On the Size of the Plasma Region

AMS (MOS): 35J60, 35B40, 35J67     

Some results for non-linear elliptic problems with mixed boundary conditions

We prove existence and uniqueness results for non-linear elliptic equations with lower order terms, L¹ data, and mixed boundary conditions that include as particular cases the Dirichlet and the Neumann problems. Mathematics Subject Classification (2000) 35J25, 35D05, 35J70, 35J60     

Branching-coalescing particle systems

We study the ergodic behavior of systems of particles performing independent random walks, binary splitting, coalescence and deaths. Such particle systems are dual to systems of linearly interacting Wright-Fisher diffusions, used to model a population with resampling, selection and mutations. We use this duality to prove that the upper invariant measure of the particle system is the only homogeneous nontrivial invariant law and the limit started from any homogeneous nontrivial initial law.Mathematics Subject Classification (2000):Primary: 60K35, 92D25; Secondary: 60J80, 60J60Research supported in part by the German Science Foundation.AcknowledgementWe thank Klaus Fleischmann who played a stimulating role during the early stages of this project and answered a question about Laplace functionals, Claudia Neuhauser for answering questions about branching-coalescing processes, Olle Häggström for answering questions on nonamenable groups, and Tokuzo Shiga for answering our questions about his work. We thank the referee for drawing our attention to the reference [SU86]. Part of this work was carried out during the visits of Siva Athreya to the Weierstrass Institute for Applied Analysis and Stochastics, Berlin and to the Friedrich-Alexander University Erlangen-Nuremberg, and of Jan Swart to the Indian Statistical Institute, Delhi. We thank all these places for their kind hospitality.     

Heat Content and a Hardy Inequality for Complete Riemannian Manifolds

Upper bounds are obtained for the heat content and the torsionalrigidity of a complete Riemannian manifold M, assuming a Hardyinequality for the Dirichlet Laplacian on M. 2000 MathematicsSubject Classification 35J05, 35K05, 58J35.     

On a perturbed anisotropic equation with a critical exponent

Abstract We study a perturbed anisotropic equation without using the knowledge of the limiting problem. This provides a different method from that introduced by Brézis and Nirenberg [4]. Our arguments use some tools recently developed in [5, 6]. Keywords: Anisotropic critical exponent, Critical level, Compactness, Nehari manifold Mathematics Subject Classification (2000): 35J25, 35J60, 35J65     

Non-linear elliptic problems having natural growth and <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript> data in Orlicz spaces

An existence result for a strongly non-linear elliptic equation with natural growth condition on the non-linearity and L¹-data is proved. Mathematics Subject Classification (2000) 35J25, 35J60     

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     

Absolute Continuity in Periodic Waveguides

We study second order elliptic operators with periodic coefficientsin two-dimensional simply connected periodic waveguides withthe Dirichlet or Neumann boundary conditions. It is proved thatunder some mild smoothness restrictions on the coefficients,such operators have purely absolutely continuous spectra. Theproof follows a method suggested previously by A. Morame totackle periodic operators with variable coefficients in dimension2. 2000 Mathematical Subject Classification: 35J10, 35P05, 35J25.     

The Paneitz Curvature Problem on Lower-Dimensional Spheres

In this paper we prescribe a fourth order conformal invariant (the Paneitz curvature) on the n-spheres, with n∊{5,6}. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we prove some existence results. Mathematics Subject Classifications (2000): 35J60, 53C21, 58J05, 35J30.     

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     

A direct uniqueness proof for equations involving the p-Laplace operator

 We provide a simple convexity argument for some known uniqueness theorems. Previous proofs were more technical and had to pay attention to the behaviour of solutions near the boundary. Received: 3 May 2002 Mathematics Subject Classification (2000): 35J20, 35J70, 49R05     

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     

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     

Some existence results for solutions to SU(3) Toda system

We study SU(3) Toda system in non-abelian relativistic self-dual gauge theory. In the range of parameters where the corresponding Trudinger-Moser inequality fails, we show the existence of the solution by a different variational formulation from Lucia-Nolasco's [15]. This work was supported by a grant of the Japan-Korea Scientific Cooperation Program - Joint Research “Mathematical analysis and mathematical science for self-interacting particles.” The second author was partially supported by Grant-in-Aid for Scientific Research (No. 16740103), Japan Society for the Promotion of Science. Mathematics Subject Classification (2000) 35B40 - 35J50 - 35J60 - 49Q99 - 58E15 - 58J05 - 70S15     

On the convergence of genetic algorithms – a variational approach

A variational approach is introduced to study the existence and uniqueness of stationary states and (exponential) stability of genetic algorithms with mutation and interactive selection.Mathematics Subject Classification (2000): 35J20 (90C30, 92D25, 35J60, 31C25)     

Bifurcation phenomena associated to a class of p-Laplacian like operators

 Global bifurcation of positive solutions for some degenerate quasilinear elliptic problems is considered. The uniform estimate of the gradient of weak solutions is given. This estimate is crucial in our arguments. Received: 10 August 2001 Supported in part by Grant-in-Aid for Scientific Research (No. 11640207), Ministry of Education, Science, Sports and Culture, Japan. Mathematics Subject Classification (2000): 35B32, 35J25, 35J70     

Some Singularly Perturbed Problems on Annuli and a Counterexample to a Problem of Gidas, Ni and Nirenberg

1991 Mathematics Subject Classification 35B05, 35J60.     

On a periodic Schrödinger equation with nonlocal superlinear part

We consider the Choquard-Pekar equation and focus on the case of periodic potential V. For a large class of even functions W we show existence and multiplicity of solutions. Essentially the conditions are that 0 is not in the spectrum of the linear part –+V and that W does not change sign. Our results carry over to more general nonlinear terms in arbitrary space dimension N2.Mathematics Subject Classification (2000):35Q55, 35Q40, 35J10, 35J20, 35J60, 46N50, 49J35, 81V70in final form: 14 November 2003     

Periodic orbits of a one-dimensional inelastic particle system

The dynamical behavior of a one-dimensional inelastic particle system with two particles of different masses traveling between two walls is investigated. Energy is added at only one of the walls, which is oscillating, while the other wall is stationary. We show that if the particle nearer to the stationary wall is slightly lighter than the other particle and collisions between particles tend to the elastic limit, there are an infinite number of stable orbits. We also show that the widely studied situation of equal masses is an extremely special case, in which all the orbits are degenerate and collapse to a single trivial orbit in which one of the particles is trapped against the stationary wall. To cite this article: J.J. Wylie, Q. Zhang, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Two-Sided Estimates of the Heat Kernel of the Schrodinger Operator

Two-sided bounds for the heat kernel of the Schrödingeroperator with a singular potential are obtained with explicitconstants. 1991 Mathematics Subject Classification 35B05, 35J10.