Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind

In this paper, we present results about the existence and uniqueness of solutions of elliptic equations with transmission and Wentzell boundary conditions. We provide Schauder estimates and existence results in H\"older spaces. As an application, we develop an existence theory for small-amplitude two-dimensional traveling waves in an air-water system with surface tension. The water region is assumed to be irrotational and of finite depth, and we permit a general distribution of vorticity in the atmosphere.


Elliptic theory.
Let Ω ⊂ R n be a connected bounded C 2,β domain for n > 1 and β ∈ (0, 1). Suppose that there exists a C 2,β hypersurface Γ that divides Ω into two connected regions such that and denote by S := ∂Ω. Let ν = (ν 1 , . . . , ν n ) be the normal vector field on the interface Γ pointing outward from Ω 1 . We define the co-normal derivative operator on Γ a ij ν i ∂ x j , and the tangential differential operator along Γ  Here, we are using · := (·)| Ω 1 − (·)| Ω 2 to denote the jump operator across Γ. We think of α = +1 as favorable and α = −1 as unfavorable. We shall assume uniform ellipticity condition on the operators L and B; that is, there exist constants λ, µ > 0 such that a ij (x)ξ i ξ j for all x ∈ Ω, ξ ∈ R n , (1.4) and µ|ξ| 2 ≤ n s,t=1 a st (x)ξ s ξ t for all x ∈ Γ and ξ ∈ R n such that ξ · ν(x) = 0. (1.5) The coefficients a ij and a st satisfy a ij = a ji , a st = a ts for all i, j, s, t = 1, . . . , n. We also assume that a ij , b i , c are in L ∞ (Ω), and a st , b s , c are in L ∞ (Γ). Note that B contains second-order tangential derivatives of u. This is characteristic of so-called Wentzell-type boundary conditions, whose study was initiated by Wentzell in [43]. They arise, for example, in stochastic equations [22] or as an asymptotic model for roughness of the boundary or other more complex geometrical effects [9]. They also appear in water waves and continuum mechanics, which is our principal interest here. For instance, the Young-Laplace Law states that at the interface between two immiscible fluids, the pressure experiences a jump proportional to the curvature. In a free boundary problem where the interface is given as the graph of an unknown function, this naturally leads to quasilinear versions of Wentzell-type conditions. More generally, the curvature of a hyperplane is the first variation of its surface area. Thus, these types of conditions are frequently encountered in free boundary problems where the shape of the interface contributes to the energy.
Transmission conditions refers to the jump operator in B. They are commonly found in multiphase problems, where physically their purpose is to enforce continuity of the normal stress across a material interface. Many researchers alternatively call these diffraction problems (see, for example, [26,5]).
We first have an a priori estimate for classical solutions in Hölder spaces.
We emphasize that this theorem only holds if we assume c ≥ 0, c > 0, and α has a favorable sign. However, for a general c, c, and α, we are still able to assert the Fredholm solvability of the problem. Letting which is a Banach space with respect to the norm u X := u C 2,β (Ω 1 ) + u C 2,β (Ω 2 ) + u C 0,β (Ω) , we have the following theorem. 1) with f = g = 0 has nontrivial solutions that form a finite dimensional subspace of X, or b) the homogeneous problem has only the trivial solution in which case the inhomogeneous problem has a unique solution in X for all f ∈ C 0,β (Ω 1 ) ∩ C 0,β (Ω 2 ) and g ∈ C 0,β (Γ).
A great deal of research has been devoted to studying elliptic problems with linear and nonlinear Wentzell boundary conditions, but they remain comparatively less wellunderstood. One of the earliest works to consider Wentzell conditions was Korman [25] who, like us, was interested in their connection to a problem in water waves. Specifically, he investigated a model describing three-dimensional periodic capillary-gravity waves where the gravity pointed upward. A Schauder theory was later provided by Luo and Trudinger [29] for the linear case. In the quasilinear setting, Luo [28] gave a priori estimates for uniformly elliptic Wentzell conditions, while later Luo and Trudinger [30] studied the degenerate case. More recently, Nazarov and Paletskikh [34] derived local Hölder estimates in the spirit of De Giorgi for divergence form elliptic equations with measurable coefficients and a Wentzell condition imposed on a portion of the boundary. See also the survey by Apushkinskaya and Nazarov [3] for a summary of the progress made on the nonlinear problem.
