Electronic Journal of Differential Equations: Conference 18, 2010.
Proceedings of the 2007 Conference on Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems . Northern Arizona University, Flagstaff, Arizona. May 23-27, 2007.
Foreword
This conference was a continuation of a very successful
conference Variational Methods: Open Problems, Recent Progress, and Numerical
Algorithms that was organized by J.M. Neuberger in Flagstaff in June 2002,
where open questions and numerical algorithms for variational methods
were the dominant theme.
The 2007 conference had a different focus and expanded agenda.
The organizing committee consisted of the following members: John M. Neuberger (conference chairman; Northern Arizona University, Flagstaff AZ, USA), Maya Chhetri (University of North Carolina at Greensboro, Greensboro NC, USA), Petr Girg (University of West Bohemia, Pilsen, Czech Republic) and Nandor Sieben (Northern Arizona University, Flagstaff AZ, USA).
The scientific committee consisted of the following members: Alfonso Castro (Harvey Mudd College, Claremont CA, USA), Goong Chen (Texas A&M University, College Station TX, USA), Pavel Drabek (University of West Bohemia, Pilsen, Czech Republic), Tim Kelley (North Carolina State University, Raleigh NC, USA), and Ratnasingham Shivaji (Mississippi State University, Mississippi State MS, USA).
The workshop was aimed at researchers, graduate students, and advanced undergraduate students working in various areas of Nonlinear Analysis, Differential Equations, and their applications. Five leading experts in their respective fields Djairo de Figueiredo (Brazil), Jean Mawhin (Belgium), James Serrin (USA), Peter Takac (Germany), Jianxin Zhou (USA) each gave two one-hour keynote presentations. There were 28 contributed 30-minute talks, a workshop on numerical methods, and plenty of time allotted for small group discussions. There were 17 student participants in this conference, some of them presenting their results for the first time. There were 43 registered participants affiliated to universities from 11 countries around the world and across the United States: Belgium (4), Brazil (1), Canada (1), Czech Republic (5), Germany (2), Greece (1), Italy (3), Mexico (2), Portugal (3), Spain (1), USA (21).
The papers in this conference proceedings comprise only of a small subset of these talks. This volume contains seven mathematical papers; one article authored by a keynote speaker, and six articles authored and/or co-authored by 13 of the 28 contributed speakers, four of them co-authored by a student. As the title of our conference suggests, most of the material concerns variational and/or topological methods as applied to nonlinear elliptic PDE. There was a focus on open problems; hence papers with conjectures were encouraged. Survey papers were also solicited, as were works featuring applications. This conference successfully brought together those doing cutting-edge nonlinear functional analysis with researchers more computationally oriented; hence some of the papers in this volume feature numerical investigations. It was our intent to survey what is known in the general sub ject area, list the important open questions, and suggest analytical and numerical techniques that might be beneficial to those seeking to make progress towards solving those open problems. We hope that the reader will find this volume an excellent starting point for considering new problems and relevant techniques in nonlinear elliptic and related PDE.
Editor's reviews of main articles
The first article by keynote speaker
Peter Takac
is an overview of the results and methods
for quasilinear elliptic boundary value problems of the second order.
This excellent survey paper is intended as a tutorial for students
and introduction to this field for
the non expert and provides an extensive bibliography referencing much of the
relevant literature. The paper treats both variational and/or topological
methods and also contains nice results that can be obtained by a combination of
both. Thus it is not strictly divided into variational and topological part.
The variational thread of explanation covers some elementary methods such as
Rayleigh method, but its core lies in special convexity methods and the method
of quadratization of the functional both developed by the author. The
topological methods are covered by lower and upper solution method, bifurcation
from infinity and some calculation of the topological degree. The paper also
contains nice introduction with useful inequalities that are scattered across
vast literature. Due to its extent of 40 pages, it can be considered as a
monograph.
The paper by Jiri Benedikt and Petr Girg treats Prufer transformation which is a useful tool for the study of second-order ordinary differential equations by means of topological methods. The authors focus on an extension of this transformation that is suitable for study of boundary value problems for the p-Laplacian in the resonant case. Though this method is widely known, their properties are difficult to find in the literature. The purpose of this paper is to establish its basic properties in deep detail. This paper is intended as possible reference where the most useful properties of this transformation can be found together with their proofs.
