Diff. Geometry, Mathematical Physics, Partial Differntial Equations Seminar: Denis Bonheure: Universite Libre de Bruxelles
Topic
A rough guide to reduction methods for strongly coupled elliptic systems
Speakers
Details
Abstract:
In this talk, I will first recall the notions of superlinearity and subcriticality for strongly coupled elliptic systems. I will present various functional frameworks and their limitations. I will then discuss two reduction methods that allow to get rid of the indefiniteness of the energy functional. These reductions to a single equation are powerful to treat basic questions for superlinear systems. For instance, I will discuss the notion of ground states, in bounded domains and in R^N, show how to get the information on the symmetry and the sign of the ground states through the definition of a convenient Nehari manifold or constrained minimization problem. I will also discuss the classical question of existence of infinitely many critical points of perturbed indefinite symmetric functionals and how one of the reduction method allow to use the notion of Morse index. Finally, I will show how these reduction methods can help in proving partial symmetry and symmetry breaking. As a paradigm, I will illustrate the ideas on the Lane-Emden system with Hénon weights.
References : B.-Ramos ANIHP 2009 - B.-dos Santos JDE 2010 - B.-Ramos-dos Santos Trans. AMS 2012 & preprint.
In this talk, I will first recall the notions of superlinearity and subcriticality for strongly coupled elliptic systems. I will present various functional frameworks and their limitations. I will then discuss two reduction methods that allow to get rid of the indefiniteness of the energy functional. These reductions to a single equation are powerful to treat basic questions for superlinear systems. For instance, I will discuss the notion of ground states, in bounded domains and in R^N, show how to get the information on the symmetry and the sign of the ground states through the definition of a convenient Nehari manifold or constrained minimization problem. I will also discuss the classical question of existence of infinitely many critical points of perturbed indefinite symmetric functionals and how one of the reduction method allow to use the notion of Morse index. Finally, I will show how these reduction methods can help in proving partial symmetry and symmetry breaking. As a paradigm, I will illustrate the ideas on the Lane-Emden system with Hénon weights.
References : B.-Ramos ANIHP 2009 - B.-dos Santos JDE 2010 - B.-Ramos-dos Santos Trans. AMS 2012 & preprint.
Additional Information
For further information please see the event page at: http://www.math.ubc.ca/Dept/Events/index.shtml?period=future&series=all.
Denis Bonheure: Universite Libre de Bruxelles
Denis Bonheure: Universite Libre de Bruxelles