Spectral shape optimization problems with Neumann conditions on the free boundary
Duration: 56 mins 10 secs
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| Description: |
Bucur, D
Monday 10th June 2019 - 11:30 to 12:30 |
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| Created: | 2019-06-10 13:11 |
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| Collection: | New trends and challenges in the mathematics of optimal design |
| Publisher: | Isaac Newton Institute |
| Copyright: | Bucur, D |
| Language: | eng (English) |
| Distribution: |
World
(downloadable)
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| Explicit content: | No |
| Aspect Ratio: | 16:9 |
| Screencast: | No |
| Bumper: | UCS Default |
| Trailer: | UCS Default |
| Abstract: | In this talk I will discuss the question of the maximization of the k-th eigenvalue of the Neumann-Laplacian under a volume constraint. After an introduction to the topic I will discuss the existence of optimal geometries. For now, there is no a general existence result, but one can prove existence of an optimal {\it (over) relaxed domain}, view as a density function. In the second part of the talk, I will focus on the low eigenvalues. The first non-trivial one is maximized by the ball, the result being due to Szego and Weinberger in the fifties. Concerning the second non-trivial eigenvalue, Girouard, Nadirashvili and Polterovich proved that the supremum in the family of planar simply connected domains of R2 is attained by the union of two disjoint, equal discs. I will show that a similar statement holds in any dimension and without topological restrictions. |
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