The model that comprises all these situations is the two-phase Euler equation. We denote by \( \mathcal{D}_{\text{in}}(t)\) the open domain of the interior phase with the surface \( §(t)=\partial \mathcal{D}_{\text{in}}(t)\) and by \( \mathcal{D}_{\text{out}}(t) = (\overline{\mathcal{D}_{\text{in}}(t)})^c\) the open domain of the exterior phase, so that

Inside of the two phases, the evolution is described by the incompressible Euler equations,

describing the conservation of momentum and mass. Here, \( u=u(t,x)\in\mathbb{R}^3\) is the velocity of the fluid, \( p=p(t,x)\in\mathbb{R}\) is the pressure, and \( \bar \rho=\bar \rho(t,x)\in\mathbb{R}_{\ge0}\) is the mass density. The latter is assumed to be constant in the respective phases so that \( \mathcal{D}_{\text{in}}(t) = \{\bar \rho(t)= \rho_\text{in}\}\) and \( \mathcal{D}_{\text{out}}(t)=\{\bar \rho(t)= \rho_{\text{out}}\}\) . We will assume that \( \rho_{\text{out}}\in\mathbb{R}_{>0}\) with \( \rho_{\text{out}}\ge \rho_{\text{in}}\) .

Across the interface, the tangential velocity component may experience a discontinuity (turning \( §(t)\) into a vortex sheet), though its normal component must be continuous,

Here \( [f]= f_{\text{in}} - f_{\text{out}}\) measures the jump across the interface of a discontinuous quantity \( f= f_{\text{out}} \mathbf{1}_{\mathcal{D}_{\text{out}}} + f_{\text{in}}\mathbf{1}_{\mathcal{D}_{\text{in}}}\) , and \( n\) is the unit normal vector pointing from \( \mathcal{D}_{\text{in}} (t)\) into \( \mathcal{D}_{\text{out}}(t)\) . The Young–Laplace equation

in which \( \bar \sigma\) is the surface tension and \( H\) is the mean curvature, relates the pressure difference to the geometry of the surface. We are using the convention that the mean curvature is the sum of the two principal radii \( H=(\kappa_1+\kappa_2)\) , chosen to be positive for convex \( \mathcal{D}_{\text{in}}(t)\) .

Finally, the two phases itself are advected by the flow,

The model reduces to the free boundary Euler equation if the interior phase is a vacuum, \( \rho_\text{in}=0\) , in which case the inner pressure is a constant, and the jump condition in (5) can be rewritten as \( p = \bar \sigma H\) , because the pressure is defined only up to an additive constant. Allowing for \( \rho_\text{in}=0\) in our choice of densities, it is enough to consider the general two-phase setting with jump condition (5) in the following.

Steady vortex rings are traveling wave solutions, moving at a constant speed \( \bar W\) in a fixed direction, say, \( e_z\) , i.e.,

In the new variables, the two phases and their interface \( §\) are frozen in time and the momentum equation becomes

In this formulation, \( \bar \rho\) is a two-valued constant. The relative velocity \( U- \bar We_z\) is now tangential to the ring’s surface,

It is not restrictive to assume a cylindrical symmetry about the \( e_z\) axis for our vortex ring, so that

There is no swirl velocity pointing in the angular direction, and for symmetry reasons, the velocity is tangential at the \( z\) -axis, \( U^r=0\) . We caution the reader that in these axi-symmetric variables, the velocity is not divergence-free with respect to \( r\) and \( z\) . Instead, it holds that

We will denote by \( \Omega\) the cross-sectional slice of the ring \( \mathcal{D}_\text{in}\) that lies in the meriodial half plane \( \mathbb{H} = \{(r,z): r>0\}\) .

In two-dimensional steady flows, a well-known advantage is that the pressure and kinetic energy density are related by the Bernoulli equation (see, e.g., Chapter 1.9 in [37]) along every streamline. Specifically, this relationship implies that

for some \( \bar \nu\in\mathbb{R}\) , which we will refer to as the Bernoulli constant. Here and in the following, \( \tau\) is the tangent on \( \partial \Omega\) obtained from \( n\) via a counter-clockwise rotation by 90 degrees, so that \( \tau=n^{\perp}\) . The above identity will be used to eliminate the pressure term in the Young–Laplace equation (5), see (24) below.

We remark for later reference that in the axisymmetric setting, the mean curvature \( H\) of the two-dimensional cylindrical surface \( \mathcal{S}\) reduces to the curvature \( \kappa\) of the one-dimensional surface \( \partial \Omega\) via

It remains to introduce two quantities that are of considerable importance in vortex theory. The first is the circulation \( \bar b\) around the vortex ring, which is computed along the boundary curve \( \partial \Omega\) of the vortex cross-section,

It describes the cumulative shear velocity around the ring and is proportional to the ring’s speed given by the Kelvin–Hicks formula (1). By Kelvin’s circulation theorem, the circulation along any (unsteady) evolution is an integral of the Euler equations.

