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Eigenvalue problems in complicated domains (Summer Semester 2017)

Aktuelle Termine

Datum Veranstaltung
Schedule
Lecture: Wednesday 11:30-13:00 SR 3.68
Problem class: Friday 8:00-9:30 SR 3.68
Lecturers
Lecturer, Problem classes PD Dr. Andrii Khrabustovskyi
Office hours: by appointment
Room 3.037 Kollegiengebäude Mathematik (20.30)
Email: andrii.khrabustovskyi@kit.edu

The lecture course deals with spectral properties of differential operators in domains with very complicated geometry (see the figures). Our tools are the methods of asymptotic analysis and perturbation theory allowing to reduce initial complicated problems to more simple ones.

In the first part of the course we treat some abstract topics:

  • Various types of resolvent convergence and their properties,
  • Spectral convergence
  • Convergence in varying Hilbert spaces
  • Min-max principle and its applications.

Then, in the second part, we apply these methods to the main object of our interest – eigenvalue problems in domains with complicated geometry. The following topics will be treated:

  • Eigenvalue problems in varying domains: general results.
  • Laplace operator in a domain with a hole. Capacity.
  • Homogenization in perforated domains.
  • Eigenvalue problems in thin domains. Dumbbell-shape domains. Quantum graphs.

Requirements: Basic knowledges in functional analysis (Banach and Hilbert spaces, linear operators, weak and strong convergences etc.) and partial differential equations (Sobolev spaces, weak solutions etc.)

References

Books

  • T.Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995 (or earlier editions)
    (resolvent convergence, spectral convergence, classical perturbation theory)
  • M.Reed, B.Simon, Methods of Modern Mathematical Physics, Academic Press, New York - San Francisco - London, 1978.
    (min-max principle)
  • O.Post, Spectral analysis on graph-like spaces. Lecture Notes in Mathematics, 2039. Springer, Heidelberg, 2012
    (thin domains; convergence in varying Hilbert spaces)

Articles

  • J.Rauch, M.Taylor, Potential and scattering theory on wildly perturbed domains, J. Funct. Anal. 18 (1975), 27-59.
    (domain with holes; capacity; homogenization in perforated domains)
  • P.Exner, O.Post, Convergence of spectra of graph-like thin manifolds, J. Geom. Phys. 54 (2005), no. 1, 77–115.
    O.Post, Spectral convergence of quasi-one-dimensional spaces, Ann. Henri Poincaré 7 (2006), no. 5, 933–973
    (thin domains; convergence in varying Hilbert spaces)
  • J.Arrieta, Neumann eigenvalue problems on exterior perturbations of the domain. J. Differential Equations 118 (1995), no. 1, 54–103.
    (dumbbell-shape domains)