Transmission boundary conditions are of great importance to physics and other applied sciences. They are also of interest from a purely mathematical perspective as they arise naturally in the weak formulation of PDEs with discontinuous coefficients. The study of transmission problems dates back to the 1950s and 1960s. Schechter [37] andSeftel' [42] investigated even-order elliptic equations on a smooth and bounded domain with smooth coefficients. Schechter obtained estimates and provided an existence for weak solutions. His strategy involved transforming the transmission problem into a mixed boundary value problem for a system of equations. On the other hand,Seftel' found a priori L p -estimates. Oleȋnik [36] also studied transmission problems for second-order elliptic equations with smooth coefficients; approximating equations were used to derive results for weak solutions. One of the most foundational work was done by Ladyzhenskaya and Ural'tseva [26], who considered second-order elliptic equations on a bounded domain and then obtained estimates for weak and classical solutions in Sobolev and Hölder spaces, respectively. In contrast to Schechter's approach, Ladyzehnskaya and Ural'tseva exploited cleverly chosen test functions to deduce their a priori estimates. More recently, Borsuk [10,11,12] has treated linear and quasilinear transmission problems on non-smooth domains.
Apushkinskaya and Nazarov [4] considered Sobolev and Hölder solutions of linear elliptic and parabolic equations for two-phase systems. However, they only examined the problem with a favorable sign α = +1 of the transmission term, and did not study Fredholm property. Note that in water wave applications, the sign is typically unfavorable. With that in mind, in this paper we make the effort to also include Schauder estimates and Fredholm solvability for α = −1 as well; see also Remarks 2.1, 2.5, and 2.7. Our approach is to view the Wentzell boundary condition as a non-local (n − 1)-dimensional elliptic equation, treating the jump in the co-normal derivative term as forcing that can be controlled using techniques from the literature on transmission problems.

1.2.
Steady wind-driven capillary-gravity water waves. Our second set of results considers an application of the above elliptic theory to a problem in water waves. In particular, we will prove the existence of small amplitude periodic wind-driven capillarygravity waves in a two-phase air-water system. One of the main novelties here is that we also allow for a general distribution of vorticity in the air region. For simplicity we take the flow in the water to be irrotational. When discussing these results, we adopt notational conventions common in studies of steady water waves which occasionally conflict with our notations in the elliptic theory part.
Let us now formulate the problem more precisely. Fix a Cartesian coordinate system (X, Y ) ∈ R 2 so that the X-axis points in the direction of wave propagation and the Y -axis is vertical. The ocean bed is assumed to be flat and at the depth Y = −d, while the interface between the water and the atmosphere is a free surface given as the graph of η = η(X, t). We then normalize η so that the free surface is oscillating around the line Water Ω 2 (t) Figure 1. The air-water system Y = 0. The atmospheric domain is assumed to be bounded in Y ; that is, the air region lies below Y = ℓ for some fixed ℓ > 0. At a given time t, the fluid domain is where Ω 1 is the air region, and Ω 2 is the water region, We also denote I(t) := ∂Ω 1 (t) ∩ ∂Ω 2 (t). Here we think of I(t) as playing the role of Γ in the notation of the previous subsection.
Let u = u(X, Y, t) and v = v(X, Y, t) be the horizontal and vertical fluid velocities, respectively, and denote by P = P (X, Y, t) the pressure. We say that this is a traveling wave provided that there exists a wave speed c > 0 such that the change of variables (X, Y ) → (x, y) := (X − ct, Y ) eliminates time dependence. The velocity field is assumed to be incompressible and, in the moving frame, (u, v, η, P ) are taken to be 2π-periodic in x.
The kinematic and dynamic boundary conditions for the lidded atmosphere problem with surface tension σ are (1.9) Note that the last condition will give rise to nonlinear Wentzell and transmission terms.
