## Abstract

This is a mathematical study of steady two-dimensional waves of prescribed period which propagate without change of form on the surface of an infinitely deep expanse of fluid that is moving under gravity and bounded above by heavy, frictionless, thin elastic sheet. The flow, which is supposed irrotational, is at rest at infinite depth and its velocity is stationary relative to a frame moving with the wave. In that frame, the elastic sheet coincides with the zero streamline and its material points move according to the equations of hyperelasticity.

From the mechanics of the surface material, the pressure on a point of the sheet in the moving frame is shown to be a function of its height, its slope and derivatives with respect to arc length of these quantities. Independently, according to classical fluid mechanics, the pressure in the fluid at the same point is determined by its height and the fluid velocity tangent to the zero streamline.

Therefore, this is a free-boundary problem: to find a non-self-intersecting curve in the plane which is the zero contour of a harmonic function (the stream function) and at which the normal derivative of the same harmonic function is a prescribed function of the shape of the curve.

With the wavelength fixed, the parameters are the density *ρ* of the undeformed sheet, the wave velocity *c*_{0}, the mean velocity *c* of the surface sheet relative to the wave and the gravitational acceleration *g*. The case of a weightless sheet, in which *ρ*=0 and *c* does not appear, is a special case. Under quite general hypotheses on the elastic response of the sheet, the existence of rapidly propagating or very slow (depending on parameter values) steady waves is established by finding a critical point of the Lagrangian. This is a saddle-point problem in the calculus of variations.

## 1. Introduction

Here, we define a hydroelastic wave as the steady irrotational periodic motion under gravity of an inviscid incompressible liquid when the top streamline is in contact with a frictionless elastic sheet. Mathematical work on such problems began with the linear theory of Greenhill (1886), and recently there have been numerical studies of large-amplitude solutions to nonlinear models, for example Forbes (1986, 1988) and Hegarty & Squire (2002). The existence question for hydroelastic waves on two-dimensional flows of infinite depth when the sheet's elastic response may be strongly nonlinear, but when its mass is neglected, was formulated by Toland (2006) as a variational problem. For a class of problems in which the stored energy is a sum of the effects of bending and stretching, the existence of waves propagating with arbitrarily large speeds was established by maximizing a Lagrangian over a class of admissible functions. In the present paper, we develop the variational approach to deal with heavy sheets and more general elastic responses.

### (a) The physical problem

We consider waves that are periodic and travel with speed *c*_{0} on the surface of an inviscid fluid which is at rest at infinite depth and occupies the region beneath a heavy thin elastic sheet. We assume that, relative to a frame moving with the wave speed, the wave profile and the fluid's Eulerian velocity field are stationary, but the material of the sheet is in motion. A steady wave is then one for which the fluid pressure and the force of gravity lead to a solution of the equations of hyperelasticity for the positions, in Lagrangian coordinates, of points on the sheet that are everywhere in contact with the fluid surface. We will see that such waves are given by non-trivial solutions of the free-boundary value problem (1.17*a*)–(1.17*d*) and that the material in the surface sheet may drift relative to the wave. The approach is to reduce (1.17*a*)–(1.17*d*) to a saddle-point problem for the natural Lagrangian associated with this solid–fluid interaction and to show that such saddle points exist (theorem 3.6 and lemma 3.7).

The fluid motion is supposed to be two-dimensional, steady, irrotational and 2*π* periodic (the velocity field is irrotational in the (*x*, *y*)-plane, 2*π* periodic in the *x*-direction and stationary with respect to a frame moving with the velocity *c*_{0} of the surface profile). The intersection of the sheet with the plane *z*=0, called the sheet section, will be supposed to behave dynamically like a uniform, thin, hyperelastic rod, as described by Antman (1995), with a stored energy function that depends on stretch and curvature. By the reference section is meant the line *y*=*z*=0, and steady travelling waves are sought which satisfy the constraint that

(★) at any moment of time the material in an interval of length 2

*π* of the reference sheet is deformed to become one period of the wave surface.

