Im Rahmen des Oberseminars Analysis/Numerik spricht
Herr Dr. Raffael Hagger (Christian-Albrechts-Universität zu Kiel)
Titel: „On the spectrum of the double-layer operator on dilation-invariant domains“
Abstract:
Let 
be a bounded Lipschitz domain. The corresponding double-layer operator 
is defined by 
where 
is the Newtonian potential. An almost 30-year old conjecture, oftentimes attributed to Carlos Kenig, implies that for every bounded Lipschitz domain the spectral radius of 
is less than or equal to 
. This conjecture has been verified for several special cases including 
-domains and polygons, but the general case remains wide open.
In this talk we will study this problem on locally-dilation-invariant domains. We say that 
is locally dilation invariant at 
if there is an αx ∈ (0, 1) and a δx > 0 such that
In other words, 
can be described locally as the graph of a Lipschitz continuous function f satisfying 
. If at every the domain is either or locally dilation invariant, we call a locally-dilation-invariant domain. This notion is motivated by a recent paper of Chandler-Wilde and Spence [1], where a stronger version of the spectral radius conjecture was refuted.
Our method can briefly be described as follows. After some standard localization techniques, a Floquet-Bloch trans-form translates the spectral radius problem to an infinite family of eigenvalue problems of compact integral operators,
which are then treated by a Nystr ̈om-type approximation. This results in finitely many numerical inequalities that can be checked by a straightforward Matlab routine.
This talk is based on joint work with S.N. Chandler-Wilde, K.-M. Perfekt and J.A. Virtanen [2].
[1] S.N. Chandler-Wilde, E. Spence: Coercivity, essential norms, and the Galerkin method for second-kind integral equations on polyhedral and Lipschitz domains, Numer. Math. 150, 299-371 (2022).
[2] S.N. Chandler-Wilde, R. Hagger, K.-M. Perfekt, J.A. Virtanen: On the spectrum of the double layer operator on locally-dilation-invariant Lipschitz domains, to appear in Numer. Math. (2023)
Der Vortrag findet statt am:
Donnerstag, den 01.06.2023 von 14.15 - 15.15 Uhr, Raum W1 0-006
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