In particular, the right hand side can be viewed as a second-order elliptic operator acting on η, while the jump in the pressure will relate to a jump in (u − c) 2 + v 2 via Bernoulli's theorem that we discuss below. We consider waves without (horizontal) stagnation, that is, we will always assume u − c < 0 in Ω. (1.10) As (u, v) is divergence free according to (1.8), we can define the pseudostream function ψ = ψ(x, y) for the flow by The level sets of ψ are called streamlines. Without stagnation (1.10), we have ψ y < 0, which implies that each streamline is given as the graph of a function of x via a simple Implicit Function Theorem argument. The boundary conditions in (1.9) show that the air-water interface, bed, and lid are each level sets of ψ. We will take ψ = 0 on the upper lid so that ψ = −p 0 on y = −d, where p 0 is defined by It can be shown that p 0 does not depend on x (see, for example, [46]). Bernoulli's theorem states that is constant along streamlines. Evaluating the jump of E on the interface gives where κ is the signed curvature of the air-water interface and Q := 2 E + g̺d . Recall that in two dimensions, the vorticity ω is defined to be If there is no stagnation (1.10), there exists a function γ, called the vorticity strength function, such that ω(x, y) = γ(ψ(x, y)) for all (x, y) ∈ Ω.
The vorticity plays a key role in the wind generation of water waves as we will discuss below. Mathematically, it substantially complicates the analysis.
Finally, we will use the following notational conventions. For any integer k ≥ 0, α ∈ (0, 1), and an open region R ⊂ R n , we define the space C k+α per (R) to be the set of C k+α (R) functions that are 2π-periodic in their first argument.
Our main theorem is an existence result for traveling capillary-gravity water waves in the presence of wind. We prove this theorem using a local bifurcation theoretic strategy that draws on the ideas of Constantin and Strauss [17], who studied rotational periodic gravity water waves in a single fluid. Indeed, following the publication of [17], traveling water waves with vorticity have been an extremely active area of research (see, for example, the surveys in [38,16]).
Our most direct influence is the work of Bühler, Shatah, and Walsh [13] on the existence of steady gravity waves in the presence of wind. These authors studied exactly the system (1.8)-(1.10) taking σ = 0. One of the main objectives of that paper was to construct waves that were dynamically accessible from an initial state where the flow is laminar and the horizontal velocity experiences a jump over the interface. More specifically, this meant that the circulation along each streamline was prescribed in order to ensure that its values in the air and water regions were distinct (see Remark 3.3). We also adopt this approach in the present paper, though the addition of surface tension necessitate many nontrivial adaptations.
1.3. History of the problem. Steady capillary and capillary-gravity waves have been the subject of extensive research. Because we are particularly interested in the role of vorticity, we will restrict our discussion to rotational waves. In this setting, progress is much more recent and begins with the work of Wahlén [44,45], who proved the existence of smallamplitude periodic capillary and capillary-gravity waves in two-dimensions for a single fluid system. As in [17], this was done for a general vorticity function γ. Contrary to the gravity wave case, Wahlén showed that with surface tension there can be double bifurcation points; this is a rotational analogue of the famous Wilton ripples [49]. Later, Walsh considered two-dimensional periodic capillary-gravity waves with density stratification [47,48].
Recently, Martin and B-V Matioc proved the existence of steady small-amplitude capillarygravity water waves with piecewise constant vorticity [31]. While they consider a one-layer model, the analysis has a similar flavor to that in the present paper. A-V Matioc and B-V Matioc also constructed weak solutions for steady capillary-gravity water waves in a single fluid [32].
The waves we construct can also be viewed as internal waves moving along the interface between two immiscible fluid layers confined in a channel. Versions of this problem have been investigated by many authors. For instance, Amick-Turner [1] and Sun [39,40] considered the existence of solitary waves in a channel where the flow is irrotational at infinity. Amick-Turner built their solitary waves as limits of periodic waves with the period tending to infinity. Sun, on the other hand, exploited the fact that the leading-order form of the wave is given by the Benjamin-Ono equation, and then used singular integral operator estimates to control the remainder. The existence of continuously stratified channel flows has also been verified in a number of regimes. Note that these are rotational, since heterogeneity in the density produces vorticity. Specifically, Turner [41] and Kirchgässner [24] investigated small-amplitude continuously stratified waves using a variational scheme and a center manifold reduction method, respectively. A large-amplitude existence theory was also provided by Bona, Bose, and Turner [8], Lankers and Friesecke [27], and Amick [2]. We remark that, in all of these works, the vorticity vanishes at infinity. Finally, internal waves with surface tension on the interface were recently considered by Nilsson [35]. In that paper, each fluid layer was assumed to be irrotational and constant density. Using spatial dynamics and a center manifold reduction, Nilsson proved the existence of both periodic and solitary wave solutions.