We suppose that the position, at time *t*, of the material point with Lagrangian coordinates (*X*, 0) in the reference section is given by(1.1)where *u* and *v* are 2*π* periodic and . (We seek solutions of the equations of hyperelasticity in this particular form.) Then, the wave profile at time *t* is the curveThe third line of the above equation shows that describes a fixed profile propagating from left to right without changing its shape at a constant speed *c*_{0}; the first line shows that the material point with Lagrangian coordinates (*X*, 0) has temporal period 2*π*/*c* relative to a frame moving with speed *d*. Thus, *c*_{0} is the wave speed and *d* is the drift velocity of the material in the surface sheet, both of which are calculated relative to the fluid at rest at infinite depth. The drift velocity *d* might be zero, in which case the material points move periodically in time with respect to the fluid at rest at infinite depth or, if *c*_{0}*d*<0, the surface sheet may drift in the opposite direction to that of wave propagation. We suppose throughout that *c*_{0}≥0 and .

#### (i) Mechanics

We study the mechanics of the sheet by assuming that its section behaves like a thin, nonlinear elastic rod under the combined effects of gravity and the pressure exerted by the fluid with which it is in contact. From classical rod theory, it follows that ** R** in (1.1) must be a solution of the system(1.2)(1.3)which coincides with eqns (1.45) and (1.46) in Antman (1995) when the sheet is thin. Here,

*ρ*≥0 is a constant representing the density of the material

*in the reference configuration*(

*ρ*=

*ρA*in Antman's notation);

**and**

*n***are the resultant contact force and couple, respectively;**

*m***is the body force**

*f**per unit reference length*, arising from the pressure in the fluid and the force of gravity on the sheet; and

**is a unit vector normal to the (**

*k**x*,

*y*)-plane.

Following Antman (1995), we define a coordinate frame at by specifying unit vectors ** a**(

*X*,

*t*) and

**(**

*b**X*,

*t*) where , and real variables

*ν*(

*X*,

*t*) and

*μ*(

*X*,

*t*), as follows (figure 1):Note that , , and the curvature of the sheet section at is

*σ*(

*X*,

*t*)=

*μ*/

*ν*. With this notation, we adopt the constitutive assumption of Antman (1995), eqn (1.19), that the resultant contact force and couple are given, respectively, by(1.4)Let

*be defined by(1.5)where with for .*

**r**Since the fluid velocity is stationary with respect to a frame moving with velocity *c*_{0}, and, owing to (1.5), the fluid pressure *P* at is a function of *X*−*ct*, and so ** f** is also a function of

*X*−

*ct*. Therefore, we suppose that

*and*

**m***, and consequently*

**n***ν*,

*μ*and

*ϑ*, are functions of

*X*−

*ct*. This yields a system to be satisfied by

**,(1.6a)(1.6b)where ′ means differentiation with respect to**

*r**s*. From now on,

**and**

*m***, which are given by (1.4), are considered as functions of**

*n**s*(=

*X*−

*ct,*where

*c*=

*c*

_{0}−

*d*).

#### (ii) Fluid pressure

Since(1.4), (1.6*a*) and (1.6*b*) yieldLet . Then,(1.7a)(1.7b)(1.7c)From (1.7*a*) and (1.7*c*), we find that(1.7d)and from (1.7*b*) and (1.7*c*) that(1.7e)If the sheet section is hyperelastic in the sense of Antman (1995), eqn (1.33*a*), there exists a function *E* such thatThe idea now is to note thatso that the travelling-wave problem (1.7*a*)–(1.7*e*) reduces to a case of (1.6*a*) and (1.6*b*) with zeroes on the right-hand sides. In other words, our travelling-wave problem is equivalent to a static hyperelastic rod problem with a different stored energy function. Note from (1.7*d*) that(1.8)In this paper, we assume that *E* has a particular form (as in Toland (2006))where ** e**, a function of stretch and curvature, is a stored energy function. Then,where we call(1.9)the pseudo-stored energy function. It is important to remember that depends on

*c*and

*ρ*if

*ρ*>0 (the dependence is omitted from the notation for convenience only). Hence,(1.10a)(1.10b)(1.10c)where and denote partial derivatives of with respect to

