Spectral analysis for linearizations of the Allen-Cahn equation around rescaled stationary solutions with triple-junction [Elektronische Ressource] / vorgelegt von Tobias Kusche
116 Pages
English
Downloading requires you to have access to the YouScribe library
Learn all about the services we offer

Spectral analysis for linearizations of the Allen-Cahn equation around rescaled stationary solutions with triple-junction [Elektronische Ressource] / vorgelegt von Tobias Kusche

Downloading requires you to have access to the YouScribe library
Learn all about the services we offer
116 Pages
English

Description

Spectral analysis for linearizations ofthe Allen-Cahn equation aroundrescaled stationary solutions withtriple-junctionDissertationzur Erlangung des Doktorgrades der Naturwissenschaften(Dr. rer. nat.) der Fakult¨at fur¨ Mathematikder Universit¨at Regensburgvorgelegt vonTobias KuscheausRegensburg2006Promotionsgesuch eingereicht am: 10.01.2006Die Arbeit wurde angeleitet von: Prof. Dr. Harald GarckePrufungsaussc¨ huß: Vorsitzender: Prof. Dr. Uwe Jannsen1. Gutachter: Prof. Dr. Harald Garcke2. Gutachter: Prof. Dr. Stanislaus Maier-Paapeweiterer Prufe¨ r: Prof. Dr. Wolfgang HackenbrochContentsList of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Vector-valued Sturm-Liouville operators 1921.1 Exponential L -bounds . . . . . . . . . . . . . . . . . . . . . . . . 211.2 Pointwise exponential bounds . . . . . . . . . . . . . . . . . . . . 251.3 The range of Sturm-Liouville sytems . . . . . . . . . . . . . . . . 291.4 Convergence of the spectrum . . . . . . . . . . . . . . . . . . . . . 322 Spectral analysis for a two-phase transition 352.1 Standing wave solutions . . . . . . . . . . . . . . . . . . . . . . . 352.2 Linearizations around standing waves . . . . . . . . . . . . . . . . 412.2.1 Spectral analysis . . . . . . . . . . . . . . . . . . . . . . . 422.2.

Subjects

Informations

Published by
Published 01 January 2006
Reads 9
Language English

Exrait

Spectral analysis for linearizations of
the Allen-Cahn equation around
rescaled stationary solutions with
triple-junction
Dissertation
zur Erlangung des Doktorgrades der Naturwissenschaften
(Dr. rer. nat.) der Fakult¨at fur¨ Mathematik
der Universit¨at Regensburg
vorgelegt von
Tobias Kusche
aus
Regensburg
2006Promotionsgesuch eingereicht am: 10.01.2006
Die Arbeit wurde angeleitet von: Prof. Dr. Harald Garcke
Prufungsaussc¨ huß: Vorsitzender: Prof. Dr. Uwe Jannsen
1. Gutachter: Prof. Dr. Harald Garcke
2. Gutachter: Prof. Dr. Stanislaus Maier-Paape
weiterer Prufe¨ r: Prof. Dr. Wolfgang HackenbrochContents
List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1 Vector-valued Sturm-Liouville operators 19
21.1 Exponential L -bounds . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2 Pointwise exponential bounds . . . . . . . . . . . . . . . . . . . . 25
1.3 The range of Sturm-Liouville sytems . . . . . . . . . . . . . . . . 29
1.4 Convergence of the spectrum . . . . . . . . . . . . . . . . . . . . . 32
2 Spectral analysis for a two-phase transition 35
2.1 Standing wave solutions . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 Linearizations around standing waves . . . . . . . . . . . . . . . . 41
2.2.1 Spectral analysis . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.2 Convergence of the ground state . . . . . . . . . . . . . . . 46
3 Spectral analysis at the triple-junction 51
3.1 Sobolev spaces with symmetry . . . . . . . . . . . . . . . . . . . . 51
3.2 Rescaled stationary solutions . . . . . . . . . . . . . . . . . . . . . 56
3.3 Linearizations around rescaled stationary solutions . . . . . . . . 59
3.3.1 Spectral analysis . . . . . . . . . . . . . . . . . . . . . . . 59
3.3.2 Exponential decay of eigenfunctions . . . . . . . . . . . . . 69
3.3.3 The range space . . . . . . . . . . . . . . . . . . . . . . . . 83
3.3.4 Convergence of the ground state . . . . . . . . . . . . . . . 85
3.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4 Discussion 97
A Measure theory 101
B Operator theory 103
C Sesquilinear forms 109
12List of symbols
B (x) p. 14 Im(z) p. 13r
kC (Ω,B) p. 16 ker(f) p. 14
kC (Ω,B) p. 16 K p. 13
k 2C (Ω,B) p. 16 L (Ω,K) p. 16
b
∞ 2C (Ω,B) p. 16 L (Ω,K) p. 16J
∞ ∞C (Ω,B) p. 16 L (Ω,K) p. 160
∞ 2C (Ω) p. 53 L (I ) p. 420,G odd
det(A) p. 15 L(H) p. 15
diam(U) p. 13 L(H ,H ) p. 151 2
dim U p. 14 lin M p. 14
div as operator p. 107 M(m,K) p. 15
div u p. 17 N p. 13
αD u α multiindex p. 17 N p. 130
kD u k∈N p. 17 O(m) p. 15
D T operator or p. 15 O(g()),→ 0 g function p. 14T
Gsesquilinear form p. 109 P p. 52Ω
Gl(m,K) p. 15 P(X) p. 13
kH (Ω,K) p. 16 r() p. 60
k,∞H (Ω,K) p. 16 R p. 13+
k,∞
H (Ω,K) p. 16 R p. 13−loc