As mentioned above, steady water waves in the presence of wind was studied by Bühler, Shatah, and Walsh in [13]. Our main contribution relative to that work is to account for capillary effects on the air-water interface. It is known that surface tension is important in the formation of wind-driven waves. Indeed, high frequency and small-amplitude capillarygravity waves are the first to form when wind blows over a quiescent body of water.
One of the most successful explanations for the mechanism behind the wind generation of water waves was given by Miles [33]. His main observation was that vorticity in the air region can create a certain resonance phenomenon that destabilizes the system. Importantly, this so-called critical layer instability can occur even when the horizontal velocity is continuous -or nearly continuous -over the interface, and therefore does not require exceedingly strong wind speeds like the Kelvin-Helmholtz model. The mathematical ideas underlying Miles's theory were recently reexamined and rigorously proved by Bühler, Shatah, Walsh, and Zeng [14]. In that work, the authors also allowed surface tension. This is somewhat important as the interface Euler problem itself is ill-posed when there is a jump in the tangential velocity and there is no surface tension (see, for example, [6]). In a forthcoming work, the author intends to study the stability of the family of waves constructed in Theorem 1.4. This will serve as a model for wind generation of water waves in the spirit of Miles, but with an initial state that is not purely laminar.
1.4. Plan of the article. We now briefly discuss the strategies we use to derive these results. The elliptic theory is proved in Section 2. Our approach is based on the work of Luo and Trudinger [29], who gave Schauder estimates for elliptic equations with Wentzell boundary conditions.
In Section 3, we construct capillary-gravity water waves where the air region is rotational. Following Bühler, Shatah, and Walsh [13], the first step in this procedure is to reformulate the interface Euler system (1.8)-(1.10) as a quasilinear elliptic equation on a fixed domain. Due to surface tension, there is now a nonlinear Wentzell condition on the image of the interface in these new coordinates. We construct the non-laminar waves using local bifurcation theory. This entails studying the spectrum of the linearized equation at a laminar flow, and here we make essential use of the elliptic theory developed in Section 2. One major difficulty that arises is that this linearized problem is of Sturm-Liouville type, but associated to an indefinite inner product. Consequently, to successfully determine the spectral behavior, we must work in Pontryagin spaces. A similar issue was encountered by Wahlén in [44,45]. Finally, we apply the Crandall-Rabinowitz local bifurcation theorem to obtain Theorem 1.4.

Elliptic Theory
To simplify subsequent calculations, it is convenient to first change variables. Fix a point x 0 ∈ Γ. Then by the assumption on Ω, there is a neighborhood U of x 0 and a C 2,β diffeomorphism that maps U to some ball B ⊂ R n so that Γ maps to {x n = 0}, Ω 1 to B ∩ {x n > 0}, and Ω 2 to B ∩ {x n < 0} (see, for example, [29]). Then it suffices to assume that Γ is the hyper-plane {x n = 0}, and consequently, Ω 1 and Ω 2 lie inside the upper-half and lower-half planes respectively.
In this case, the co-normal derivative operator simplifies to and the Wentzell and transmission condition on Γ becomes We also denote by ∇ ′ the tangential gradient on Γ in this case.
We then cover Γ by a finite number of spheres in which the estimate in [21, Theorem 6.2] for B on Γ can be applied. This ensures the existence of a positive constant C = C(n, β, L, B, µ) such that Similarly, we have Next, we use a basic elliptic estimate for the Dirichlet problem in Ω ′ k with boundary condition u k | Γ (see, for example, [21, Theorem 6.6]), to obtain Moreover, we have the following interpolation for some ǫ > 0. Finally, evaluating (2.3) with k = 1, 2 and summing, using u | Γ = 0 and the estimates (2.1), (2.2), (2.4) and choosing appropriate ǫ give Then we have the estimate where the L ∞ norms are taken over Ω 1 and Ω 2 .