The second quantity is the intensity of the interior vortex (or inner circulation),

where the last identity can be easily verified by an application of Stokes’ theorem. It is evident that in the regular setting, in which the tangential velocity does not exhibit a jump discontinuity, both quantities agree. In our construction, we will consider the situation in which the fluid is irrotational outside of the ring and the vorticity is uniform in its interior. Because it is the potential vorticity \( r^{-1}\text{curl} U\) that is transported by the axisymmetric Euler equations rather than the vorticity itself (see, for instance, Chapter 1.5 in [37]), it is natural to assume that \( \text{curl} U = \bar \xi r\) for some \( \bar \xi\in\mathbb{R}\) . In the case where the cross-sectional radius \( \bar \epsilon\) of the vortex ring is small compared to the inner radius \( R\) , so that \( \Omega\) is comparable to \( B_{\bar \epsilon}(R,0)\) , it must then hold that

The goal of this work is to find an axisymmetric traveling wave vortex ring solution to the free-boundary Euler equation (2), (3), (4), (5), (6), with cross-sectional domain \( \Omega\) (in \( \mathbb{H}\) ) that is close to a disc \( B_{\bar \epsilon}(R,0)\) . We restrict to small cross-sectional radii, \( \epsilon=\bar \epsilon/R\ll1\) .

We present an initial version of our main result, where we intentionally remain somewhat non-specific. For a mathematically precise statement, please refer to Theorem 3.

The theorem provides the first rigorous construction of vortex rings with surface tension, as well as hollow vortex rings. The asymptotic formula for the speed in (17) generalizes the Kelvin–Hicks formula. Specifically, as mentioned earlier, in the one-fluid case where \( \rho_{\text{in}} =\rho_{\text{out}}\) and \( \bar \sigma = 0\) , the circulation (11) and vortex intensity (12) agree, \( \bar a \approx \bar b\) . This allows us to recover the Kelvin–Hicks formula (1) with the constant \( c = 1/4\) . Similarly, if the interior is irrotational (\( \bar a = 0\) ) or a vacuum (\( \rho_{\text{in}} = 0\) ), we find that \( c = 1/2\) . The formula including surface tension was derived (non-rigorously) in Section 4 of [38] for a hollow vortex.

The uniqueness statement made in the theorem is intentionally kept vague. We kindly refer the impatient reader to Theorem 3 for more details on the solution class.

The conditions on the (positive) surface tension in (14) and (15) ensure that \( \bar\sigma_{ \epsilon}\) grows at least as fast as \( 1/ \epsilon\) for \( \epsilon\ll1\) , but slower than \( 1/ \epsilon^2\) . The lower bound is necessary to make sure that the linearized operator (see (56)) does not degenerate, while the upper bound is necessary for technical reasons. A discrete set of values is excluded if \( \bar\sigma_{ \epsilon}\) is near the lower end of this range. For these values, the linear operator associated with the jump condition (5) will degenerate, which we suspect results in a saddle point bifurcation. For a thorough explanation of this exclusion, we kindly refer the reader again to Theorem 3 and the subsequent discussion. The growth condition in (16) is rather mild and is satisfied by any reasonable monotone function in the specified range.

We consider it as important to notice that the choice \( \epsilon \bar \sigma_{ \epsilon}\sim |\log { \epsilon} |\) is admissible under the conditions (14), (15), and (16). This scaling is critical for rings with small cross-sectional radius, \( \epsilon=\bar \epsilon/R\ll1\) , as it balances both kinetic and surface energy in the two-phase Hamiltonian

associated with the Euler system (2), (3), (4), (5), (6). Indeed, the kinetic energy of a thin ring grows asymptotically as \( |\log \epsilon|\) , similar to that of a two-dimensional vortex patch, while the surface area decreases proportionally to \( \epsilon\) . Another relevant scaling is \( \epsilon \bar\sigma\sim 1\) , which is the critical scaling under which the limiting problem in 2D is scaling-invariant. We will elaborate on this later.

To simplify the notation in the following, we will interpret all differention symbols with respect to the cylindrical variables \( (r,z)\) , so that, for instance, \( \nabla = (\partial_r,\partial_z)^T\) , \( \nabla\cdot = \nabla\cdot\) , and \( \text{curl} = -\nabla^{\perp}\cdot\) .

