Table Of ContentSELF-SIMILAR VOIDING SOLUTIONS OF A SINGLE LAYERED
MODEL OF FOLDING ROCKS
T. J. DODWELL∗, M. A. PELETIER†, C. J. BUDD∗, AND G. W. HUNT∗
Abstract. In this paper we derive an obstacle problem with a free boundary to describe the
formation of voids at areas of intense geological folding. An elastic layer is forced by overburden
pressure against a V-shaped rigid obstacle. Energy minimization leads to representation as a non-
linearfourth-orderordinarydifferentialequation, forwhichweprovetheirexistsauniquesolution.
DrawingparallelswiththeKuhn-Tuckertheory,virtualwork,andideasofduality,wehighlightthe
physical significance of this differential equation. Finally we show this equation scales to a single
1 parametricgroup,revealingascalinglawconnectingthesizeofthevoidwiththepressure/stiffness
1 ratio. This paper is seen as the first step towards a full multilayered model with the possibility of
0 voiding.
2
Key words. Geological folding, voiding, nonlinear bending, obstacle problem, free boundary,
n Kuhn-Tuckertheorem
a
J
AMS subject classifications. 34B15,34B37,37J55,58K35,70C20,70H30,74B20,86A60
7
2
1. Introduction. The bending and buckling of layers of rock under tectonic
] plate movement has played a significant part in the Earth’s history, and remains of
S major interest to mineral exploration in the field. The resulting folds are strongly
P
influenced by a subtle mix of geometrical restrictions, imposed by the need for layers
.
n tofittogether,andmechanicalconstraintsofbendingstiffness,inter-layerfrictionand
i workeddoneagainstoverburdenpressureinvoiding. Anexampleofsuchafoldisseen
l
n inFigure1.1,herethevoidingisvisiblethroughtheintrusionofsoftermaterial(dark
[
in this figure) between the harder layers (shinier in the figure) which have separated
1 while undergoing intense folding.
v
5
2
3
5
.
1
0
1
1
:
v
i
X
r
a Fig. 1.1. A photograph of a geological formation from Millock Haven, Cornwall, UK, demon-
stratingtheformationofvoids,visiblebytheintrusionofsoftermaterial,whileharderlayersundergo
intense geological folding. Scale is approximately 5m across.
Consider a system of rock layers, of constant thickness, initially lying parallel
to each other that are then buckled by an external horizontal force, while being held
togetherbyanoverburdenpressure. Ifrocklayersdonotseparateduringthebuckling
∗BathInstituteofComplexSystems,UniversityofBath,BA27AY
†Institute of Complex Molecular Systems and Department of Mathematics and Computer Sci-
ence,TechnischeUniversiteitEindhoven,POBox513,5600MBEindhoven,TheNetherlands
‡Correspondingauthor: C.J.Budd([email protected])
1
2 Dodwell. TJ.et al.
process it is then inevitable that sharp corners will develop. To see this, consider a
single layer buckled into the shape of a parabola with further layers, of constant
thickness, lying on top of this. Moving from the bottom layer upwards, geometrical
constraints mean that the curvature of the individuals layer tightens until it becomes
infinite, marking the presence of a swallowtail singularity [1]. Beyond this singularity
the layers interpenetrate in a non-physical manner. This process is illustrated in
1456 J. A. Boon et al.
Figure 1.2, showing how the layers would continue through the singularity if they
were free to interpenetrate.
(a) (b)
–0.75
–0.80
–0.85
–0.90
–0.95
–1
0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.35 0.40 0.45 0.50 0.55 0.60 0.65
Fig. 1.2. A close-up view of the propagation of a sine wave, demonstrating the physically-
Figure5.Close-upofapropagationofacosinewave(a)usingtheLagrangianmethod,showingthe
unrealisable swallowtail catastrophe.
self-intersectingcurvesattheswallowtail.(b)UsingtheLSM,showingthelocalV-shapednatureof
thecurves.