Proof. We will follow very closely the classical arguments when proving this maximum principle in the interior. Rewriting L in non-divergence form gives choosing σ ≥ 1 large enough so that σ 2 − τ σ ≥ 1, and without loss of generality, because of the boundedness of Ω, assuming Ω lies between {x 1 = 0} and where u + := max(u, 0) and d := diam Ω. Then Then since c ≥ 0, sup On the other hand, by construction u − v ≤ 0 on ∂Ω k . Therefore, the maximum principle implies u ≤ v in Ω k , and that there exists a positive Next, since u| S = 0, if u| Γ ≤ 0 for all x ∈ Γ, then If we suppose that u attains its local maximum at some point x 0 ∈ Γ and u(x 0 ) > 0, then by the positive definiteness of the matrix (a st ), By the positive-definiteness of the matrix (a ij ), we have and hence ∂ N u(x 0 ) ≥ 0. Then the condition on Γ gives Therefore, sup and hence we obtain the desired estimate (2.5) by using sup Remark 2.3. Note that if Ω is periodic in one variable, the lemma still holds by modifying the proof to assume that Ω lies between two hyperplanes parallel to the periodic direction.
Using the notation of a Hölder seminorm, we have the following simple lemma whose proof will be omitted: Now we can derive the existence and uniqueness of solution in Hölder spaces.
Next, recalling the definition of X as in (1.7 Then Y is a Banach space with respect to the norm Thus, problem (2.6) can be written as where L θ : X → Y , so the solvability of the problem (2.6) for arbitrary f ∈ C 0,β (Ω 1 ) ∩ C 0,β (Ω 2 ) and g ∈ C 0,β (Γ) is then equivalent to the invertibility of the mapping L θ . We note that L 0 and L 1 are bounded operators.
On the other hand, by Lemma 2.4, Lemma 2.2, and estimate (2.7), we have for some ǫ > 0, and hence where the constant C does not depend on θ. Thus, by the method of continuity (see, for example, [21, Theorem 5.2]), the surjectivity of L 1 , which we are investigating, is equivalent to that of L 0 which is the problem (2.8) Finally, we recall that is the unique solution to B ′ ϕ = g on Γ for a given g ∈ C 0,β (Γ), then by [21, Lemma 6.38] we can make an extension to have ϕ ∈ C 2,β (Ω 1 ) ∩ C 2,β (Ω 2 ). Now we have a Dirichlet problem which has a unique solution u k ∈ C 2,β (Ω k ) by [21, Theorem 6.14]. Therefore, by Lemma 2.4, we conclude that there is a unique solution in C 0,β (Ω) ∩ C 2,β (Ω 1 ) ∩ C 2,β (Ω 2 ) to the system (1.1).
Remark 2.5. As a consequence of Theorem 1.2, we see that L 1 = (L, B) is a Fredholm operator of index 0 despite the sign of the transmission term. Indeed, for θ ∈ [0, 1], consider the following linear operator where L θ : X → Y and with coefficients a st , b s , and c satisfying the hypotheses of Theorem 1.2. Note that the sign of the transmission term is unfavorable. It is clear that the map θ → L θ ∈ L(X, Y ) is continuous. Then Schauder estimate from Theorem 1.1 and Remark 2.1 give for some small ǫ > 0, so Choosing ǫ > 0 small, we have for some constant C > 0 independent of θ, which implies that L θ has finite dimensional null space and closed range. Thus, L θ is semi-Fredholm. If θ < 1 2 , the map L θ is invertible by Theorem 1.2 and hence has index 0. By the continuity of the index, it also holds for θ ≥ 1 2 , which means that we have Fredholm index 0 regardless of the sign of the transmission term.

Fredholm property.
In light of Remark 2.5, it suffices to take α = +1. To simplify our notation, we write L for L 1 , which is the problem we are considering. If L and B do not satisfy the conditions c ≥ 0 and c > 0, it is still possible to assert a Fredholm alternative, which we formulate as in Theorem 1.3.