*ν*and

*σ*, respectively, and and are partial derivatives with respect to

*ν*and

*μ*, respectively. Thus, (1.8) is equivalent to(1.11)where and

*γ*is a constant. Under hypotheses (H1)–(H3) in §2

*a*, this means that , where

*ϖ*is defined by (2.4) and

*γ*=

*γ*(

*σ*,

*n*) is chosen to ensure that (★) in §1

*a*is satisfied. In other words,

*γ*depends on the functions

*η*and

*σ*via the formula(1.12)where

*ς*denotes arc length along one period of , the sheet section. Now, for any real numbers

*γ*

_{1}and

*σ*

_{1}, let(1.13)(1.14)Note that

*ϖ*, defined by (2.4), and depend on

*c*, and hence on

*c*

_{0},

*d*and

*ρ*. From (1.10

*b*) and (1.11),(1.15)and it follows from (1.7

*e*) that the pressure exerted on the sheet by the fluid beneath must be determined by its position and curvature through the formula(1.16)in which

*γ*=

*γ*(

*σ*,

*η*), is given by (1.15) and

*ς*is arc length.

Note that this formula for *P* depends only on the *geometric shape* of the sheet section and does not require knowledge of the displacement of the material points of the reference section. In the physical domain, the unknown region occupied by the liquid is characterized by the kinematic requirement that the surface is a streamline and the dynamic condition that the pressure in the fluid and the effect of gravity yield the force needed to deform the sheet. Therefore, the existence of a steady hydroelastic wave with speed *c*_{0}, when the drift velocity of the surface elastic sheet is *d* and (★) holds, means the existence of a non-self-intersecting smooth curve in the plane which is 2*π* periodic in the horizontal direction and for which there exists a solution of the following system:(1.17a)(1.17b)(1.17c)with the dynamic (pressure) boundary condition(1.17d)where *P*(*ς*) is given by (1.16). The question now is ‘are there any non-trivial solutions of (1.17*a*)–(1.17*d*)?’ In Toland (2006), the free-boundary value problem (1.17*a*)–(1.17*d*) with *ρ*=0 was inferred from the Lagrangian variational principle that comes from the Hamiltonian system of Zakharov (1968), as formulated by Dyachenko *et al.* (1996), when travelling waves are sought. In §2, we return to that variational viewpoint to prove the existence of travelling waves in the general case.

### (b) Hypotheses and main result

In the following sections, we make hypotheses (H1)–(H4) on ** e** that lead to a saddle-point formulation of the hydroelastic-wave problem, and additional hypotheses (H5) and (H6) that yield the existence of a non-trivial solution. Roughly speaking, they may be summarized as follows:

the stored energy function

** e** in (2.1) has the property that , where

*κ*≥0 is a constant, and the mapping(H)is strictly convex and tends to ∞ as

*t*→∞ or 1/

*α*, and as |

*σ*|→

*β*, where

*αβ*<1.

Suppose that the elasticity of the surface sheet is given by a stored energy function ** e** that satisfies (H1)–(H6) and

*g*>0 is given. For given wave velocity

*c*

_{0}and sheet drift velocity

*d*, let

*c*=

*c*

_{0}−

*d*(see §1

*a*). A consequence of our existence results, theorem 3.6 and lemma 3.7, is the following. Suppose that

*c*

^{2}

*ρ*≤

*κ*and that the stored energy function

**satisfies (3.11). Suppose also thatThen, there exists a 2**

*e**π*-periodic curve of class

*C*

^{2,1/2}such that the

*C*

^{2,1/2}-solution

*ψ*of the Dirichlet problems (1.17

*a*)–(1.17

*c*) also satisfies the dynamic boundary condition (1.17

*d*).

A particular case occurs when

*ρ*is sufficiently large, depending onand*e**g*, and*c*is sufficiently small, depending on*ρ*and,Hence, when*e**ρ*is sufficiently large, there exist waves for all*c*_{0}and*d*with sufficiently small (*c*_{0}and*d*may have opposite signs and either*c*_{0}or*d*may be zero).The case

*c*=*c*_{0}=*d*=0 and*ρ*is sufficiently large is included. This limiting case corresponds to*static hydroelastic waves*in which there is no motion in the surface sheet or in the fluid and elastic stresses are balanced by hydrostatic pressure.Another limiting case is when

*ρ*=0 and*c*_{0}is sufficiently large.

