LisMath Seminar 

26/04/2021, 12:00 — 12:30 — Online
Matthias Hofmann, LisMath, Faculdade de Ciências, Universidade de Lisboa

Spectral theory, clustering problems and differential equations on metric graphs

We present our thesis work dealing with several topics in PDE theory on metric graphs. Firstly, we present our framework and present existence results for nonlinear Schroedinger (NLS) type energy functionals as generalizations and unification of various results obtained by several authors, most notably from [1] and [2], among others. Secondly, we consider spectral minimal partitions of compact metric graphs recently introduced in [3]. We show sharp lower and upper estimates for various spectral minimal energies, estimates between these energies and eigenvalues of the Laplacian and discuss their asymptotical behaviour. Thirdly, we present Pleijel's theorem on the asymptotics of the number of nodal domains $\nu_n$ of the $n$-th eigenfunction(s) of a broad class of operators of Schroedinger type on compact metric graphs. Among other things, these results characterize the accumulation points of the sequence $(\frac{\nu_n}{n})_{n\in\mathbb N}$, which are shown always to form a finite subset of $(0,1]$. Finally, we introduce a numerical method for calculating the eigenvalues for a special operator in the beforementioned class, the standard Laplacian, based on a discrete graph approximation.

References

[1] Adami, Serra and Tilli, Journal of Functional Analysis 271 (2016), 201-223

[2] Cacciapuoti, Finco and Noja, Nonlinearity 30 (2017), 3271-3303

[3] Kennedy et al, Calculus of Variations and Partial Differential Equations 60 (2021), 61