∞k
R p. 13H (Ω,K) p. 17 +
k Re(z) p. 13H (I ) p. 42odd
k sign(z) p. 13H (Ω) p. 53G

k supp(u) p. 14H (Ω) p. 53G
mS(R ) p. 15I n p. 15
K
T p. 60I p. 41
∗T T operator p. 15im(f) p. 13
34 as operator p. 107 [A,B] p. 25
4u p. 17 T ⊂S T, S operators p. 15
%() p. 60 ⊂⊂ p. 13
nρ(T) p. 103 ⊗ T T operator p. 15i ii=1
∂σ(T) p. 103 p. 18
∂ν
kσ (T) p. 103 ∂U ∈C p. 18d
σ (T) p. 103 ∇ as operator p. 107e
σ (T) p. 103 ∇u p. 17p
t
≥ for operators p. 111 u →u t sesquilinear form p. 109n

|.| p. 13 p. 13U
ck.k p. 14 U p. 13X
k.k k p. 16C (Ω)
k.k k p. 17H (Ω)
k.k k,∞ p. 17H (Ω)
k.k 2 p. 16L
J
k.k ∞ p. 16L (Ω)
k.k 2 p. 16L (Ω)J
k.k p. 15L(B ,B )1 2
k.k p. 15tr
h.,.i p. 13
h.,.i p. 162L
h.,.i p. 162L
J
h.,.i p. 15
tr
h.,.i p. 14X
w
−→ p. 14
[T] T operator p. 17
[V] V function or p. 105 f.
real number p. 13
4Introduction
Phase-field models
Via formal calculations, Bronsard and Reitich, [BR], studied the asymptotic be-
havior of the vector-valued Ginzburg-Landau equation
∂ T 2 u = 2 4u −(DW(u )) (1)
∂t
∂ u| = 0 or u (x,t)| =h(x) (2)∂Ω ∂Ω
∂ν
u (x,0) =g(x) (3)
nas → 0. We consider this equation on an open domain Ω ⊂ R and for
m mu : Ω×R → R , where m,n ≥ 2. The potential W : R → R is smooth+
and attains its minimum value zero at exactly three distinct pointsa,b, andc, so
as to model a three-phase physical system. Instead of equation (1), we can also
consider the vector-valued Allen-Cahn equation
T∂2 2 ˆ v = 4v − DW(v ) . (4)
∂t
Equation (1) equals (4) via
1ˆW(x) := W(x),
2
and
1 v (x,t) :=u x, t .
22
The question is how the solution u of (1), (2), and (3) behaves as → 0. The
phase-field parameter > 0 represents the thickness of the transition layer be-
tweendifferentphases. Therefore, weexpectthatu approachesasharpinterface
model as → 0. One such sharp interface model is the mean-curvature flow.
nRoughly speaking, this is a family (Γ ) of smooth manifolds inR such thatt t∈[0,t]
the signed distance function d(.,t) of Γ fulfillst

