**UBC Mathematics Department**

*http://www.math.ubc.ca*

## 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).

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