Models have dealt with these singularities by for instance limiting the number
of layers [4, 7], using the concept of viscosity solutions [1], or postulating a simpli-
take an initial function f(x,y,0)ZyKcos(2px). This is not a signed distance
fied geometry of straight limbs punctuated by sharp corners, as is observed in kink
function,meaningthatgradientsaresteeperandthereforehardertoapproximate
banding [10, 18]. These approaches, however, disregard the resistance of the layers to
accurately numerically, but it does give an accurate and easy to implement
bending, which is exipneitcitaeldzetroobleeveeslpseecti.aWllyerneolewvacnotmcpluotseetthoetrheesuslitnignuglalaryiteyr.sHGeraendwecomparethese
t
therefore introduce twhiethptrhoepesortluytioofneslapsrteidcicstteidffnbeyssthienLtoagtrhaengmiaondefollrimngu,laatniodnc.oUmsibnignethis,theexact
itwithaconditionopfanroanm-ienttreicrpeeqnueattriaotnioonf.tAhesalayreesrualtt,tthheetliamyeerst wisilglivneontfibtytogether
completely, but do work against overburdern pressure and create voids.The folding of
xZsC 2psin 2ps t 1C4p2sin2 2ps K1=2;
rocks is a complex process with many interacðting faðctorÞs.ÞðIn a multilayðeredÞÞmodel it
is clear that work needs to be done toysZlidceosth2epslayCerts1ovCer4epa2cshino2t2hpers inK1t=h2e: presence
ð Þ ð ð ÞÞ
of friction, to bend the individual layers and finally to separate the layers (voiding).
Thecurvatureofthereferencecurvetakesitsmaximumvalueof4p2atthepoint
InordertounderstansZd1th/e2ianntderahcetnicoenabestinwgeuelnartihtyeporcoccuerssswohfebnentZdin1g/4apn2d, vxZoid1i/n2g,wyZe K1C1/4p2.
will not consider theAeffcleocstes-uopfofrficthtieonsiningutlhairsitpyaopfetrhebuextawctill(mleualvtei-vthailsuetdo)tshoelustuiobnjeacrtisingfromthe
of later work. Lagrangian description is plotted in figure 5a. We now compare the Lagrangian
The process ofsvooluidtiionngwisithilltuhsattradteerdiviendbFyiguusrieng1t.3h,ewLShMich.Ashcoawlcsulaatiloanboursaintogrtyhemethodfor
recreationoffoldinghrZoc0k.s01obatnadinDetdZb0y.0c0o5mispprreessseinngteldatinerfiaglluyreco5nbfi(nheedreltahyeercsororfepspaopnedr.ingclose-upof
As we move throughthethseinsgaumlapriltey,itshsehocwurnv)a.tOubreserovfetthheatla,yinercsonintrcarsetatsoetuhnetLilaagrapnoginiatndescription,
the LSM has deleted the self-intersecting part of the curves and the resulting
is reached where the work against pressure in voiding balances the work in bending
curveshaveanapparentgradientdiscontinuityatthecentre.ThelocalV-shaped
and the layers separate. A number of features of the voiding process can be seen in
nature of these curves is very similar to that of the layers in the chevron folding
this figure. It is clear that the voids have a regular and repeatable form and that a
pattern illustrated in figure 1b. Indeed, if we take G to be the V-shaped curve
0
typicalvoidoccurswGhehn{a(xs,myo):oxtZhlsa,yyeZrosfKpa1p/2er}stehpeanrathteesrfersoumltinongecawlchuiclahtihoansoaftnheearlayersG using
corner-shape. th0eLSMisgiveninfigjure6ajandaclose-upinfigure6b.WeseethattheLStMhas
In this paper wesupcrceessesnfutllaysciomppedlifiweidthentheerggyr-abdaiseendt sminogduellaroiftyv,oriedpinrogdiuncsipnigretdhebsyelf-replicating
the processes observfeedatiunreFiogf.p1a.3ra.llTelhfeomldiondgelincotnhsisisctsasoefwahseirneglaellellaaysteircslGaytehravweitehxaactly the same
shape and the same arc-length. We note that in figure 6b, the corner is slightly
verticaldisplacementw(x)forceddownwards,andbent,intoacorner-shapedobstacle
smoothedduetotheerror.Thiseffectcanbereducedbyrefiningthemesh.
of shape f(x) w(x) by a uniform overburden pressure q (see Fig. 1.4). The corner
≤ As a further measure of the accuracy of the calculation of the propagating
is defined to have infinite curvature at the point, x=0. For x sufficiently large, the
cosinereferencecurveG h{(x,y):xZs,yZcos(2ps)},0%s%1,weplotthetotal
0 | |
layer and obstacle are in contact so that w = f. However, close to x = 0 the layer
arc-length of the resulting curves. The choice of reference curve ensures that
[q]Z0. Hence, from theorem 2.1, the total arc-length of the curve G stays
t t
Proc.R.Soc.A(2007)
Self-similarvoidingsolutionsoffoldingrocks 3
and obstacle separate, leading to a single void for those values of x for which w >f.
