorrect. The necessary corrections are proposed.     

On the Size of the Plasma Region

AMS (MOS): 35J60, 35B40, 35J67     

The scalar curvature problem on the four dimensional half sphere

In this paper, we consider the problem of prescribing the scalar curvature under minimal boundary conditions on the standard four dimensional half sphere. We provide an Euler-Hopf type criterion for a given function to be a scalar curvature to a metric conformal to the standard one. Our proof involves the study of critical points at infinity of the associated variational problem.Received: 2 August 2003, Accepted: 10 May 2004, Published online: 16 July 2004Mathematics Subject Classification (2000): 35J60, 35J20, 58J05K. El Mehdi: elmehdik@ictp.trieste.it     

Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions

Summary. One of the most important problems in numerical simulations is the preservation of qualitative properties of solutions of the mathematical models by computed approximations. For problems of elliptic type, one of the basic properties is the (continuous) maximum principle. In our work, we present several variants of the maximum principles and their discrete counterparts for (scalar) second-order nonlinear elliptic problems with mixed boundary conditions. The problems considered are numerically solved by the continuous piecewise linear finite element approximations built on simplicial meshes. Sufficient conditions providing the validity of the corresponding discrete maximum principles are presented. Geometrically, they mean that the employed meshes have to be of acute or nonobtuse type, depending of the type of the problem. Finally some examples of real-life problems, where the preservation of maximum principles plays an important role, are presented.The first author was supported by the Hungarian Research Fund OTKA under grant no. F034840The second author was supported by the Agora Center under Grant InBCT of TEKES, Finland, and by the Academy Research Fellowship no. 208628 from the Academy of FinlandMathematics Subject Classification (2000): 35B50, 35J65, 65N30, 65N50     

Nonlinear Anisotropic Elliptic and Parabolic Equations in R<Superscript>N</Superscript> with Advection and Lower Order Terms and Locally Integrable Data

We prove existence and regularity results for distributional solutions in R^N for nonlinear elliptic and parabolic equations with general anisotropic diffusivities as well as advection and lower-order terms that satisfy appropriate growth conditions. The data are assumed to be merely locally integrable. Mathematics Subject Classifications (2000) 35J60, 35K55.This work was supported by the BeMatA program of the Research Council of Norway and the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282. This work was done while M. Bendahmane visited the Centre of Mathematics for Applications (CMA) at the University of Oslo, Norway.     

On heat conductors with a stationary hot spot

We consider a heat conductor having initial constant temperature and zero boundary temperature at every time.The hot spot is the point at which temperature attains its maximum at each given time. For convex conductors, if the hot spot does not move in time, we prove symmetry results for planar triangular and quadrangular conductors.Then, we examine the case of a general conductor and, by an asymptotic formula, we prove that, if there is a stationary critical point, not necessarily the hot spot, then the conductor must satisfy a geometric condition. In particular, we show that there is no stationary critical point inside planar non-convex quadrangular conductors. Mathematics Subject Classification (2000) Primary 35K05, 35K20, 35J05; Secondary 35J25, 35B38, 35B40     

On the existence and Morse index of solutions to the Allen-Cahn equation in two dimensions

In this paper we study transition layers in the solutions to the Allen-Cahn equation in two dimensions. We show that for any straight line segment intersecting the boundary of the domain orthogonally there exists a solution to the Allen-Cahn equation, whose transition layer is located near this segment. In addition we analyze stability of such solutions and show that it is completely determined by a geometric eigenvalue problem associated to the transition layer. We prove the existence of both stable and unstable solutions. In the case of the stable solutions we recover a result of Kohn and Sternberg [13]. As for the unstable solutions we show that their Morse index is either 1 or 2. Mathematics Subject Classification (2000) 35J60, 35Q72, 35J20, 35P15, 35P20, 35B25, 35B35, 35B40, 35B41     

A Variational Method For The Existence of Bounded Solutions of a Sublinear Forced Oscillator

We prove that, for every bounded and measurable forcing p(t),the differential equation has bounded solutions with arbitrarily large amplitude. In generalit is not possible to say that all solutions are bounded, asshown by an example due to Littlewood. The proof is based on a variational method which can be seenas a dual version of Nehari's method for boundary value problemson compact intervals. 2000 Mathematics Subject Classification34C11, 34B14, 49J35.     

An improved strongly sub-feasible SSLE method for optimization problems and numerical experiments

In this paper, the super-linearly and quadratically convergent strong sub-feasible method [J.L. Li, J.B. Jian, A superlinearly and quadratically convergent strongly subfeasible method for nonlinear inequality constrained optimization, OR Transactions, 7 (2) (2003) 21-34] for nonlinear inequality constrained optimization is improved, such that the iterative points can get into the feasible region after a finite number of iterations. As a result, a strict restricted condition can be overcome. Another two contributions of this paper are that a new bidirectional Armijo line search is presented and a lot of numerical comparison results are reported.     

Boundary Hemivariational Inequalities of Hyperbolic Type and Applications

In this paper we examine two classes of nonlinear hyperbolic initial boundary value problems with nonmonotone multivalued boundary conditions characterized by the Clarke subdifferential. We prove two existence results for multidimensional hemivariational inequalities: one for the inequalities with relation between reaction and velocity and the other for the expressions containing the reaction–displacement law. The existence of weak solutions is established by using a surjectivity result for pseudomonotone operators and a priori estimates. We present also an example of dynamic viscoelastic contact problem in mechanics which illustrate the applicability of our results.Mathematics Subject Classifications (2000). 34G20, 35A15, 35L85, 35L70, 74H20