Davide Zucco's seminar

[Emanuele Tuillier Illingworth]

SEMINAR ANNOUNCEMENT
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Davide Zucco
(Politecnico di Torino)

Title: Optimizing the first Dirichlet eigenvalue of the Laplacian with 
an obstacle.

Abstract:
Inside a fixed bounded domain $\Omega$ of the plane, we look for the 
best compact connected set $K$, of given perimeter, in order to
maximize the first Dirichlet eigenvalue of the Laplacian 
$\lambda_1(\Omega\setminus K)$. We discuss some of the qualitative 
properties
of the maximizers, passing toward existence, regularity and geometry. 
Then we study the problem in specific domains: disks, rings, and, more 
generally, disks with several holes. In these situations, we prove 
symmetry and, in some cases non symmetry results, identifying the 
explicit solution. We choose to work with the outer Minkowski content as 
the “good” notion of perimeter. Therefore, we are led to prove some new 
properties for it as its lower semicontinuity with respect to the 
Hausdorff convergence and the fact that the outer Minkowski content is 
equal to the Hausdorff lower semicontinuous envelope of the classical 
perimeter.

Venue: Wednesday 1 March at 11:00 at SISSA, lecture room 133