We study both the resulting shape of this elastic layer and the size of the voiding
region. This investigation is the first part of a more general study of the periodic
multi-layered voiding pattern seen in Fig. 1.3.
To study this situation we construct a potential energy functional V(w) for the
system,derivedinSection2,whichisgivenintermsoftheverticaldisplacementw(x)
and combines the energy U required to bend the elastic layer and the energy U
B V
required to separate adjacent layers and form voids. The potential energy function is
then given by
B (cid:90) ∞ w2 (cid:90) ∞
V =U +U xx dx+q (w f)dx, where w f. (1.1)
B V ≡ 2 (1+w2)5/2 − ≥
−∞ x −∞
The resulting profile is then obtained by finding the minimiser of V over all suitably
regular functions w f. This constrained minimization problem is closely related to
≥
manyotherobstacleproblems,ascanbefoundinthestudyoffluidsinporousmedia,
optimal control problems, and the study of elasto-plasticity [8].
Whileobstacleproblemsareoftencastasvariationalinequalities[12],hereweuse
the Kuhn-Tucker theorem for its suitability when interpreting the results physically.
In Section 2 we prove various qualitative properties of constrained minimizers, and
usetheEuler-Lagrangeequationtoderiveafourth-orderfree-boundaryproblemthat
they satisfy.
Inaddition,weshowthatstationarityimpliesthatacertainquantity(the‘Hamil-
tonian’) is constant in any region of non-contact (Section 3). This property extends
the well-known property of constant Hamiltonian in spatially invariant variational
problems, going back to Noether’s theorem. However, we also give a specific inter-
pretation of both the fourth-order differential equation and the Hamiltonian in terms
of horizontal and vertical variations, with clear analogues with the concept of virtual
work. Here horizontal and vertical variations define virtual displacements on the sys-
tem, and the resulting ODEs describe the required load balance at a given point of
a stationary solution. In Section 3.3 we show how integration of the Euler-Lagrange
equation and the Hamiltonian gives vertical and horizontal force balances for the
system, where individual terms can be identified with their physical counterpart.
Fig. 1.3. A laboratory experiment of layers of paper constrained and loaded. In this figure
the black lines are for illustrative purposes, and are produced by inserting a single black layer of
paperbetween25layersofwhite. Theresultingdeformationshowstheformationofvoidswhenthe
imposed curvature becomes too high. Note the regular and repeatable nature of the voids.
4 Dodwell. TJ.et al.
w
q (cid:96) (cid:96)
− +
s
f =k|x|
x
Fig. 1.4. This figure shows the setup of the model discussed in this paper. An overburden
pressureq forcesanelasticlayerwintoanotherlayerf withacornersingularity. (cid:96)+ and(cid:96)− define
the first points of contact either side of the centre line. In this paper the layer is described both by
Cartesian coordinates (x,w) measured from the centre of the singularity, and intrinsic coordinates
characterised by arc length s and angle ψ.
Section 4 gives a shooting argument that shows there exists a unique solution to
this obstacle problem. These can be rescaled to form a one-parameter group, which
gives the main result of Section 5:
Theorem1.1. Givenk >0sothatf =k x,thereexistsaconstantβ =β(k)>0
| |
such that for all q > 0 and B > 0, the horizontal size of the void (cid:96) and the vertical
shear force at the point of contact Bw ((cid:96) ) scales so that
xxx
−
(cid:16)q (cid:17)−1/3 β (cid:16)q (cid:17)2/3
(cid:96)=β Bw ((cid:96) )= B
B xxx − − (1+k2)5/2 B
InSection6weshowthattheseanalyticalresultsagreewiththenumerics,aswell
as with physical intuition. As the ratio of overburden pressure to bending stiffness
becomeslarge, thesizeofthevoidtendstozero, givingadeformationwithstraighter
limbsandsharpercorners. Byallowingthelayerstoformavoid,themodeliscapable
of producing both gently curving and sharp-cornered folds, without violating the
elastic assumptions. Understanding this local behavior at areas of intensive folding
may be seen as a first step to a multilayered model with the possibility of voiding.
