An introduction to variational methods

In this talk, we will introduce the variational approach to the study of equations.
To get started, we will review the spectral theory of symmetric real matrices and the min-max principle for eigenvalues.
We will then move on to the spectral theory of the Laplacian on bounded open sets of ℝⁿ with Dirichlet boundary conditions
and use the min-max characterisation of eigenvalues to deduce Courant's nodal domain Theorem.
We will see that a rich and somewhat surprising behaviour may be observed when one replaces smooth domains of ℝⁿ
by unidimensional domains called metric graphs, showing the role of unique continuation principles in the classical theory.
In the last part of the talk, we will see how variational methods may also be applied to study solutions of some nonlinear elliptic PDEs,
using suitable constrained minimisation problems.