Thus, depending on the density of the surface sheet, very slow or very fast hydroelastic waves are proved to exist, and the drift can have either sign depending on circumstances.

## 2. Variational principle

In this section, we begin the variational formulation of (1.17*a*)–(1.17*d*) and introduce hypotheses in pursuit of an existence theory. The treatment of kinetic energy and gravitational potential energy of the fluid will be the same as in the Lagrangian formulation by Toland (2006), but the formula for potential energy (more precisely for the pseudo-potential energy that derives from (1.9)) of a sheet when *c*_{0}, *d* and *ρ* are given will need further study.

### (a) Energy of a surface sheet

The notation is that of (1.6*a*) and (1.6*b*). Let and *ρ*≥0 be given. Suppose that *s*∈[0,2*π*] denotes a point of the reference sheet and let denote its position after deformation, so that is the stretch at . Let denote the curvature of at . Then, the pseudo-elastic energy (recall (1.9)) in one period of is(2.1)the gravitational potential energy is(2.2)and the pseudo-potential energy of the sheet is . We assume that, for some *α*>1, *β*>0 and *κ*≥0, the *C*^{3}-function satisfies the following.

(∂/∂

*ν*)(*ν*^{2}*e*_{1}(*ν*,*σ*)−(*κ*/2)*ν*^{2}) is bounded below by a positive constant.*e*_{12}is bounded.*e*_{1}(*ν*,*σ*)→∞ as*ν*→*α*and*ν*^{2}*e*_{1}(*ν*,*σ*)→−∞ as*ν*→0.

Here, , and similarly for higher derivatives.

Our existence theory is for waves with *c*^{2}*ρ*≤*κ*. If the elastic sheet has zero density, what follows with *κ*=*ρ*=0 is an extension of Toland (2006) to more general elastic responses when mass is neglected.

Hypothesis (H1) is equivalent to the assumption that the second derivative of the map is bounded below on (1/*α*, ∞) by a positive constant multiple of *t*^{−2}, an observation exploited in lemma 3.2.

A simple example occurs when for suitable functions *s* and *b* (see Toland 2006). □

By (H1)–(H3), there exists *α*^{*}∈(0, 1) and such that(2.3)Hence, if *c*^{2}*ρ*≤*κ*, we may define a function ( depends on *c* and *ρ*) by(2.4)By the implicit function theorem, is a *C*^{2} function and is increasing for each *σ*∈(−*β*, *β*). By (2.3), since *c*^{2}*ρ*<*κ* and *α*^{*}∈(0, 1),Since for all |*σ*|<*β*,(2.5)and because (2.6)

Owing to (H2), may be extended as a function which is uniformly continuous on compact subsets of . Thus, by (H1), has a continuous extension to . Hence, for any *c* and *ρ* with *c*^{2}*ρ*≤*κ*,(2.7)(2.8)uniformly for |*σ*|≤*β*. □

Under these hypotheses, we now derive (1.11) and (1.12) from a variational principle when a *parametrization* of the wave surface (not the displacement of the material points of the reference section) is given.

Suppose that is a parametrization of a surface and that the curvature at is given by . Since there is no friction between the fluid and the sheet, and owing to constraint (★) from §1*a*, we assume that the pseudo-energy of the sheet is given by when , and *δ* is a critical point of the functional ( is fixed) with respect to all diffeomorphisms *δ* of [0,2*π*] with *δ*(0)=0. Therefore, for given , the energy in the sheet section is characterized as a critical value (critical with respect to *Χ*, the inverse of *δ*) of the energy which, after the change of variables *τ*=*δ*(*x*), is(2.9)Here, ′ denotes differentiation with respect to *τ*. A critical point *Χ* must satisfy the Euler–Lagrange equation(2.10a)with the boundary conditions *Χ*(0)=0, *Χ*(2*π*)=2*π*. Hence, by (2.4),and, for given , we require such that(2.10b)If the length of one period of is in the interval [2*π*,2*απ*), then the existence of a unique(2.11)satisfying (2.10*b*) follows from (2.5) to (2.7) and the dominated convergence theorem, since is a monotone function of *γ*. The pseudo-energy in a period of the deformed sheet is then(2.12)and, owing to remark 2.2 and (2.11), the stretch is given pointwise by(2.13)when , for some *α*_{M}>0. Note that equations (2.10*a*) and (2.10*b*), which were derived from the above variational principle, coincide with (1.11) and (1.12), which came directly from mechanics.

