Unit:
Τομέας Μαθηματικής ΑνάλυσηςLibrary of the School of Science
Author:
Αργυροπούλου Ευτυχία
Dissertation committee:
Ι. Στρατής Καθηγητής (Επιβλέπων) , Α. Γιαννακόπουλος Καθηγητής, Γ. Μπαρμπάτης Αναπλ. Καθηγητής
Original Title:
Μερικές διαφορικές εξισώσεις και προβλήματα της επιστήμης των υλικών
Translated title:
Partial differential equations and problems of the science of materials
Summary:
The main objective of this thesis is the homogenization of partial differential
equations (mainly Maxwell’s equations) describing electromagnetic phenomena in
complex media. In particular, we study the homogenization of Maxwell’s
equations focusing on the periodic unfolding method in complex media under
Drude-Born-Fedorov type, local in time, constitutive relations.
Firstly, we formulate Maxwell’ s problem as an evolution initial value (Cauchy)
problem in a Hilbert space supplemented with the constitutive relations of a
bianisotropic medium (the most general linear medium in electromagnetics).
Further, we analyze the notion of homogenization and we apply it as examples to
equations of elliptic type in divergence form and to Maxwell’s system in
bianisotropic media.
We present also the method of periodic unfolding in the case of an elliptic
partial differential equation and in the main part of this work we consider the
problem of the well-posedness of the time-dependent Maxwell’s equations in a
Drude-Born-Fedorov type environment considering the fields to be elements of an
appropriate Hilbert space. In order to prove the existence and uniqueness we
apply the Faedo-Galerkin method and for the continuous dependence from the
initial data we use semigroup theory for operators. The rest of the main part
of the thesis deals with the homogenization of the considered problem, using
the periodic unfolding method.
In the last chapter, we examine the time-harmonic Maxwell problem in a
bianisotropic cavity, which we study by transforming it to an eigenvalue
problem.
Keywords:
Maxwell's equations, Homogenization, Chiral meterial, Unfolding operator, Drude-Born-Fedorov