The paper by Marco Ghimenti and Anna Maria Micheletti study a semilinear elliptic problem with a subcritical power nonlinearity, the underlying spatial domain being a sym- metric, smooth connected compact Riemannian manifold of a dimension greater than one embedded in an Euclidean space. The main result gives lower estimate on the number of solutions (u, -u) which change sign exactly once. A lower bound on the number of such pairs is expressed in terms of equivariant Ljusternik-Schnirelmann category.
The next paper by Christopher Grumiau and Christophe Troestler is also a research article, which is focused on subcritical superlinear Lane-Emden problem on a ball or an annulus. In particular, it is shown that, for p close to 2, least energy nodal solutions are odd with respect to an hyperplane which is their nodal surface. It was previously known that least energy nodal solutions cannot be radial. On the other hand, it was also known that they have at least some residual symmetry, namely that they possess the Schwarz foliated symmetry. Unfortunately, such sym- metry does not guarantee that the zero set of the solution is an hyperplane passing through the origin as is widely believed. It is shown in this paper that this is true for p close to 2.
In the paper by Gabriela Holubova and Petr Neesal, the authors study the structure c of the Fucik spectra for the linear multi-point differential operators. They introduce a variational approach in order to obtain a robust and global algorithm which is suitable for the exploration of unknown Fucik spectrum structure. Their method is applied to the four-point selfadjoint differential operator of the fourth order which is closely connected to the nonlinear model of a suspension bridge with two towers. Finally, they numerically reconstruct the Fucik spectra in the case of four-point non-selfadjoint ordinary differential operators of the second order. The resulting pictures demonstrate their non-trivial and interesting structure. Since the boundary value problems treated in this paper are closely related to practical applications (suspended bridges), the presented numerical results rise many important open questions and motivate further analytical research in this direction.
In the paper by David Medina and Pablo Padilla, the authors present a geometric framework to study invariant sets of dynamical systems associated with differential equations. This frame-work is based on extremizing properties of invariant sets for an area functional. This approach yields existence results for heteroclinic and periodic orbits. The authors also implement this approach numerically by means of the steepest descent method.
Finally, the paper by Nikolas S. Papageorgiou and Eugenio M. Rocha treats a nonlinear e Neumann boundary-value problem driven by the p-Laplacian with a nonsmooth potential (hemivariational inequality). Using variational techniques based on the smooth critical point theory and the second deformation theorem, the authors establish existence and a multiplicity results (not necessarily assuming the coercivity of the Euler functional).
List of keynote lectures and talks
Keynote speakers:
Acknowledgments
This conference would not have been possible without support from NSF DMS
Grant No. DMS 0653868 (Variational and Topological Methods: Theory,
Applications, Numerical Simulations, and Open Problems; Northern Arizona University).
The organizers would also like to thank Northern Arizona University's
Department of Mathematics and Statistics, Dean of the College of Arts
and Sciences, and Provost for their generous contributions and
to Department of Mathematics, University of West Bohemia, Pilsen,
Czech Republic, for its partial financial support
of the event from the Research Plan MSM 4977751301.
We would like to thank the Electronic Journal of Differential Equations
for the opportunity to publish and disseminate this proceedings
electronically worldwide free of charge.
Finaly, we would like to thank the principal speakers and participants
for their valuable contributions to the success of the conference and
publication of the proceedings.
Special Issue Editors:
Maya Chhetri (Associate Editor) Department of Mathematics and Statistics University of North Carolina at Greensboro Greensboro, NC 27402, USA. email: maya@uncg.edu |
Petr Girg (Main Editor) Department of Mathematics, University of West Bohemia Plzen, CZ-30614, Czech Republic email: pgirg@kma.zcu.cz |
John M. Neuberger (Associate Editor) Department of Mathematics and Statistics Northern Arizona University, Flagstaff, AZ 86011, USA. email: john.neuberger@nau.edu |
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