We finish the section by noting the effect on (2.12) of adding *a j*, , to . The left-hand side of (2.10

*a*) is unchanged, but

*gρa*is added to the right-hand side. Owing to (2.10

*b*), this means subtracting

*gρa*from

*γ*and therefore there is no effect on the function ). Consequently, the addition of

*a*to has no effect on the first term in (2.12). The effect on the second term is to add 2

**j***πgρa*, owing to (2.10

*b*). Therefore, the total effect (which might have been anticipated from the mechanics) is the addition of a constant 2

*πgρa*to (2.12). □

### (b) Critical-point formulation

We have seen that, when the wave speed *c*_{0} is given, the pseudo-potential energy of the heavy elastic sheet on the surface of a steady hydroelastic wave may be calculated from its *shape*, when its density *ρ* and drift velocity *d* are known. Now, we recall (Dyachenko *et al.* 1996; Toland 2006) that the energy of the fluid underneath may also be calculated from the shape. This leads to a formulation of our problem as one for critical points of a Lagrangian functional, the values of which depend on parametrizations of the surface (as opposed to Lagrangian descriptions of displacements of the reference sheet). We begin by introducing a family of such parametrizations and derive a formula for the Lagrangian. The technical analysis owes a lot to Toland (2006) and Shargorosky & Toland (in press).

#### (i) Parametrized curves

Let denote the usual Lebesgue space of 2*π*-periodic functions on , which are *p*th-power locally integrable, and , the Sobolev space of functions whose *k*th weak derivative lies in , 1≤*p*≤∞, . For , its conjugate function (or Hilbert transform) is defined almost everywhere byand is a bounded linear operator on and , 1<*p*<∞, . For and *τ*∈[0,2*π*], let(2.14a)Then, *Ω*(*w*)∈ and, by a classical result (e.g. Martinez-Avendano & Rosenthal 2007, lemma 2.7.1), . Let(2.14b)(2.14c)(2.14d)(2.14e)Recall from Toland (2006), §4, that(2.14f)and note that *L*(*w*)≥2*π* (Jensen's inequality) is the length of (*w*). Let(2.14g)An observation from Shargorosky & Toland (in press), exploited by Toland (2006), is that for (*w*) to be the surface profile of a hydrodynamic wave we need that . Moreover, there is no Lagrange multiplier associated with membership of , at a point which is sufficiently regular, owing to the following observation.

*Suppose, for p>1, that* *and* *is bounded. Then, w is an interior point of* *in* .

For with , the weak curvature of (*w*) at is defined by(2.14h)where ′ denotes weak differentiation. Note that . Let(2.15)(2.16)We will seek critical points of the Lagrangian (strictly speaking, pseudo-Lagrangian owing to the involvement of the pseudo-elastic energy of the sheet) in the set . (The use of _{0} emerges in lemma 3.1).

#### (ii) Lagrangian

For any , a curve is parametrized by (2.14*e*) and the terms in (2.12) may be identified with those in (2.14*a*)–(2.14*h*) as follows:Therefore, for , (2.12) has the form(2.17a)where depend on *c* and as in (2.4), and *γ*(*σ*(*w*), *w*) is the constant determined by(2.17b)The unique *γ* satisfying (2.17*b*) is estimated in §5*b*. Thus, is the functional that gives the energy (pseudo-elastic+gravitational) due to the sheet of a hydroelastic travelling wave. In addition,(2.17c)is the gravitational potential energy of the fluid below the sheet and(2.17d)is its kinetic energy (see Toland (2006)). We are interested in the existence of non-constant regular critical points of the Lagrangian functional(2.18)for which(2.19)

*Suppose that c*^{2}*ρ*≤*κ and* *is such that* *is bounded. Let U be a ball about* *in* *such that* *is uniformly bounded for* , *and* . *Then*, *is C*^{2}.

