UBC Mathematics Department
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Colloquium Abstract: Professor Nassif Ghoussoub, Department of Mathematics, UBC

Perturbations from Symmetry in Non-homogenous Boundary Value Problems and Hamiltonian Systems

The most basic -- symmetric but non-linear -- partial differential equation \Delta u+|u|^{p-1}u=0 on a bounded domain \Omega of {\bf R}^n with a homogenous boundary condition (i.e., u=0 on \partial \Omega ) often has an infinite set of solutions. What happens if the symmetry is broken in a most elementary way? like for \Delta u +|u|^{p-1}u=f on \Omega (with f\neq 0) or when the Dirichlet boundary condition is non-homogenous (i.e., u=u_0\neq 0 on \partial \Omega ). The situation is then more complicated and much more interesting.

Partial results can be obtained for PDEs, but complete solutions are available for the cases of Hamiltonian systems and the second order systems of the Calculus of Variations (\'a la Bolza).