2. A voiding model close to a geometric singularity.
2.1. The modelling. We consider an infinitely long thin elastic layer, of stiff-
ness B, whose deformation is characterized by its vertical position w(x) as a function
of the horizontal independent variable x R. Overburden pressure, from the weight
∈
of overlying layers, acts perpendicularly to the layer with constant magnitude q per
unit length. The layer is constrained to lie above the a V-shaped obstacle, defined by
the function f(x) = k x, i.e. w f. Although we appear to solve the problem for
| | ≥
an infinitely thin layer, the analysis is the same for any layer of uniform thickness up
to changes of stiffnes B. In all cases w(x) defines the ceterline of the layer, and f(x)
defines the shape the layer would take in the absense of voids. This is only possible
in this special case since f has straight limbs, and can therefore be propagated for-
wards and backwards without change. The setup and parameters of the model are
summarised in Fig. 1.4.
Self-similarvoidingsolutionsoffoldingrocks 5
The contact set of a function w is the set Γ(w) = x R : w(x) = f(x) ,
{ ∈ }
the non-contact set Γc(w) is its complement, and we define the two contact limits
(cid:96) =inf x>0:u(x)=f(x) and (cid:96) =sup x<0:u(x)=f(x) .
+ −
{ } { }
We now derive a total potential energy function for the system, described by the
displacement w.
2.1.1. Bending Energy. Classic bending theory (e.g. [17, Ch. 1]) gives the
bending energy over a small segment of the beam ds as dU = Bκ(s)2ds, where κ is
B 2
curvature. Integrating over all s we find
B (cid:90) ∞ B (cid:90) ∞ w2 ds B (cid:90) ∞ w2
U = κ2ds= xx dx= xx dx
B 2 2 (1+w2)3dx 2 (1+w2)5/2
−∞ −∞ x −∞ x
The quadratic dependence on w implies that a sharp corner has infinite bending
xx
energy. Thisisthebasicreasonwhyatanyfiniteoverburdenpressuretheelasticlayer
will show some degree of voiding.
2.1.2. Work done against overburden pressure in voiding. The overbur-
den pressure acting on the layer is q per unit length, therefore considering displace-
ments w for which w f the work done by overburden pressure in voiding is given
≥
by q(w f)dx, and integrating over all x gives
−
(cid:90) ∞
U =q (w f)dx.
V
−
−∞
We see that if q is large, then U becomes a severe energy penalty.
V
2.1.3. Total potential energy. The total potential energy function is the sum
of bending energy and work done against overburden pressure,
B (cid:90) ∞ w2 (cid:90) ∞
V = xx dx+q (w f)dx (2.1)
2 (1+w2)5/2 −
−∞ x −∞
Thesolutionsofthesystemarethenminimizersoftheenergyfunctional(2.1)subject
to the constraint w f.
A natural space≥on which to define V is the complicated-looking H2 (R) (f +
L1(R)). Here H2 (R) is the space of all functions with second derivativloecs in∩L2(K)
loc
for any compact set K R. Finiteness of the first term in V requires (at least)
w H2 (R), and well-de⊂finedness of the second term requires that w f L1(R).
∈ loc − ∈
However, under the conditions w f and V(w)< these conditions are automati-
≥ ∞
cally met, and therefore we will not insist on the space below.
2.2. Constrained minimization of total potential energy.
2.2.1. Properties and existence of minimizers. Before deriving necessary
conditions on minimizers of (2.1) under the condition w f, we first establish a few
≥
basic, but important, properties. These are that a constrained minimizer exists, is
necessarilyconvexandsymmetric,andhasasingleintervalinwhichitisnot incontact
with the obstacle. We will prove uniqueness using different methods in Section 4.
Wewritew# fortheconvexhullofw,i.e. thelargestconvexfunctionv satisfying
v w. If w f, then since f is convex, it follows that w# f.
