|Politecnico di Torino|
|Anno Accademico 2009/10|
Self-adjoint extensions of differential operators and asymptotic modeling of defect (didattica di eccellenza)
Dottorato di ricerca in Matematica Per Le Scienze Dell'Ingegneria - Torino
Il corso sarą tenuto dal Prof. Sergey NAZAROV - Institute of Mechanical Engineering Problems - V.O. Bol'shoy - St-Petersburg - Russia
Contenuto del corso
Asymptotic analysis of boundary value problems in domains with singular perturbed boundaries. Applications to spectral problems, shape optimization problems, acoustic and elastic waveguides, strapped modes.
Il corso della durata di 20 ore si terrą nel mese di ottobre 2010.
The first lecture will take place on
THURSDAY, OCTOBER 7, FROM 4.00 TO 6.00 P.M.
in Aula Buzano, Dipartimento di Matematica, Politecnico.
The schedule of future lectures will be fixed with the participants.
For more information, please, write to
The topics of the course will be selected from the following ones:
Asymptotic analysis of boundary value problems in domains with singular perturbed boundaries.
Asymptotic procedures in the methods of compound and matched asymptotic expansions.
Integral characteristics of defects in harmonic analysis.
Capacity and polarization matrices.
Remarks on elasticity.
The Kondratiev norms and weighted Sobolev and H\"older spaces. Sharp estimates of asymptotic remainders.
Asymptotics of energy functionals. Shape functionals and topological derivatives, shape optimization problems. Open questions.
Spectral problems. Asymptotics of eigenvalues and eigenfunctions. The justification procedures.
Individual and collective asymptotics.
Symmetric and self-adjoint unbounded operators in Hilbert space. Closed quadratic forms.
The discrete and essential spectrum.
Self-adjoint extensions. The Neumann formulas. A simple introduction to the Birman--Krien--Vishik theory.
Self-adjoint extensions of differential operators.
Formulas for parameters of extensions
and the polarization matrices. Modeling of defects.
Remarks on elasticity and damage of solids.
Remarks on junctions of domains with different limiting dimensions. Open questions.
Self-adjoint extensions and spectral problems. Asymptotics of eigenvalues in the discrete spectrum. Problems of shape optimization.
Acoustic and elastic waveguides, trapped modes.
Asymptotics of eigenvalues below
the continuous spectrum. Open questions. The justification procedure and some introduction to the theory of spectral measure.
Eigenvalues of quantum waveguides. Discrete spectrum in a triple waveguide.
Eigenvalues embedded into continuous spectrum of a regularly perturbed waveguide.
|Statistiche superamento esami|
Programma provvisorio per l'A.A.2009/10