The proof is straightforward and identical to that of Toland (2006), §6, adjusted to deal with the definition of that now involves the sheet density. ▪

The relation between regular critical points of and steady hydroelastic waves is similar to that in Toland (2006), §7. If the hypotheses of theorem 2.5 hold and , and if ** U** is the region below extended periodically in the

*x*-direction, let be the harmonic function that satisfies the Dirichlet problem(2.20)Then, yields a solution of the steady hydroelastic-wave problem (1.17

*a*)–(1.17

*d*).

## 3. Existence theory

Owing to theorem 2.5 and the remark following it, we seek critical points of in where (*w*) is non-self-intersecting. The existence of hydroelastic waves was established by Toland (2006) under the assumption that *ρ*=0 and . In that case, and the function has an especially simple formwhere . Since the constraint (2.17*b*) does not appear explicitly, a relatively uncomplicated application of the direct method of the calculus of variations leads to the existence of a maximizer. However, in the general case, the same approach would lead to a maximization problem for with the non-affine constraint (2.17*b*), which might not be respected by the weak limit of a maximizing sequence. Consequently, a weak limit might not be a maximizer of the constrained problem. In this section, we find a way around that difficulty. The first step is to reduce the problem to one in which critical points with zero mean are sought. Suppose that has zero mean and . Then, from remark 2.3,LetThen, lemma 3.1, which shows that it is sufficient to find non-trivial maximizers of in , defined in (2.16), is now immediate.

*Suppose that* *has the property that* *for all* . *Then*,*is not a constant and* .

### (a) A saddle-point problem

For , let , , and note from (H1) that is a convex function of *t*. Here, we suppose in addition that

.

Letand let be defined by(3.1)Note from corollary 5.1(d) that *γ* is defined on when (H1)–(H3) hold and the properties of *γ* are given in corollaries 5.1–5.3. Then, the functional to be maximized on is(3.2)where(3.3)and, bearing in mind that depends on *c* and *ρ*,(3.4)The key to an existence theory for critical points of is an interpretation of the question as a saddle-point problem. For , let

*Suppose (H1)–(H4) hold and let* *be fixed. Then, the functional* *is concave in γ for fixed* . *Moreover, for fixed* , *is maximized at the point γ which satisfies* *(3.1)* *and* *is a convex function of* .

Let and let be as in (H4). Then, for ,However, for a fixed real number ,where is the Legendre transform of . Also, if , thenIn all cases, is a concave function of *γ* for fixed *b* and , and a convex function of for *γ* and fixed (see Ekeland & Temam (1976), ch. III, lemma 3.1, or by direct calculation). It is immediate that is a concave (possibly identically equal to ) in *γ* for fixed functions and . Since is convex in *u* for fixed *γ* and *w*, it follows that, for fixed *w*,Now, note from the definition of that, for ,which is bounded when is bounded below. It follows that, when and are fixed functions, is a differentiable function of . Moreover,We have already observed, as a consequence of (H1)–(H3), that such a *γ* exists and is unique. The concavity of as a function of *γ* means that the unique *γ* which satisfies (3.1) is a maximizer. For fixed *w*, the value of this maximum is given by , which is therefore a convex function of . ▪

*Let (H1)–(H4) hold and* *be fixed. Then,* *is convex and weakly lower semi-continuous on* *with the topology of* , .

Since is convex, it will suffice (Ekeland & Temam (1976), ch. I, corollary 2.2) to show that it is strongly lower semi-continuous on in . Therefore, suppose that in and, without loss of generality, that pointwise almost everywhere. Then,Since , it follows from (2.7) that . Therefore, since is bounded below by corollary 5.1(d), we may suppose, without loss of generality, that . It follows, from the dominated convergence theorem and the pointwise almost everywhere convergence of and the definition of (corollary 5.1(d) again), that . Now, by Fatou's lemma,

This completes the proof. ▪

### (b) Maximization

SupposeThe hypothesis fulfils two roles. First, it guarantees that is a Jordan curve as required in (2.19) by bounding the curvature below and the length above so that cannot self-intersect (see Toland (2006), remark 8.2). Second, it leads to the following compactness result. Let .