≤ ≥ ≥
Theorem 2.1. For any w, V(w#) V(w), and any constrained minimizer w is
convex. For all x R, k w (x) k.≤
x
∈ − ≤ ≤
6 Dodwell. TJ.et al.
Proof. First we note that if w H2 (R), also w# H2 (R). Indeed, by consid-
∈ loc ∈ loc
ering expressions of the form
(cid:90) x2
w#(x ) w#(x )= w#(x)dx,
x 2 − x 1 xx
x1
it follows that the measure w# is Lebesgue-absolutely continuous, and satisfies 0
w# w . Since w L2(Rxx), it follows that w# L2(R). Then w# H2(K) fo≤r
alxlxco≤m|paxcxt|K R byxxin∈tegration. xx ∈ ∈
Defining th⊂e set Ω := x R : w#(x) = u(x) , the function w# is twice differ-
{ ∈ }
entiable almost everywhere on Ω, with a second derivative w# equal to w almost
xx xx
everywhere on Ω. On the complement Ωc, w# =0 by [9, Theorem 2.1].
xx
Substituting w# into (2.1) shows that V(w) V(w#), with equality only if
≥
w# =w. Since w minimizes V, we have w =w#, and therefore w is convex.
The restriction on the values of w follows from the monotonicity of w and the
x x
fact that w f tends to zero at .
− ±∞
As a direct consequence of Theorem 2.1,
Theorem 2.2. The non-contact set Γc(w) of a minimizer w is an interval con-
taining x=0, and for all x (cid:96) and x (cid:96) we have w(x)=f(x).
+ −
≥ ≤
Note that this statement still allows for the possibility that (cid:96) = .
±
±∞
Proof. Suppose that x ,x >0 are such that w =f at x=x and at x=x . By
1 2 1 2
convexity of w we then have w = f on the interval [x ,x ]. If the contact set Γ(w)
1 2
is bounded from above, then by the convexity of w, there exists ε > 0 and a > 0
such that w(x) a+(k+ε)x for all x R, implying that U (w) = . Therefore
V
≥ ∈ ∞
Γ(w) [0, ) is an interval, and if it is non-empty, then it is necessarily extends to
∩ ∞
+ . Similarly, Γ(w) ( ,0] is an interval, and if non-empty it extends to .
∞ ∩ −∞ −∞
Finally, note that x = 0 can not be a contact point, since the condition w f
would imply that w H2 (R). Therefore the non-contact set Γc(w) is an inte≥rval
(cid:54)∈ loc
that includes x=0.
Theorem 2.3. Any minimizer w is symmetric, so that w(x)=w( x).
−
Proof. We proceed by using a cut-and-paste argument. If w is a minimizer, then
it follows from Theorems 2.1 and 2.2 that w is convex and for all x (cid:96) and x (cid:96) ,
+ −
≥ ≤
w(x) = f(x). Therefore w ((cid:96) ) = k, and the intermediate value theorem states
x ±
±
that there exists xˆ ((cid:96) ,(cid:96) ) such w (xˆ) = 0. If V (w) V (w), then we
− + x [−∞,xˆ] [xˆ,∞]
∈ ≥
define the function
(cid:26)
w(x+xˆ) k xˆ if x<0;
w˜ = − | |
w( (x+xˆ)) k xˆ if x>0.
− − | |
If V (w) V (w), then we define w˜ as
[−∞,xˆ] [xˆ,∞]
≤
(cid:26)
w( (x+xˆ)) k xˆ if x<0;
w˜ = − − | |
w(x+xˆ) k xˆ if x>0.
− | |
In either case w˜ H2 , w˜ is symmetric, and V(w˜) V(w).
∈ loc ≤
Since w˜ is a minimizer, w˜ solves a fourth-order differential equation in its non-
contact set Γ(w˜)c (which includes x = 0; see (2.6) and Remark 2.9). By standard
Self-similarvoidingsolutionsoffoldingrocks 7
uniquenesspropertiesofordinarydifferentialequations(e.g.[5]),wandw˜areidentical
on both sides of x=0, and remain such until they reach the constraint f. Therefore
w w˜ and is therefore symmetric.