It is a standard approach to rewrite the steady Euler system as an elliptic problem. Since \( rU(r,z)\) is divergence-free by (8), there exists a function \( \psi\) such that \( U = r^{-1}\nabla^{\perp}\psi\) in \( \mathbb{R}^3\) . This function is commonly referred to as the stream function. It is determined by the following elliptic Dirichlet problem:

Here the last condition encodes the fact that \( u\) should be non-singular at the \( z\) -axis. We remark that the Dirichlet boundary condition at the \( z\) -axis is consistent with the requirement that the velocity must be tangential there so that \( \partial_z \psi=0\) . As mentioned in the introduction, the vorticity \( \text{curl} U\) exhibits a singularity at the surface \( \partial \Omega\) of the vortex ring’s cross-section, characterized by the (negative) jump in the tangential velocity component. The magnitude of the shear velocities along this surface is not known a priori, and determining them (implicitly) will be part of our analysis.

As outlined above, to simplify the problem, we focus on uniform vorticity distributions inside of the ring, leading to the assumption \( \text{curl} U = r\bar \xi\) in \( \Omega\) for some constant \( \bar \xi \in \mathbb{R}\) , and we suppose that the fluid outside of the ring is irrotational, so that \( \text{curl} U=0\) in \( \overline{\Omega}^c\) . We thus relax (18) to

where the complement is taken with respect to the half plane \( \mathbb{H}\) . From the tangential flow condition in (7) we also derive a Dirichlet boundary condition at the surface. Since \( \psi - \bar W/2 r^2\) must be constant along \( \partial \Omega\) , there exists a \( \bar \gamma\in\mathbb{R}_{\ge 0}\) so that

In the literature, \( \bar \gamma\) is referred to as the flux constant. The non-negativity of \( \bar \gamma\) is a consequence of the maximum principle.

Next, we reformulate the Young–Laplace equation using the stream function. By replacing the pressure jump in (5) with the jump in the squared shear velocity, as given by the Bernoulli-type equation (9), and expressing the velocity in terms of the stream function, we derive the following updated jump condition:

Notice, that we used the identity \( \tau\cdot e_z = n\cdot e_r\) to rewrite the speed term.

Taking into account equations (19), (20), (22), (23), and (24), the elliptic problem for the stream function becomes an overdetermined boundary value problem. For a given domain \( \Omega\) and fixed velocities \( \bar W\) and flux constants \( \bar\gamma\) , the inner domain Dirichlet problem (22) and (23), together with the outer domain Dirichlet problem (22), (19), and (20) would already uniquely determine \( \psi\) . Therefore, the requirement that the jump condition (24) is also satisfied imposes a condition on the shape of \( \Omega\) .

Our analysis shows that this problem is uniquely solvable for a class of domains \( \Omega\) close to \( B_{\bar \epsilon}(R,0)\) . It also determines the unknowns \( \bar W, \bar \gamma\) , and \( \bar \nu\) .

We now present a second still non-specific version of our main theorem, which leads to Theorem 1 by choosing \( U=r^{-1} \nabla^\perp\psi\) .

Analyzing overdetermined boundary problems is a classical problem in mathematical physics, particularly in potential theory, fluid dynamics, and differential geometry. A well-known example is the Schiffer problem, where the eigenvalue problem for the Neumann Laplacian is studied for geometries that permit constant Dirichlet data (see page 688 of [39] or [40]). Closely related is the overdetermined Dirichlet eigenvalue problem proposed by Sicbaldi [41], or in a more general nonlinear form by Berestycki, Caffarelli, and Nirenberg [42]. See also the work by Dom
specialChar{39}inguez-V
specialChar{39}azquez, Enciso, and Peralta-Salas [43].

In the context of (one-phase) vortex rings, overdetermined boundary problems similar to ours, but involving a regular stream function, have been studied since Fraenkel’s work [12]. Alt and Caffarelli also explored free fluid problems using a variational method [44]. While many related contributions exist, we cannot cite them all here; however, a good, though still non-comprehensive, overview is provided in [43]. Our problem differs from those mentioned above in that it involves a jump condition for the normal derivative, rather than a (nonlinear) Neumann condition. To the best of our knowledge, such conditions have not been investigated in other settings.

We solve the overdetermined boundary problem in a perturbative setting. The key observation is that, in the limit of vanishing cross-sectional radii, \( \epsilon \to 0\) , the vortex ring locally resembles a cylinder, and the cross-section \( \Omega\) approximates the disk \( B_{\bar \epsilon}(R,0)\) . After an appropriate change of variables (i.e., using blow-up variables), the limiting problem becomes thus a circular vortex patch in the plane, where the jump condition (24) is trivially satisfied.

In the next subsection, we will perform this change of variables and then formulate our precise result.

For a ring of small cross-sectional radius, we make the ansatz

where \( \Omega_{\theta} \) is the domain enclosed by the image \( \chi_{\theta}(\partial B_1(0))\) of the unit circle under the map

By construction, \( \theta\) describes the radial signed distance from the boundary \( \partial \Omega_{\theta}\) to the unit circle \( \partial B_1(0)\) . It will be regular and small so that the cross-section is nearly spherical. See Figure 1 for an illustration.