4d(x,t) = d(x,t), t∈ [0,T], x∈ Γ.t
∂t
5A precise definition is given in [AS]. For m = 1, i.e. the scalar Allen-Cahn
equation, de Mottoni and Schatzman, [deMS], proved that there exist initial
data for the Allen-Cahn equation such that the corresponding solutions converge
to the minima ofW uniformly outside each tubular neighborhood of Γ as→ 0.t
Essentially, the proof is a rigorous justification of formal asymptotic expansion,
i.e. it is supposed that in a tubular neighborhood of (Γ ) the solution u ist t∈[0,T]
approximately given by the asymptotic expansion
N
X d (x,t) iu (x,t) = u ,x,t , t∈ [0,T], x∈ Γ (δ).i tA
i=0
Note that Γ (δ) := {x ∈ Ω : dist(x,Γ ) < δ}. The function d is the modifiedt t
distance function, i.e.
NX
id (x,t) =d(x,t)+ d (x,t), t∈ [0,T], x∈ Γ (δ). i t
i=1
If one puts u into the Allen-Cahn equation, expands the term DW (u ) viaA A
Taylor expansion, and arranges the terms according to their -power, the results
are equations for the u of the formi
L u =R (d ), (5)0 i i−1 i−1
whereR (d ) depends only on known quantities and the functiond whichi−1 i−1 i−1
2is not determined so far. The operator L has domain H (R,C) and is given by0
00 2L u =−u +D W(θ )u.0 0
The function θ is the unique increasing solution of0
00−θ +DW(θ) = 0, θ(0) = 0,
that connects the two distinct minima of W. Equation (5) has a solution if and
only if
⊥R (d )∈ ker(L ) .i−1 i−1 0
This determines d , as dim ker(L ) = 1. As the solutions of (5) decay at ani−1 0
exponential rate, the approximate solution u can be extended to Ω. The resultA
is a family of approximate solutions (u ) such thatA ∈(0,1)

d(x,t)
2u (x,t) =θ +O( ), x∈ Γ (δ),0 tA
and
∂2 2 k u − 4u +DW (u ) =O , → 0.A A A∂t
6The integerk∈N grows with the lengthof the asymptotic expansion. Important
fortheproofoftheconvergenceu →u istoanalyzethebehaviorofthesmallestA
eigenvalue λ of the operator1
2d 2L =− +D W(θ ) (6) 02dz

1 12that is equipped with Neumann boundary conditions in L − , ,C . This

deliversthe[deMS]-estimatefortheAllen-Cahnoperator, i.e. thesmallesteigen-
value of
2 2 − 4+D W (u )A
2behaveslikeO( ),→ 0. Theoperatorthatisgivenbythedifferentialexpression
2 2 − 4+D W (u )A
is called Allen-Cahn operator. It represents the linearization of the Allen-Cahn
equation around the approximate solution u .A
Concerning the vector valued Allen-Cahn equation, Bronsard and Reitich proved
shorttimeexistencefortheproblemofthreecurvesΓ movingbymeancurvaturei
suchthatthethreecurvesmeetatatriple-junctionm(t), andtheotherendpoint
of each curve lies on the boundary of Ω - cf. figure 1.
G
1
G
3
G
2
Figure 1: Three-phase boundary motion.
Via formal asymptotic expansion, Bronsard and Reitich obtained the evolu-
tion laws of three-phase boundary motion derived by material scientists. At the
triple junction m(t), they used the expansion
N
X x−m(t) iu (x,t)≈ u ,t .i

i=0
For the function u , the expansion leads to the equation0
T
−4u +(DW(u )) = 0.0 0
7Moreover, in directions tangentially to the interfaces, one expects that u ap-0
proaches the standing wave solution that connects two minima of W. The ex-
istence of such an u was rigorously proved in the work of [BGS], details given0
in chapter 3. This is the first step in the proof of rigorous convergence to the
limiting flow. If one pursues the formal calculation to determine the u ’s, he isi
led to equations of the form
L u =R . (7)0 i i−1
The function R depends only on known quantities, and L is given by thei−1 0
differential expression
2−4+D W(u ).0
The operator L was introduced in [BGS]. It’s domain is given by the set of0
22 2all elements in (H (R ,C)) that are equivariant with respect to the symmetry
group G of the equilateral triangle. A byproduct of the proofs in [BGS] is that
L is self-adjoint and positive semidefinite.0
Target of the endevours
Now, we consider the case m = 2. In [BGS], they proved the existence of a
solution θ of0
00 t−θ +(DW(θ )) = 000
which connects two distinct global minima of W and fulfills
sup|u (x,y)−θ (x)|→ 0, y→∞. (8)0 0
x∈R
In this work, we show that the convergence in (8) produces a strong connection
oddbetween the essential spectrum of L and the spectrum of the operators L ,0
odd≥ 0. The operator L is given by the restriction of the vector valued version
of L (cf. (6)) to a certain subspace.
,odd oddSet λ = minσ(L ). The first main result of this work is the following1
Theorem (Theorem 3.1).
Theorem Suppose dim ker (L ) = 1. Then the following statements hold:0
1. We have
,odd
minσ (L ) = liminfλ > 0,e 0 1
→0
and
oddσ(L )⊂σ (L ).e 00
8