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Titles and abstracts
Abbas Bahri
(Department of Mathematics, Rutgers University, New Brunswick)
Flow lines on a space of dual Legendrian curves
Thomas Bartsch
(Mathematisches Institut, Universitaet Giessen)
Extremal sign changing solutions of Dirichlet problems
Abstract.-
We present some recent results on properties of sign changing
solutions obtained via variational methods.
Fabrice Bethuel
(Laboratoire d'analyse numérique,
Université Pierre et Marie Curie - Paris VI)
Variational methods for equations arising in superfluidity and
superconductivity
Haim Brezis
(Laboratoire d'analyse numérique,
Université Pierre et Marie Curie - Paris VI
and Department of Mathematics, Rutgers University, New Brunswick)
Lifting in Sobolev spaces: the full picture
Abstract.-
We give a complete answer to the question whether a function
u in the Sobolev space Ws,p with values in
S1 can be written as
u = ei f for some f in the same
Sobolev space.
We will see that there are obstructions of topological as well as
analytical nature.
Kung-Ching Chang
(School of Mathematical Sciences, Peking University, Beijing)
Some results on semilinear elliptic systems
Abstract.-
Many results in the study of semilinear elliptic equations are extended to
systems.
Especially, the bifurcations, the sub-and super-solutions,
Ambrosetti-Prodi type result, Amann-Zehnder type result and the
characterization of mountain pass points.
An extended version of the Hess-Kato theorem for the principal eigenvector
and the Maslov type index are introduced.
Mónica Clapp
(Instituto de Matemáticas,
Universidad Nacional Autònoma de México)
Variational methods for perturbed symmetric functionals
Abstract.-
Functionals which are invariant under a group of symmetries
often have many critical values.
These come from the many "linkings" provided by the symmetries.
Following an idea of Bolle we give conditions for those linkings to be
preserved by a path of functionals which starts with a symmetric one, for
arbitrary group actions.
These conditions depend on the growth of the critical values of the
symmetric functional.
Using some Borsuk-Ulam type results we obtain good lower bounds for these
values for many groups.
Norman Dancer
(School of Mathematics and Statistics,
University of Sydney)
Peak solutions
Djairo G. De Figueiredo
(IMECC, UNICAMP, Campinas)
A priori estimates of positive solutions of nonlinear elliptic systems
Abstract.-
We study semilinear elliptic systems with dependence on the gradient and
with superlinear growth with respect both to the dependent variables and
their gradients.
Manuel Del Pino
(Departamento de Ingenieria Matemática,
Universidad de Chile, Santiago)
Eigenvalue problems related to the onset of superconductivity
Maria J. Esteban
(CE.RE.MA.DE., Université de Paris IX Dauphine)
Some nonlinear problems in relativistic quantum mechanics
Abstract.-
In this talk I will make a review of some methods devised
to solve variational problems which are very indefinite due to the
total unboundedness of the Hamiltonian involved in the corresponding
Euler-Lagrange equations.
New min-max techniques are necessary to tackle the new difficulties
encountered here.
Patricio L. Felmer
(Departamento de Ingenieria Matemática,
FCFM, Universidad de Chile, Santiago)
Concentration phenomena in elliptic PDE
Alexander D. Ioffe
(Department of Mathematics, Technion, Haifa)
Slope of De Giorgi-Marino-Tosques and the Ljusternik-Graves theorem
Abstract.-
It will be shown that the concept of the slope introduces by
De Giorgi, Marino and Tosques in 1980 provides for the most adequate
instrument for dealing with metric regularity and related problems
and that the quantitative estimates for local covering and regularity
for mappings into Banach spaces obtained in terms of slopes are exact.
Relationship between slopes and subdifferentials will be also considered.
Waclaw Marzantowicz
(Faculty of Mathematics and Computer Science,
A. Mickiewicz University of Poznán)
Periodic solutions of nonlinear problems with positive oriented
periodic coefficients
Abstract.-
We study nonlinear ODE problems in the plane, or Euclidean complex space,
with the right hand side being polynomial, or rational, function of the
space variable with non-constant periodic coefficients in the time variable
t.
As the coefficients functions we admit only functions with vanishing Fourier
coefficients for negative indices - called the positive oriented periodic
functions.
Using an interpretation of this class of functions as the boundary traces of
functions which are holomorphic in the open unit disc, we can employ
techniques of the complex analysis and power series.