≡
Corollary 2.4. Since w is symmetric, (cid:96) = (cid:96) =(cid:96).
+ −
−
Finally, these assembled properties allow us to prove the existence of minimizers:
Theorem 2.5. There exists a minimizer of V subject to the constraint w f.
≥
Proof. Let w be a minimizing sequence. By Theorems 2.1 and 2.3 we can
n
assume that w is convex and symmetric, and we therefore consider it defined on
n
R+. By the convexity, since w (x) f(x) 0 as x , the derivative w
n n,x
− → → ∞
converges to f(cid:48)( ) = k as x ; therfore the range of w is [0,k]. Since by
n,x
convexity(cid:82)∞(w∞ f)dx w (→0)2∞/2k,theboundednessofV(w )impliesthatw (0)
0 n− ≥ n n n
is bounded.
From the upper bounds on w it follows that w is bounded in L2(R+);
n,x n,xx
combinedwiththeboundsonw (0)andw (0)=0, thisimpliesthatasubsequence
n n,x
convergesweaklyinH2(K)tosomewforallboundedsetsK [0, ). Sincetherefore
⊂ ∞
w converges uniformly on bounded sets, it follows that w (0)=0 and that
n,x x
(cid:90) ∞ w2 (cid:90) ∞ w2
liminf n,xx dx xx dx.
n→∞ (1+w2 )5/2 ≥ (1+w2)5/2
0 n,x 0 x
Similarly, uniform convergence on bounded sets of w , together with positivity of
n
w f, gives by Fatou’s Lemma
n
−
(cid:90) ∞ (cid:90) ∞
liminf (w f)dx (w f)dx.
n
n→∞ − ≥ −
0 0
Therefore V(w) liminfV(w ), implying that w is a minimizer.
n
≤
2.2.2. The Euler-Lagrange equation. We now apply the Kuhn-Tucker theo-
rem [14, pp. 249] to derive necessary conditions for minimizers of (2.1) subject to the
constraint w f. Since any minimizer w is symmetric by Theorem 2.3, we restrict
ourselvestosy≥mmetricw,andthereforeconsiderw definedonR+ withthesymmetry
boundary condition w (0)=0.
x
Theorem 2.6. Let q,B,k >0. Define the set of admissible functions
=(cid:8)w f +H2(R+) L1(R+):w (0)=0(cid:9). (2.2)
x
A ∈ ∩
If w minimizes (2.1) in subject to the constraint w f, then it satisfies the sta-
A ≥
tionarity condition
(cid:90) ∞(cid:20) w 5 w2 w (cid:21) (cid:90) ∞
B xx ϕ B xx x ϕ +qϕ dx= ϕdµ, (2.3)
(1+w2)5/2 xx− 2 (1+w2)7/2 x
0 x x 0
for all ϕ H2(R+) L1(R+) satisfying ϕ (0)=0, where µ is a non-negative measure
x
∈ ∩ (cid:82)∞
satisfying the complementarity condition (w f)dµ=0.
0 −
Proof. FortheapplicationoftheKuhn-Tuckertheoremwebrieflyswitchvariables,
and move to the linear space X :=H2(R+) L1(R+), taking as norm the sum of the
∩
respectivenormsofH2andL1. Foranyw ,wedefinethevoidfunction v :=w f,
∈A −
which is an element of X; the two constraints v (0) := w (0) f (0+) = k and
x x x
− −
8 Dodwell. TJ.et al.
v := w f 0 are represented by the constraint (v) 0, where : X Z :=
R R −H2(≥R+) is given by G ≤ G →
× ×
v (0)+k
x
(v):= vx(0) k .
G − −
v
−
We also define Vˆ(v):=V(v+f).
If w satisfies the conditions of the Theorem, then the corresponding function
v X minimizes Vˆ subject to (v) 0. The functionals Vˆ :X R and :X Z
∈ G ≤ → G →
are Gateaux differentiable; since is affine, v is a regular point (see [14, p. 248]) of
G
the inequality (v) 0. The Kuhn-Tucker theorem [14, p. 249] states that there
G ≤
exists a z∗ in the dual cone P∗ = z∗ Z∗ : z∗,z 0 z Z with z 0 of the
{ ∈ (cid:104) (cid:105)≥ ∀ ∈ ≥ }
dual space Z∗, such that the Lagrangian
():=Vˆ()+ (),z∗ (2.4)
L · · (cid:104)G · (cid:105)
is stationary at v; furthermore, (v),z∗ =0.
