Department of Mathematics - Master Calendar

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HADES

Event Type
Seminar/Symposium
Sponsor
Department of Mathematics
Location
243 Altgeld Hall
Date
Mar 21, 2023   1:00 - 2:00 pm  
Speaker
Richard S. Laugesen
Contact
Xiao Liu
E-Mail
liu328@illinois.edu
Views
13
Originating Calendar
Events - Department of Mathematics

Speaker: Richard S. Laugesen

Title: Maximizers beyond the hemisphere for the second Neumann eigenvalue 

Abstract:  Among membranes of given area vibrating freely on a sphere, which shape achieves the highest fundamental tone? Equivalently, on which insulated subdomain of given size in a sphere will the temperature equilibrate most rapidly? Motivated by symmetry, one conjectures the answer is a spherical cap.

We prove the conjecture for all but the largest domains by showing that the first positive Neumann eigenvalue mu_2 of the spherical Laplacian  is maximal for a spherical cap among simply connected domains covering up to 94% of the 2-sphere. The previous best result, by Bandle, was 50%, which means her maximizing caps lay in a hemisphere.

Further questions include…what if the domain has holes? and is the next positive eigenvalue mu_3 maximal for the union of two disjoint caps, analogous to the two-disk result for planar domains?

(Joint work with Jeffrey Langford, Bucknell University.)


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