Geometric Structures seminar - Giorgio Saracco - Tuesday June 18th, 2024 , 14:00 (Rome time) - room 133 at SISSA

[Samuel Borza]

Dear All,

This is to announce the next seminar from the series "Geometric Structures".

Speaker: Giorgio Saracco (University of Florence)

Title: Existence of minimizers of Cheeger's functional among convex sets
Time: Tuesday June 18th, 2024 , 14:00 (Rome time)
Venue: room 133 - SISSA main building

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Abstract:
Given any open, bounded set in R^N, the Cheeger inequality states that its first eigenvalue of the Dirichlet p-Laplacian is suitably bounded from below by the p-th power of the so-called Cheeger constant of the set. A natural question is whether this inequality is sharp and if the infimum of the ratio of these two quantities is attained (at least when restricting to suitable classes of competitors) by some set.

Parini proved existence of minimizers among convex sets in the linear case p=2, limitedly to the planar case N=2. The result was later extended to general p by Briani—Buttazzo—Prinari, still for N=2. They conjecture that existence of minimizers among convex sets should hold regardless of the dimension. Together with Aldo Pratelli, we positively solve the conjecture. The proof exploits a criterion proved by Ftouhi paired with some cylindrical estimate on the Cheeger constant.
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More information can be found here: https://sites.google.com/view/geometric-structures/

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Samuël Borza