Qualitative properties of solutions to partial differential equations: existence, nodal properties, symmetries
KeywordsPDE, variational method, min-max, bifurcation
The research subjects proposed by the Numerical Analysis team concern the analysis of non-linear partial differential equations (PDE), in particular second order PDEs with a variational structure but also fourth order PDEs. Questions of interest are:
- the existence of solutions, especially those that can be obtained using a min-max variational principles;
- the symmetries of positive solutions or solutions that enjoy a variational characterization;
- the uniqueness and the multiplicity of certain types of solutions, for example positive solutions;
- the existence of branches of bifurcation, in particular when related to the questions of multiplicity and symmetry breaking.
Numerical simulations are also performed in order to explore these questions and to derive conjectures.