Talk by Sedef Özcan
On July 24h, 2024, Sedef Özcan (Dokuz Eylül University, Izmir) gave a talk about "Torsional Rigidity for the Laplacian on compact metric graphs with delta-Couplings" as part of the research seminar Analysis of the FernUniversität in Hagen. This lecture is partially supported by the COST action Mathematical models for interacting dynamics on networks.
Abstract
Show Abstract
Initially, torsional rigidity is defined as the L1-norm of the solution υ of
−∆υ(x) = 1, x ∈ Ω,
υ(z) = 0, z ∈ ∂Ω.
Pólya noticed that torsional rigidity is actually a geometric constant that depends on the shape and size of a given domain. Pólya demonstrated that among all open bounded domains with the same area, the circular domain possesses the greatest torsional rigidity. Pólya’s approach is based on the variational characterization of T(Ω)
T(Ω) = sup_{u∈H^1_0(Ω)} (||u||_1^2)/(||∇u||_2^2).
Mugnolo and Plümer developed the theory of torsional rigidity for the Laplacian on metric graphs with at least one Dirichlet vertex. In this talk, we will mention the torsional rigidity for the Laplacian on metric graphs with δ coupling conditions. We obtain a variational characterization of torsional rigidity, enabling the derivation of surgical principles. Using these
principles, we establish upper and lower bounds on torsional rigidity. Additionally, we explore potential manifestations of Kohler-Jobin-type inequalities in the context of δ-conditions.
