Stability analysis of geometric evolution equations with tripel lines and boundary contact [Elektronische Ressource] / vorgelegt von Daniel Depner

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• Mathematics

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• Publié par: universitat_regensburg
• Publié le: 01 janvier 2010
• Nombre de lectures: 11
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Stability Analysis of
Geometric Evolution Equations with
Triple Lines and Boundary Contact
DISSERTATION ZUR ERLANGUNG DES DOKTORGRADES
DER NATURWISSENSCHAFTEN (Dr. rer. nat.)
AN DER NWF I - MATHEMATIK
¨DER UNIVERSITAT REGENSBURG
vorgelegt von
Daniel Depner
Regensburg, April 2010Promotionsgesuch eingereicht am 13. April 2010.
Die Arbeit wurde angeleitet von Prof. Dr. H. Garcke.
Pru¨fungsausschuss: Vorsitzender: Prof. Dr. B. Amann
1. Gutachter: Prof. Dr. H. Garcke
2. Gutachter: Prof. Dr. K. Deckelnick (Universita¨t Magdeburg)
weiterer Pru¨fer: Prof. Dr. G. Dolzmann
Ersatzpru¨fer: Prof. Dr. H. AbelsContents
1 Introduction 1
2 Facts about Hypersurfaces 9
2.1 Differential operators and curvature terms . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Evolving hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4 Evolution of area and volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Evolution Equations with Boundary Contact 40
3.1 Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Mean curvature flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.1 Resulting partial differential equation . . . . . . . . . . . . . . . . . . . . 44
3.2.2 Linearization around a stationary state . . . . . . . . . . . . . . . . . . . 45
3.2.3 Conditions for linearized stability . . . . . . . . . . . . . . . . . . . . . . . 54
3.3 Volume preserving mean curvature flow . . . . . . . . . . . . . . . . . . . . . . . 63
3.4 Surface diffusion flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4.1 Linearized stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4.2 Some comments on nonlinear stability . . . . . . . . . . . . . . . . . . . . 88
3.5 Examples for stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4 Triple Lines with Boundary Contact 99
4.1 Mean curvature flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.1.1 Geometric properties of the flow . . . . . . . . . . . . . . . . . . . . . . . 101
4.1.2 Parametrization and resulting partial differential equations . . . . . . . . 105
4.1.3 Linearization around a stationary state . . . . . . . . . . . . . . . . . . . 110
4.1.4 Conditions for linearized stability . . . . . . . . . . . . . . . . . . . . . . . 121
4.2 Surface diffusion flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.2.1 Geometric properties of the flow . . . . . . . . . . . . . . . . . . . . . . . 129
4.2.2 Parametrization and resulting partial differential equations . . . . . . . . 134
4.2.3 Linearization around a stationary state . . . . . . . . . . . . . . . . . . . 136
4.2.4 Conditions for linearized stability . . . . . . . . . . . . . . . . . . . . . . . 138
i5 Appendix 154
5.1 Normal time derivative of mean curvature . . . . . . . . . . . . . . . . . . . . . . 154
5.2 Normal time derivative of the normal . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.3 Facts about the vector product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Bibliography 162
iiChapter 1
Introduction
The subject of the present work is the study of geometric evolution laws for evolving hyper-
surfaces with boundary contact and triple lines. The considered hypersurfaces lie inside a fixed
◦bounded region and are in contact with its boundarythrough a 90 angle. In case of triple lines
they also meet each other with some prescribed angle conditions, see Figure 1.1 for a sketch of
the arising situations for curves in the plane.
Γ2
Ω ΩΓ
Γ1
Γ3
(a) one hypersurface (b) three hypersurfaces
Figure 1.1: A sketch of the arising situations.
The geometric evolution laws that we want to consider are the mean curvature flow
V = H, (1.1)
the surface diffusion flow
V = −ΔH (1.2)
and the volume preserving mean curvature flow
V = H−H. (1.3)
Here V is the normal velocity of the evolving hypersurface, H is the mean curvature, Δ is the
Laplace-Beltrami operator and H is the average mean curvature. Our sign convention is that
H is negative for spheres provided with outer unit normal. For a review concerning geometric
evolution equations, in particular for the mean curvature flow, we want to refer the reader to
the work of Deckelnick, Dziuk and Elliott [DDE05].
1CHAPTER 1. INTRODUCTION
Meancurvatureflow(1.1) wasfirststudiedbyBrakke [Bra78]fromapointofviewofgeometric
measure theory. Gage and Hamilton [GH86] showed that convex curves in the plane under
this flow shrink to round points and Grayson [Gray87] generalized this result to embedded
plane curves. Huisken [Hui84] generalized the result of [GH86] to show that convex, compact
hypersurfacesretain their convexity and become asymptotically round. Finally we mention that
2this flow is the L -gradient flow of the area functional, it is area decreasing and for curves in
the plane it is therefore also called curve shortening flow.
