___________________JOINT COLLOQUIUM______________________ Speaker: Prof. Dorin Bucur (Université de Savoie) Title: Variational approach for isoperimetric inequalities Date: Friday, 31st of October 2008 Time: 2:00 pm Venue: RC-4082, The Red Centre, UNSW A group will leave from the outside foyer on the 2nd floor at 12:20 p.m. to travel over to the University of New South Wales. Please let me know if you would like to join us, and if you can offer or need a lift. Abstract: In this talk, we are concerned with isoperimetic inequalities involving eigenvalues of elliptic operators. As an example, we discuss some classical problems, like the Faber-Krahn inequality for the first Dirichlet eigenvalue of the Laplacian, from both classical and variational points of view. We will point out the main ingredients for proving that the optimal shape is radial: prove the existence of an optimal domain, prove mild regularity of the free boundary, use a cut and reflect argument in order to prove symmetry. Recent advances and open problems will presented. The colloquium will take a break next week due to our school planning day. The following week we will hear from Prof Dick Hain of Duke University.