Proof of Theorem 1.3. For all σ, τ ∈ R, notice that for u ∈ X, (f, g) ∈ Y , where L σ,τ u := (L + σ)u, (B + τ )u . From Theorem 1.2, the mapping L σ,τ u : X → Y is invertible for σ and τ sufficiently large. Now, applying L −1 σ,τ to both sides, we obtain u = L −1 σ,τ (f + σu, g + τ u| Γ ) which can be written as (2.9) We claim that K is a compact operator. Let {(f m , g m )} ⊂ Y be bounded, and define u m := K(f m , g m ) ∈ Y 1 . We want to show that {u m } has a convergent subsequence in Y 1 . By definition of u m and K, we have where u m ∈ X, f m ∈ Y 1 , g m ∈ Y 2 . Thus, by Theorem 1.1, there exists a positive constant C = C(n, β, L, B, λ, µ) such that Note that the estimate holds for Ω k since S ∩ Γ = ∅. Since C 0,β (Ω) ⊂⊂ C 0 (Ω) and C 2,β (Ω k ) ⊂⊂ C 0,β (Ω k ), k = 1, 2, using estimates as in the proof of Theorem 1.2, we find that Then the inequality (2.10) becomes or we can write this to be which is equivalent to u m X ≤ C (f m , g m ) Y , so u m X is bounded in X. Since X ⊂⊂ Y 1 , we conclude that {u m } contains a subsequence {u m k } such that u m k → u in Y 1 , which proves the claim that K is a compact operator. Applying the Fredholm Alternative, equation (2.9) always has a solution u ∈ X provided the homogeneous equation (I − K)u = 0 has only the trivial solution u = 0. When this condition is not satisfied, the kernel of I − K is a finite dimensional subspace of Y 1 . Since the solutions of (2.9) are in one-to-one correspondence to the solutions of (1.1), we therefore can conclude the alternative stated in the theorem.
Remark 2.7. If we change the boundary term B to where the signs of the second-order term is switched, we obtain the same results as in Theorems 1.1, 1.3, and Proposition 2.6. For Lemma 2.2 and Theorem 1.2 to be valid, we have to assume in addition that α = −1 and c < 0, which means the signs of the second-order term and zeroth-order term must be opposite.

Steady capillary-gravity waves in the presence of wind
In this section, we will apply the results found above to investigate the existence of steady wind-driven water waves. There exists a well-known change of variables due to Dubreil-Jacotin that maps Ω to a strip (see [20]). We change variables (x, y) ∈ Ω → (x, −ψ) =: (q, p) ∈ D. We recall that ψ is the (relative) pseudostream function for the flow defined by (1.11), along with the boundary conditions ψ = 0 on the upper lid, ψ = −p 0 at the bed, and ψ ∈ C 0,α (Ω) ∩ C 2,α (Ω 1 ) ∩ C 2,α (Ω 2 ) for a fixed α ∈ (0, 1). Thus, the problem is now posed in a union of rectangles D = D 1 ∪ D 2 ⊂ R 2 , where the air region is mapped to D 1 := {(q, p) ∈ D : 0 < q < 2π, p 1 < p < 0}, and the water region is mapped to With that in mind, we have definitions for the lid, the free surface, and the ocean bed respectively as follows Under this change of coordinates, the Euler problem (1.8)−(1.10) becomes the following height equation where h(q, p) is the height above the bed of the point (x, y), where x = q and (x, y) lies on {−ψ = p}, and the depth operator d is defined to be Note that ρ in the above equation is for (q, p)-coordinates after the transformation. The equivalence of (3.1) to the original system (1.8)-(1.10) can be proved following [15,Lemma A.2].
Our objective is to find solutions (h, Q) ∈ S ′ , where and h p > 0 in D because of no stagnation condition (1.10). Recall that the space C k,α per (R) is the set of C k,α (R) functions that are 2π-periodic and even in their first coordinate. The presence of surface tension σ is manifested as the nonlinear second-order term in the boundary condition.
