 |
TOPOLOGICAL METHODS
IN
NONLINEAR ANALYSIS |
TABLE OF CONTENTS
| Title and Author(s) |
Page |
|
Bifurcation of Solutions of Elliptic Problems: Local and Global Behaviour
Jose L. Gamez and Juan F. Riuz
| ABSTRACT.
Here we study the local behavior of the continua in the
case of Neumann boundary conditions, pointing out some qualitative
differences with the Dirichlet case. We also combine local and
global behavior of the bifurcating sets to obtain existence of
solutions and study of their sign for some related
problems.
|
|
203
|
|
|
Non-autonomous quasilinear elliptic equations
Matteo Franca
| ABSTRACT.
In this paper we investigate positive radial
solutions of the following equation
\Delta_{p}u+K(r) u|u|^{\sigma-2}=0
where $r=|x|$, $x \in \RR^n$, $n>p>1$, $\sigma
={n p}/({n-p})$ is the Sobolev critical exponent and
$K(r)$ is a function strictly positive and bounded.
This paper can be seen as a completion of the work started in \cite{9},
where structure theorems for positive solutions are obtained for potentials $K(r)$
making a finite number of oscillations.
Just as in \cite{9}, the starting point is to introduce a dynamical system
using a Fowler transform. In \cite{9} the results are obtained using invariant manifold theory
and a dynamical interpretation of the Pohozaev identity;
but the restriction ${2 n}/({n+2}) \le p\le 2$ is necessary in order to ensure
local uniqueness of the trajectories of the system.
In this paper we remove this restriction, repeating the proof using a modification
of Wazewski's principle; we prove for the cases $p>2$ and
$1We also introduce a method to prove the existence of Ground States with fast decay for potentials
$K(r)$ which oscillates indefinitely. This new tool also shed some light on the role played
by regular and singular perturbations in this problem, see \cite{10}.
|
|
213
|
|
|
Eigenvalues and bifurcation for elliptic equations with mixed Dirichlet-Neumann boundary conditions related to Caffarelli-Kohn-Nirenberg inequalities
Eduardo Colorado and Irened Peral
| ABSTRACT.
This work deals with the analysis of
eigenvalues, bifurcation and H\"older continuity of solutions to
mixed problems like
\ceses
-\div (|x|^{-p\gamma} |\grad u|^{p-2}\grad u) = f_{\lan}(x,u) ,
&u > 0\ \inn\Om ,\\
u = 0 &\on\Sigma_1,\\
|x|^{-p\gamma}|\nabla
u|^{p-2}\dfrac{\p u}{\p \nu} = 0 &\on \Sigma_2,
\endcases
involving some potentials related with the
Caffarelli--Kohn--Nirenberg inequalities, and with different
kind of functions $f_\lan (x,u)$.
|
|
239
|
|
|
Combining fast, linear and slow diffusion
Julian Lopez-Gomez and Antonio Suarez
| ABSTRACT.
Although the pioneering studies of G. I. Barenblatt (\cite{8}) and
A\.~G\.~Aronson and L\.~A\.~Peletier (\cite{7}) did result into a huge
industry around the porous media equation, none further study
analyzed the effect of combining fast, slow, and linear diffusion
simultaneously, in a spatially heterogeneous porous medium.
Actually, it might be this is the first work where such a problem
has been addressed. Our main findings show how the heterogeneous
model possesses two different regimes in the presence of a priori
bounds. The minimal steady-state of the model exhibits a genuine
{\it fast diffusion behavior}, whereas the remaining states are
rather reminiscent of the purely {\it slow diffusion model}. The
mathematical treatment of these heterogeneous problems should
deserve a huge interest from the point of view of its
applications in fluid dynamics and population evolution.
|
|
275
|
|
|
Singularly perturbet Neumann problems with potentials
Alessio Pomponio
ABSTRACT.
The main purpose of this paper is to study the existence of single-peaked solutions of the
Neumann problem
\cases
-\varepsilon^2 \dv \left(J(x)\nabla u\right)+V(x)u=u^p
& \text{in }\O,
\displaystyle \dfrac{\de u}{\de \nu}=0
& \text{on }\de\O,
\endcases
where $\O$ is a smooth bounded domain of $\RN$, $N\ge 3$, $1
|
|
301
|
|
|
Optimal Feedback Control in the Problem of the Motion of a Viscoelastic Fluid
Valeri Obukhovski, Pietro Zecca and Victor Zvyagin
| ABSTRACT.
We study an optimization problem for the feedback control system emerging
as a regularized model for the motion of a viscoelastic fluid subject
to the Jeffris--Oldroyd rheological relation. The approach includes systems
governed by the classical Navier--Stokes equation as a particular case.
Using the topological degree theory for condensing multimaps we prove
the solvability of the approximating problem and demonstrate the convergence
of approximate solutions to a solution of a regularized one. At last we show
the existence of a solution minimizing a given convex, lower semicontinuous
functional.
|
|
323
|
|
|
Global Existence of Solutions of the Free Boundary Problem for the Equations of Magnetohydrodynamic Incompressible Viscous Fluid
Piotr Kacprzyk
| ABSTRACT.
Global motion of magnetohydrodynamic fluid in a domain bounded by
a free surface and under the external electrodynamic field is proved. The
motion is such that velocity and magnetic field are small in~$H^3$-space.
|
|
339
|
|
|
Some applications of groups of essential values of cocycles in topological dynamics
Mieczyslaw K. Mentzen
| ABSTRACT.
A class of examples showing that a measure-theoretical
characterization of regular cocycles in terms of essential
values is not valid in topological dynamics is constructed. An
example that in topological dynamics for the case of non-abelian
groups, the groups of essential values of cohomologous cocycles
need not be conjugate is given. A class of base preserving
equivariant isomorphisms of Rokhlin cocycle extensions of
topologically transitive flows is described. In particular, the
topological centralizer of Rokhlin cocycle extension of minimal
rotation defined by an action of the group ${\Bbb R}^m$ is
determined.
|
|
357
|
|