Thisstationaritypropertyis(cid:104)eGquivale(cid:105)ntto(2.3). ThederivativeofVˆ inadirection
ϕ X gives the left-hand side of (2.3); the right-hand side follows from the Riesz
∈
representation theorem [16, Th. 2.14]. This theorem gives two non-negative numbers
λ and λ and a non-negative measure µ such (a,b,u),z∗ = λ a+λ b+(cid:82)∞udµ
fo1r all a,2b R and u X. Therefore (cid:48)(ϕ),z(cid:104)∗ = (cid:82) ϕ(cid:105)dµ fo1r any2ϕ X0 with
∈ ∈ (cid:104)G (cid:105) − ∈
ϕ (0)=0.
x
In addition, the complementarity condition G(v),z∗ = 0 implies (cid:82)∞vdµ =
(cid:82)∞ (cid:104) (cid:105) 0
(w f)dµ=0.
0 −
Thisstationaritypropertyallowsustoprovetheintuitiveresultthatallminimiz-
ers make contact with the support f:
Corollary 2.7. Under the same conditions the non-contact set, Γ(w)c, is
bounded, i.e. (cid:96)< .
∞
Proof. Assume that the contact set Γ(w) is empty, implying µ 0. In (2.3)
take ϕ (x) := ϕ(x n) for some ϕ C∞(R) with (cid:82) ϕdx = 0. Sinc≡e w f L1
n − ∈ c (cid:54) − ∈
and w L2, we have w (x) k as x ; therefore, as n , the translated
xx x
∈ → → ∞ → ∞
function
w
y xx (y+n)
(cid:55)→ (1+w2)5/2
x
converges weakly to zero in L2, implying that the first term in (2.3), with ϕ=ϕ ,
n
(cid:90) ∞ w (cid:90) ∞ w
xx ϕ dx= xx (y+n)ϕ (y)dy
(1+w2)5/2 n,xx (1+w2)5/2 xx
0 x −n x
vanishes in the limit n . The second term vanishes for a similar reason. In the
→ ∞ (cid:82)
limit n we therefore find q ϕdx=0, a contradiction.
→∞
The boundedness of the non-contact set now allows us to apply a bootstrapping
argument to improve the regularity of a minimizer w, and derive a corresponding
free-boundary formulation:
Self-similarvoidingsolutionsoffoldingrocks 9
Theorem 2.8. Under the same conditions as Theorem 2.6, the function w has
the regularity w C∞(Γ(w)c) C2(R+), w is bounded, and w is a measure;
xxx xxxx
∈ ∩
the Lagrange multiplier µ is given by
µ=q(cid:96)δ +qH( (cid:96))L, (2.5)
(cid:96)
· −
where H is the Heaviside function, and L is one-dimensional Lebesgue measure. In
addition, w and µ satisfy
(cid:20) w w w w 5 w3 35 w3 w2 (cid:21)
B xxxx 10 x xx xxx xx + xx x +q =µ (2.6)
(1+w2)5/2 − (1+w2)7/2 − 2(1+w2)7/2 2 (1+w2)9/2
x x x x
in R+.
Finally, w also satisfies the free-boundary problem consisting of equation (2.6) on
(0,(cid:96)) (with µ=0), with fixed boundary conditions
w (0)=0 and w (0)=0, (2.7)
x xxx
and a free-boundary condition at the free boundary x=(cid:96),
w((cid:96))=k(cid:96), w ((cid:96))=k, and w ((cid:96))=0. (2.8)
x xx
Before proving this theorem we remark that by integrating (2.6) we can obtain
slightly simpler expressions. From integrating (2.6) directly, and applying (2.7), we
find
w 5 w2 w
B xxx B xx x +qx=qxH(x (cid:96)) for all x>0. (2.9)
(1+w2)5/2 − 2 (1+w2)7/2 −
x x
By substituting the free boundary conditions at x=(cid:96) into (2.9) we also find that the
limiting values of w at x=(cid:96) are given by
xxx
q
w ((cid:96) )= (1+k2)5/2 (cid:96), w ((cid:96)+)=0. (2.10)
xxx − − B xxx
In addition, by multiplying (2.9) by w and integrating we also obtain
xx
B w2 B
xx +q(xw w)= w (0)2 qw(0). (2.11)
2 (1+w2)5/2 x− 2 xx −
x
Note that the right-hand side of (2.9) does not contribute to the the integral since
w = 0 for all x (cid:96). Substituting the boundary conditions (2.8), we derive the
xx
≥
condition
1
Bw (0)2 =qw(0), (2.12)
2 xx
so that the previous equation becomes
B w2
xx +q(xw w)=0. (2.13)