We will prove the following theorem stated in the Dubreil-Jacotin variables, which implies Theorem 1.4. Theorem 3.1 (existence). Let p 1 < 0, ℓ > 0, and atmospheric vorticity function γ ∈ C 0,α ((p 1 , 0)) be given. Then there exists σ 0 ≥ 0 such that for each σ > σ 0 , there is a continuous curve C loc ⊂ S ′ of solution to (3.1) with the following properties: Remark 3.2. In fact, there is a necessary and sufficient condition that we call local bifurcation condition (LBC), which will be given explicit in Lemma 3.9. In particular, (LBC) always holds for σ sufficiently large. When σ is small, a local bifurcation argument can still be carried out, but the eigenvalue of the linearized problem may not be simple. In this case, a more sophisticated analysis is required (see, for example, [45,44,47]).

Laminar solutions.
We first consider laminar flows which are solutions of the height equation (3.1) that are independent of q. Physically, this entails a wave where all of the streamlines are parallel to the bed. These will serve as the trivial solution curve when we apply the Crandall-Rabinowitz theorem to obtain Theorem 3.1.
Let us define Γ rel by .
Remark 3.3. Γ rel is called the (pseudo) relative circulation and is given by where H 1 denotes one-dimensional Hausdorff measure. Note that circulation around a closed loop is conserved for the time-dependent problem by Kelvin's circulation law. For periodic domains, this includes the circulation along the streamlines {ψ = −p}. If the waves we construct are to be viewed as generated dynamically by the wind, the circulation along each streamline must agree with the initial configuration.
For laminar flows, since h does not depend on q, we can write h = H(p), where H satisfies the following ODE: Note that d(H) = H(p 1 ). The above equation can be solved explicitly, but we still need some compatibility conditions to ensure continuity across the interface.

4)
and Moreover, the depth of the fluid at parameter value λ is Since the laminar flow is independent of the surface tension σ, the proof of Lemma 3.4 can be obtained by similar arguments as in [13,Lemma 4.2], which we will omit. Note that differentiating (3.5) with respect to λ gives is concave and has a unique maximum at λ 0 satisfying 3.2. Linearized problem. Next, let us consider the linearization of the height equation (3.1) at one of the laminar solutions (H(·; λ), Q(λ)) constructed in Lemma 3.4: Since we seek solutions that are 2π-periodic and even in q, we first consider m of the form m(q, p) = M (p) cos(nq), for some n ≥ 0. If n = 0, m does not depend on q and the linearized problem (3.8) becomes This equation can be solved explicitly. Using the boundary condition at p = p 0 and the continuity of M across the interface, we find that in the water region For the air region, we first observe that and hence, M p must vanish at least once inside (p 1 , 0) by Rolle's Theorem. We also have, from the above ODE, that a 3 M p is constant, so we conclude that M p ≡ 0 in (p 1 , 0). Finally, the jump condition gives which implies that there can be a zero-mode solution if and only if λ = λ 0 , where λ 0 is defined according to (3.7).
On the other hand, if n > 0, the linearized problem (3.8) becomes To investigate the ODE (3.9), we consider the following general eigenvalue problem In particular, we are interested in the case µ = −n 2 . The eigenvalue problem (P µ ) closely resembles a Sturm-Liouville equation, but the eigenvalue occurs both in the interior and boundary conditions. Moreover, the associated inner product defining the relation between eigenfunctions is indefinite. For that reason, it is natural to reformulate it in a Pontryagin space. Here we follow the general approach of Wahlén [44,45] and Walsh [47].
With that in mind, we introduce the complex Pontryagin space (see [7,23]) We understand that the L 2 -inner product is taken over (p 0 , 0). On H, there is also an associated Hilbert space inner product, given by ũ,ṽ H = [Jũ,ṽ], where are complete subspaces with dim H + = 1 or dim H − = 1.
We omit the proof of this proposition, as it is elementary. In fact, we have explicitly that H = H + ⊕ H − , where H + := L 2 ((p 0 , 0)) × {0}, Next, define the linear operator K : D(K) ⊂ H → H by Thus, there exists a nontrivial solution of (P µ ) if and only if µ is an eigenvalue of K. Moreover, it is clear that D(K) is dense in H, and the operator K is closed. Recalling the convention u = u| p + 1 − u| p − 1 and using integration by parts, we can show which implies that K is symmetric, and in fact, self-adjoint. The next proposition provides a condition under which the operator K is positive, that is, [Kũ,ũ] > 0 for all non-zerõ u ∈ D(K).
