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Torsional Rigidity of a Radially Perturbed Ball

Mette Iversen
(University of Bristol)
Torsdag, 25 oktober, 2012, at 16:15-17:00, in Aud. D3 (1531-215)
We recently studied the question of minimising convex combinations of the first two eigenvalues of the Dirichlet Laplacian with a volume constraint, i.e.

$\inf\{\alpha \lambda_{1}(\Omega)+ (1-\alpha) \lambda_{2}(\Omega)| \, \Omega$ open in ${\bf R}^n, \ |\Omega|\leq 1 \}$. 

The proof that minimisers are connected made use of an estimate of the first eigenvalue of a radially perturbed ball. This talk will look at this problem, and at the corresponding result for the torsional rigidity. The torsional rigidity for an open set $\Omega \in {\bf R}^n$ is given by the variational characterization
$P(\Omega):=4\sup_{u \in H^1_0(\Omega)}\{(\int_{\Omega}u)^2/\int_{\Omega}|\nabla u|^2\}.$

Thus the study of the torsional rigidity by variational methods naturally links to the study of the first eigenvalue of the Dirichlet Laplacian, and the result is obtained by a choice of test function.
Kontaktperson: Søren Fournais