Eigenvalue problems in complicated domains (Summer Semester 2017)
 Lecturer: PD Dr. Andrii Khrabustovskyi
 Classes: Lecture (0163800), Problem class (0163810)
 Weekly hours: 2+1
Current Events
Date  Event 

5.9.2017, 10:00 
Schedule  

Lecture:  Wednesday 11:3013:00  SR 3.68 
Problem class:  Friday 8:009:30  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
 Minmax 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. Dumbbellshape 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, SpringerVerlag, 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.
(minmax principle) 
O.Post, Spectral analysis on graphlike 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), 2759.
(domain with holes; capacity; homogenization in perforated domains) 
P.Exner, O.Post, Convergence of spectra of graphlike thin manifolds, J. Geom. Phys. 54 (2005), no. 1, 77–115.
O.Post, Spectral convergence of quasionedimensional 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.
(dumbbellshape domains)