*When* *and* , *the sequence* *is bounded in* *for all* *and*(3.5)*Moreover, there is a subsequence, also denoted by* , *and* , *such that*(3.6)(3.7)(3.8)

See Toland (2006), lemma 8.4 and the proof of theorem 8.6. ▪

Let(3.9)From (5.1), (3.10) and (H5), it follows that is finite. If and , then *w* is not zero, and hence not a constant.

*Suppose (H1)–(H5) hold and* . *Then, there exists a non-constant function* *with* , *for all* , *and* .

Let be such that . It follows from corollary 5.1(e) that if , then and hence thatby remark 2.2. Therefore, by (H5),(3.10)Hence, if is a maximizing sequence for , it follows from (5.1) and (3.2) that lies in a compact subset of and . By lemma 3.4, there exists with in andTherefore, in (3.2). In addition,whereThe fact that is a maximizing sequence for and (3.5) holds means thatTherefore, because in . In addition, implies that is bounded away from 0 and . From corollary 5.2 with , we find that as . That now follows from the dominated convergence theorem and remark 2.2. A similar argument yields that .

Since, by lemma 3.3, is weakly lower semi-continuous on in , this shows that , is bounded and in (3.9). The fact that implies that is non-zero and therefore non-constant. Hence, and . Since and , which is closed and convex in and hence weakly closed in , it follows that . This yields the required result. ▪

*Suppose* , *(H1)–(H5) hold and* *in* *(3.9)*. *Then, there exists a non-constant function* *such that* , *is bounded and* *for all* . *Moreover, the curve* *is the graph of a function and, in particular, is non-self-intersecting*.

This is immediate from lemma 3.1 and theorem 3.5. ▪

The next result identifies a range of parameters for which .

*Suppose that* *and*(3.11)*Then*, *if**where* . *Consequently, conclusions (i)–(iii) of* *§1b* *hold*.

The proof, which involves the calculation of to as when , is given in §5*c*. ▪

The function satisfies (3.11), for example, when it has a minimum value zero when and is even in for each . □

## 4. *A priori* bounds

To show that when is given by theorem 3.6, it suffices to mimic the proof of Toland (2006), §8.3, and we confine ourselves to observing its key features in the current setting. To this end, suppose thatwhere the convergence in (H6) is uniform for owing to (H2). To reach the conclusion that , we compare the values of and , for a certain function , in order to show that, if , cannot be a maximizer of defined bywhere *J* is defined by (3.4). (Note that here we work with and not to avoid the need to construct a comparison function with .) We begin by comparing values of *J*. To do so, let , be such that , , and , a function of arc length, be defined by(4.1)Then, with regarded as a function of ,say. Here, is defined in terms of , but now we recall an observation about how *w* may be determined by an *L*-periodic function . Suppose that is given and that a simple curve parametrized by arc length is given by(4.2a)(4.2b)(4.2c)(4.2d)

*When* *and* *is given by* *(4.2a)–(4.2d)*, *there exists* *such that**In particular,* *for some positive constants a and b. Moreover,* *and* .

See Toland (2006), §4.3. ▪

Suppose that satisfies (4.2*a*)–(4.2*d*) with , and that corresponds to as in lemma 4.1. Then, . Let . It follows from the definition (2.14*h*) of thatwhere the obvious analogues of (4.1) define and . Therefore,Now, by corollary 5.3 and the definition of and ,and so, since and are bounded,owing to (5.2). A similar calculation leads to the estimateSuppose, seeking a contradiction, that and, for , letAt least one of these sets is non-empty for all . Suppose that . Since has zero mean, there exists such that(4.3)has non-zero measure for all . Now choose (depending on ) in such thatLetand note from (H6) that as . The proof of Toland (2006), theorem 8.9, shows that, for each and sufficiently close to , there exists an admissible function and a corresponding ( here is the function constructed by Toland (2006)), depending on , such thatandCombining these observations and letting , we find that is not a maximizer of over the class of admissible functions if .