2 (1+w2)5/2 x−
x
10 Dodwell. TJ.et al.
Proof of Theorem 2.8. Once again we switch variables to the void function,
v :=w f and define the functions
−
v 5 v2 (v +k)
g =B xx and h= B xx x ,
(1+(v +k)2)5/2 −2 (1+(v +k)2)7/2
x x
by (2.3) we make the estimate
(cid:90) (cid:90) (cid:90)
gϕ = hϕ + (µ q)ϕ h ϕ + µ q ϕ
xx x 2 x 2 TV ∞
R+ − R+ R+ − ≤(cid:107) (cid:107) (cid:107) (cid:107) (cid:107) − (cid:107) (cid:107) (cid:107)
C( ϕ + ϕ ),
x 2 x 1
≤ (cid:107) (cid:107) (cid:107) (cid:107)
where the total variation norm ν is defined by
TV
(cid:107) (cid:107)
(cid:110)(cid:90) (cid:111)
ν :=sup ζdν :ζ C(R+), ζ < .
TV ∞
(cid:107) (cid:107) R+ ∈ (cid:107) (cid:107) ∞
Settingϕ =ψ,itfollowsthatg isweaklydifferentiable,andg L2+L∞. From
x x
∈
Theorem 2.2 and Corollary 2.7, v 0 g =0 and therefore g L2. By
((cid:96),∞) x ((cid:96),∞) x
| ≡ ⇒ | ∈
calculating g explicitly, we may write
x
v2 (v +k)
vxxx =(1+(vx+k)2)25gx+51+xx(vx+k)2 . (2.14)
x
(cid:124) (cid:123)(cid:122) (cid:125)
∈L1
Theorem 2.1 shows that (1 + (v + k)2)5/2 L∞, therefore v L1, so that
x xxx
∈ ∈
v L∞, which in turn shows that v L2 by (2.14). We also see that since
xx xxx
∈ ∈
2v v (v +k) v3 v3 (v +k)2
h = xx xxx x xx +7 xx x , (2.15)
x −(1+(v +k)2)7/2 −(1+(v +k)2)7/2 (1+(v +k)2)9/2
x x x
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)
∈L2 ∈L∞
wehaveh L2. Wenowlooktosimilarlyboundv . Inthesenseofdistributions,
x xxxx
∈
we have
g = h +µ q, (2.16)
xx x
− −
and since h has bounded support, this is an element of , the set of measures with
x
M
finite total variation. We can now write
∈L2 ∈L∞ ∈L∞
(cid:122) (cid:125)(cid:124) (cid:123) (cid:122)(cid:125)(cid:124)(cid:123) (cid:122) (cid:125)(cid:124) (cid:123)
53v v (v +k)+ v3 35 v2 (v +k)2
v = (1+(v +k)2)5/2 g + xxx xx x xx xx x
xxxx (cid:124) x(cid:123)(cid:122) (cid:125) (cid:124)(cid:123)x(cid:122)x(cid:125) 2 (1+(vx+k)2 − 2 (1+(vx+k)2)3/2
continuousandbounded ∈M
Since v has finite total variation, v is bounded. Calculating (2.16) explicitly
xxxx xxx
we find
(cid:20) v (v +k)v v 5 v3 (cid:21)
B xxxx 10 x xx xxx xx +
(1+(v +k)2)5/2 − (1+(v +k)2)7/2 − 2(1+(v +k)2)7/2
x x x
(cid:20)35 v3 (v +k)2 (cid:21)
+B xx x +q =µ.
2 (1+(v +k)2)9/2
x