Surface diffusion flow (1.2) was first proposed by Mullins [Mu57] to model motion of inter-
faces where this motion is governed purely by mass diffusion within the interfaces. Davi and
Gurtin [DG90] derived the above law within rational thermodynamics and Cahn, Elliott and
Novick-Cohen [CEN96] identifiedit asthe sharpinterface limit of a Cahn-Hilliard equation with
degenerate mobility. An existence result for curves in the plane and stability of circles has been
shown by Elliott and Garcke [EG97] and this result was generalized to the higher dimensional
case by Escher, Mayer and Simonett [EMS98]. Cahn and Taylor [CT94] showed that (1.2) is
−1the H -gradient flow of the area functional and we finally mention that for closed embedded
hypersurfaces the enclosed volume is preserved and the surface area decreases in time as can be
seen for example in [EG97] or [EMS98].
The volume preserving mean curvature flow (1.3) was considered for example in the work of
Huisken [Hui87] and in Escher and Simonett [ES98]. The idea behind this flow is to overcome
the lack of volume conservation in the mean curvature flow by enforcing it with the help of a
nonlocal term.
We will examine the above evolution laws with boundary conditions by considering evolving
hypersurfaces Γ that meet the boundary of a fixed bounded region Ω or even intersect each
other at triple lines inside of this region. In the case of thesurface diffusion flow these boundary
conditions were derived by Garcke andNovick-Cohen [GN00] as the asymptotic limit of a Cahn-
Hilliard system with a degenerate mobility matrix. At the outer boundary this yields natural
◦boundary conditions given by a 90 angle condition and a no-flux condition, i.e. we require at
Γ∩∂Ω
Γ⊥∂Ω, (1.4)
n ∇H =0. (1.5)∂Γ
Here∇ is the surface gradient andn is the outer unit conormal of Γ at boundary points. The∂Γ
conditions (1.4) and (1.5) are the natural boundary conditions when viewing surface diffusion
−1(1.2) with outer boundary contact as the H -gradient flow of the area functional.
For the evolution law (1.2) for one evolving curve in the plane with boundary conditions (1.4)
and (1.5) Garcke, Ito and Kohsaka gave in [GIK05] a linearized stability criterion for spherical
arcsresp. lines, whicharethestationarystatesinthiscase. In[GIK08]thesameauthorsshowed
nonlinear stability results for the above situation.
For the mean curvature flow (1.1), one can also consider situations where an evolving hyper-
surface is attached to an outer fixed boundary. In this case, instead of the two conditions (1.4)
and (1.5), only an angle condition has to be fulfilled. This is due to the fact that surface diffu-
sion is a fourth order and mean curvature flow is a second order geometric evolution law. For
the stability analysis for mean curvature flow (1.1) with boundary condition (1.4) we refer to
[EY93, ESY96], where the results heavily depend on maximum principles.
2When we now draw our attention to the appearance of triple lines, we want to change the
consideredevolutionlawsslightlybyincludingsomeconstantsthatallowdifferentcontact angles
between the hypersurfaces. We assume that three evolving hypersurfaces Γ either fulfill thei
weighted mean curvature flow
V = γ H , (1.6)i i i
or the weighted surface diffusion flow
V = −m γ ΔH , (1.7)i i i i
each fori =1,2,3. Here theconstantsγ,m >0arethesurfaceenergydensityandthemobilityi i
of the evolving hypersurface Γ . If the three evolving hypersurfaces meet at a triple line L(t),i
we require that there the following conditions hold.
∠(Γ (t),Γ (t)) =θ , ∠(Γ (t),Γ (t)) =θ , ∠(Γ (t),Γ (t)) =θ , (1.8)1 2 3 2 3 1 3 1 2
γ H +γ H +γ H =0, (1.9)1 1 2 2 3 3
m γ ∇H n =m γ ∇H n =m γ ∇H n , (1.10)1 1 1 ∂Γ 2 2 2 ∂Γ 3 3 3 ∂Γ1 2 3
where the quantity ∠(Γ (t),Γ (t)) denotes the angle between Γ (t) and Γ (t) and the anglesi j i j
θ ,θ ,θ with 0 < θ < π are related through the identity θ +θ +θ = 2π and Young’s law,1 2 3 i 1 2 3
which is
sinθ sinθ sinθ1 2 3
= = . (1.11)
γ γ γ1 2 3
We can show that Young’s law (1.11) is equivalent to
γ n +γ n +γ n =0, (1.12)1 ∂Γ 2 ∂Γ 3 ∂Γ1 2 3
which is the force balance at the triple line.
For the derivation of the conditions (1.8)-(1.10) at the triple line, we refer to Garcke and
Novick-Cohen [GN00]. Th