Proposition 3.6. K is self-adjoint with simple eigenvalues. Moreover, it has a maximal negative semidefinite subspace invariant under K that has dimension one 1 . For λ > λ 0 , the operator K is positive with a unique negative eigenvalue.
Proof. It follows from the above discussion that K is self-adjoint. Since H is a π 1 -space, [23,Theorem 12.1'] implies that K has a maximal negative semidefinite subspace invariant under K that has dimension one. By an argument similar to [44,Lemma 3.8] and [45,Lemma 2], we see that K has discrete spectrum and its eigenvalues are geometrically simple. Next, since K has a maximal invariant negative semidefinite subspace which is of dimension one, it has at least one eigenvalue of negative-semidefinite type. By this, we mean the restriction of [·, ·] to the eigenspace corresponding to an eigenvalue is a negative semidefinite inner product. We caution that this does not say anything about the sign of the eigenvalue itself. Let µ be a general eigenvalue of K with corresponding non-zero eigenvectorũ. Taking the complex conjugate of the equation Kũ = µũ, we see thatμ is also an eigenvalue of K (note that the coefficients of the operator K are real). Thus, either µ is real, or both µ and its complex conjugate are eigenvalues. For the latter case, the corresponding eigenvectorũ must be neutral, that is, [ũ,ũ] = 0. This follows from the observation that On the other hand, if µ ∈ R is an eigenvalue with corresponding eigenvectorũ such that Finally, suppose λ > λ 0 . By the Cauchy-Schwarz inequality, Thus, [Kũ,ũ] > 0, that is, K is positive. Then [ũ,ũ] = 0, soũ is non-neutral. Hence all eigenvalues are real when λ > λ 0 .
If µ is a negative semidefinite eigenvalue with the corresponding eigenvectorũ, then µ[ũ,ũ] = [Kũ,ũ] > 0, and hence, it follows that µ < 0. This means that any real negative semidefinite eigenvalue of K must be negative. In fact, there is only one such eigenvalue. Indeed, if ν is another eigenvalue with corresponding eigenvectorṽ, then which implies µ = ν or [ũ,ṽ] = 0. Since any maximal invariant semidefinite subspace of K is one dimensional, we must have µ = ν. We have therefore shown K has a unique negative eigenvalue.
First, we want to show that −1 is in the range of ν. This is because we want our solutions to be 2π-periodic in q, and the null space of F m (λ * , 0) is spanned by ϕ 1 (p) cos(q) (see Lemma 3.12), which is the case where n = 1 and hence µ = −n 2 = −1 in (3.9).

3.3.
Proof of local bifurcation. We are now prepared to prove Theorem 3.1. As stated above, our approach is based on the classical theory of Crandall-Rabinowitz on local bifurcation from simple (generalized) eigenvalues. Specifically, we will treat the family of laminar flows as our trivial solutions. Suppose the solution to the height equation (3.1) can be decomposed as h(q, p) = H(p; λ) + m(q, p) and Q = Q(λ). Then substituting it into the equation gives . It is clear that F(λ, 0) = 0 for all λ > 0. Let us record the Fréchet derivative of F with respect to m at (λ * , 0).
Proof of Theorem 3.1. Suppose conditions (3.2) and (LBC) are satisfied. Then F(λ, 0) = 0 for all λ > 0 and F m , F λ , F λm exist and are continuous, which means parts (i) and (ii) are confirmed. Moreover, Lemma 3.12 and Lemma 3.13 give dimension 1 for the null space of F m (λ * , 0) and co-dimension 1 for the range of F m (λ * , 0). Thus, F m (λ * , 0) has Fredholm index 0, and hence part (iii) is justified. Lastly, the transversality condition in Lemma 3.14 fulfills part (iv). Therefore, the local bifurcation result follows directly from Theorem A.1.
Finally, back to our objective, we note that the existence of a solution of class S ′ to the height equation (iii) D 2 F(λ * , 0) is a Fredholm operator of index 0. In particular, the null space is onedimensional and spanned by some element w * . (iv) D 1 D 2 F(λ * , 0)w * / ∈ R(D 2 F(λ * , 0)).
If D 2 2 exists and is continuous, then the curve is of class C 1 .