Τομέας Μαθηματικής Ανάλυσης
Library of the School of Science

2015-11-27

2015

Αντωνόπουλος Παναγιώτης

Αλικάκος Νικόλαος Καθηγητής (επιβλέπων), Λάππας Διονύσιος Αναπλ. Καθηγητής, Μπαρμπάτης Γεράσιμος Αναπλ. Καθηγητής

Ποιοτική μελέτη λύσεων μη γραμμικών ελλειπτικών συστημάτων

Greek

Qualitative study of solutions of non linear elliptic systems

Abstract. In chapter 1, we present some basic notions related to the vector
Allen-Cahn equation.
In chapter 2, under appropriate hypotheses, we rigorously derive the Plateau
angle conditions at triple junctions of diffused interfaces in four dimensions,
starting from the vector-valued Allen-Cahn equation with a triple-well
potential. Our derivation is based on an application of the divergence theorem
using the divergence-free form of the equation via an associated stress tensor.
In chapter 3, a maximum principle is established for minimal solutions to the
system , with a potential W vanishing at the boundary of a closed convex set
, which is either smooth or coincides with a point {α}.
In chapter 4, we establish two necessary conditions for the existence of
bounded one dimensional minimizers u: the potential W must have a global
minimum supposed to be 0 without loss of generality, and as . Furthermore,
non constant minimizers connect at two distinct components of the set . We
also prove, when the previous assumptions are satisfied, the existence of
nontrivial minimizers and we show the existence of heteroclinic, homoclinic and
periodic orbits in analogy with the scalar case. Finally, we study the
asymptotic convergence of these solutions.

Young's law, Maximum principle, Minimizers

No

0

Yes

49

75

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https://pergamos.lib.uoa.gr/uoa/dl/object/1309283