This leads to existence theorems which relates the number of solutions with
the number of zeros of averaged right hand side function, allows to construct
examples of problems without periodic solutions (including already known) and
finally gives a multiplicity theorem for problems linear at the infinity.
Jean Mawhin
(Institut de Mathématiques,
Université Catholique, Louvain-La-Neuve)
Asymmetric oscillators, Landesman-Lazer conditions and unbounded
solutions
Abstract.-
We consider the equation
$$
x''+ \mu x^{+} - \nu x^{-} = f(x)+g(x)+e(t),
$$
where $x^{+}=\max\{x,0\}$; $x^{-}=\max\{-x,0\}$, in a situation of
resonance for the period $2\pi$, i.e. when
$1/\sqrt{\mu}+1/\sqrt{\nu}=2/n$ for some integer $n$.
We assume that $e$ is $2\pi$-periodic, that $f$ has limits
$f(\pm \infty)$ at $\pm\infty$, and that the function
$g$ has a sublinear primitive.
Denoting by $\varphi$ a solution of the homogeneous equation
$x''+\mu x^{+}-\nu x^{-}=0$, we show that the behavior of the solutions
of the full nonlinear equation depends crucially on whether the function
$$
\Phi(\theta)={n \over \pi}\left[{f(+\infty) \over \mu}-
{f(-\infty) \over \nu}\right]
+{1 \over 2\pi}\int_{0}^{2\pi} e(t)\varphi (t+\theta)\,dt
$$
is of constant sign or not.
In particular, existence results for $2\pi$-periodic, for
subharmonic solutions, and for unbounded solutions, based on the function
$\Phi$, are given.
This is a joint work with C. Fabry.
Marian Mrozek
(Instytut Informatyki,
Uniwersytet Jagiellónski, Kraków)
An algorithmic approach to the Conley index theory
Abstract.-
We introduce a class of representable sets which is
closed under the operations of set theoretical union,
intersection, difference and topological interior and closure.
We use this class to construct an algorithm which verifies
if for a given dynamical system a given set is an isolating
neighborhood.
In case of a positive answer the algorithm
constructs an index pair.
Louis Nirenberg
(Courant Institute, New York University)
Elliptic systems in composite materials
Abstract.-
In studying composite materials one considers weak solutions of elliptic
systems on a domain made up of subregions.
The coefficients of the systems are smooth in each subregion but change
discontinuously on going to another subregion.
It is important to obtain uniform estimates on the derivatives of the
solutions in each subregion - independent of the number of subregions
and of the distances between them.
Various estimates are prescribed. The talk will be expository.
Rafael Ortega
(Departamento de Matemática Aplicada,
Facultad de Ciencias, Universidad de Granada)
Piecewise linear oscillators with twist dynamics
Abstract.-
Consider the equation x''+g(x)=p(t)
where g is a piecewise linear restoring force and p is
a periodic function.
The existence of periodic solutions can be studied using
the Lazer-Leach's condition or the theory of "jumping
nonlinearities" (Dancer-Fucik).
The same conditions are useful to show that the equation has a dynamics
of twist type in a neighbourhood of infinity.
Then one can apply KAM theory to prove the boundedness of all solutions
or the existence of subharmonic and quasi-periodic solutions.
Eric Séré
(CE.RE.MA.DE., Université de Paris IX Dauphine)
The Dirac-Fock model
Abstract.-
The Dirac-Fock equations are the relativistic analogue of the well-known
Hartree-Fock equations.
The solutions can be found as critical points of a strongly indefinite
functional.
I will present some recent existence and multiplicity results.
Robert Turner
(Department of Mathematics,
University of Wisconsin-Madison)
Waves in Natural Systems
Abstract.-
We will present an overwiew of several systems in which traveling waves
of fixed shape propagate and discuss the interaction of models,
differential equations, and computations.
We begin with work on waves in stratified fluids, focusing on the use
of center manifolds to approach the study of small waves.
Next we introduce subject of traveling waves in biological systems,
beginning with the pivotal work of Hodgkin and Huxley on Action Potentials.
We continue with Calcium waves and contraction waves in the musculature
of nematodes.
Michel Willem
(Institut de Mathématiques,
Université Catholique, Louvain-La-Neuve)
Critical Neumann problem with a weight
Abstract.-
We study the existence of least energy solutions and concentration phenomena
for a critical Neumann problem with a weight.
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