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SELF-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 system, 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 stiffness 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 overburden pressure acting on the layer is q per unit length, therefore considering displacements 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]),wandwãreidentical 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 theorem [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