*Suppose that (H1)–(H6) hold. Then,* , *where* *as in* *(4.1)*. *Consequently*, *and* *are in* *and* . *Let* *be the region below* *and let* *be the corresponding solution of* *(2.20)*. *Then*, .

From the preceding section, we know that , for some , where relates to (H1), (H5) and (H6). Since (H6) holds, the regularity of is immediate because, when is written as a function of in a -neighbourhood of , it has a maximizer at (see the argument of Toland (2006), corollary 8.10). It now follows that . Hence, since and . Since and is continuous and nowhere zero, is a boundary of of class . The result therefore follows from Gilbarg & Trudinger (1983), lemma 6.18. ▪

## 5. Technicalities

The maximization argument and the demonstration of *a priori* bounds need a few obvious inequalities from Toland (2006) and some elementary corollaries of the hypotheses.

### (a) Inequalities

For ,(5.1)Now, let and put . Then, (H2) implies that is bounded on and it follows that is bounded since, owing to (H1), is bounded below by a constant determined by . Therefore, for and (5.2)The constant depends on and only.

### (b) Corollaries of (H)

Suppose (H1)–(H3) hold and, for and , letNote that, for , (2.10*b*) is the requirement that

*Suppose (H1)–(H3) hold*, *and* .

*is continuous and strictly decreasing in*.*as*..

*There exists a unique**with*.*Moreover*,*is bounded below by a positive constant*,*say, independent of u and w*.*Suppose that*,*and*.*Then*,*implies that*.

Part (i) is immediate from (H1) and part (ii) follows from (2.6) and (2.7). Since is strictly increasing in , and sincepart (iii) follows from (2.5).

It follows from the intermediate-value theorem that there exists with . But because . Since is bounded below independently of *w* and *u*, and is bounded, part (iv) follows from remark 2.2.

Finally, to prove (v) note thatSince , the hypotheses of part (v) imply that as . However, implies that is bounded and it follows from remark 2.2 that . This completes the proof. ▪

*For* *and* , *let* *in* *corollary 5.1(d)*, . *Then, when (H1)–(H3) hold,**where C depends on* . *Hence, by* *(5.2)*,

The result follows from the fact thatandwhere by corollary 5.1(d). Since is defined by (2.4), it follows from (H1) thatwhere depends on and is bounded below on compact intervals. Since and , , and since (5.2) holds, the result follows. ▪

*Let (H1)–(H4) hold*. *Suppose, for* , *that* , , *is bounded and* . *Let* . *Then*,*where, in the notation of* *lemma 4.1*,*and C depends on* .

By definition,Therefore,The remainder of the proof is identical to that of the preceding result. ▪

### (c)

Here, we give hypotheses on the parameters in the problem that ensure that , as required in theorem 3.6. Recall that , where is the wave speed and *d* is the drift speed of the elastic sheet. Suppose that satisfies (3.11). Throughout, we use the abbreviation for , with subscripts, such as or , denoting the partial derivatives, such as and . Since andfor all and , it follows that(5.3)where and and denote the values of and of its partial derivatives at . Note from (H1) that . The goal is to calculate to second order in as , when . In this case,It is then immediate that as in (3.2),(5.4)Now, we estimate , defined by (2.17*b*), when . When , since . For , let . Then, since , we find thatwhich yields that the dependence of on is quadratic as ,(5.5)Thus, let(5.6)Since and (5.6) holds, when and , the second term on the right-hand side of (3.4) can be writtenThe first term on the right-hand side of (3.4) with has the formsay. Now, and (5.6) holds. Hence, with , as ,Finally, note that and (5.6) mean that, as ,Since , it follows from (5.5) thatCombining these observations, we find that andwhere is given by (5.3). This proves lemma 3.7.

## Acknowledgments

I am very grateful to Pavel Plotnikov (Russian Academy of Sciences, Novosibirsk) for his keen interest in this work and for his useful comments on various drafts.

## Footnotes

- Received March 9, 2007.
- Accepted May 30, 2007.

- © 2007 The Royal Society