The non-dimensional parameter \( \epsilon:=\frac{\bar\epsilon}{R}\) measures the ratio of the inner ring radius \( R\) to the cross-section radius. In order to simplify the notation slightly, we will often write \( B\) for \( B_1(0)\) . Moreover, whenever it is convenient, we will parametrize the circle \( \partial B\) by the torus \( \mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}\) , and we will be somewhat sloppy in our notation when considering objects defined on it. Indeed, we will sometimes abuse the notation by identifying \( \theta(x)\) with \( \theta(\alpha)\) , where \( \alpha \in \mathbb{T}\) denotes the angle between \( x\) and the \( x_1\) -axis, so that \( x\cdot e_1 = \cos \alpha\) . Similarly, we will write \( \chi_\theta(\alpha) \) for \( \chi_\theta(x)\) whenever it seems comfortable. With this notation, \( \partial_{\alpha}\) will always be the tangential derivative along the circle.

We will work with the geometric conditions

The first prescribes symmetry across the radial axis (which is natural as the problem is symmetric in \( z\) ), the second fixes the volume of the perturbed ball, and the third condition sets the center of the domain to the origin (with respect to the new variables). The second and third conditions are justified since \( \bar\epsilon\) and \( R\) are given. The last condition is there to make sure \( \Omega_\theta\) is well-defined.

We will choose our perturbation variables in the Banach manifold with boundary

where \( k\in \mathbb{N}\) is an arbitrary number with \( k\geq 5\) and \( \ell \in(\frac{1}{2},1)\) is arbitrary. The restriction \( k\geq 5\) is most likely not optimal, but difficult to improve and mostly comes from Lemma 7. In particular, it implies that \( \partial \Omega_\theta\) is (at least) \( C^4\) because of the Sobolev embedding \( H^5(\partial B)\hookrightarrow C^4(\partial B)\) . The purpose of the last condition in (27) is to estimate broken terms such as e.g. \( \theta^2\epsilon^{-1}\) near \( 0\) .

We will also consider the sets

of all \( \theta\) such that (26) holds, which is a Banach manifold too (and in fact smooth around \( 0\) , as we will see in Subsection 2). For notational convenience, we will furthermore write

It is convenient to split the elliptic problem (19), (20), (21), (22), (23) into an inhomogeneous interior contribution and the remaining homogeneous problem, which are independent of each other. More specifically, slightly abusing our notation convention, we consider \( \psi_{\text{in}} = \psi-\frac{\bar W}2r^2-\bar \gamma \) , which solves the elliptic Dirichlet problem inside of the domain,

The outer part \( \psi_{\text{out}}=\psi-\psi_{\text{in}}\mathbf{1}_{\Omega}\) , defined on all of \( \Omega\) , then fulfills

together with the boundary conditions

In these new variables, the circulation condition (11) reads

It is readily checked that \( \psi_{\text{out}}=\frac{\bar{W}}{2}r^2+\bar \gamma\) also inside of \( \Omega\) . Using this decomposition, the jump condition (24) simply reads as

where the first normal derivative is the inner normal derivative.

We note that \( \psi_{\text{in}}\) is independent of \( \bar b\) and 1-homogenous in \( \bar \xi\) , while \( \psi_{\text{out}}\) is independent of \( \bar \xi\) and 1-homogenous in the triple \( (\bar b, \bar W, \bar \gamma)\) .

Because we aim at constructing vortex rings of small cross-sectional radius, \( \epsilon\ll1\) , it is customary to rescale the elliptic problem (30), (31), (32), (33) using blow-up variables. For this purpose, we consider the mapping \( q\) defined by

which maps \( \Omega\) onto some \( \Omega_{\theta}\) . We observe that \( r = R(1+\epsilon x_1)>0\) enforces that \( x_1 >-1/\epsilon\) , and thus

Moreover, we will consider

as motivated by (13) and the scaling of the circulation, which implies that \( \bar a R\approx a\) up to lower order terms. Finally, we take the opportunity to non-dimensionalize the problem. Setting

and writing \( \varphi_{\text{in}} = 4\pi a^{-1}\psi_{\text{in}} \circ q^{-1}\) and \( \varphi_{\text{out}}=b^{-1}\psi_{\text{out}}\circ q^{-1}\) , we are now considering the overdetermined free boundary problem

where \( \kappa_{\theta}\) is the curvature of the one-dimensional boundary \( \partial \Omega_{\theta}\) (see (10) for the derivation of that term). The factor of \( 4\) in (39) is simply introduced for notational convenience in the later part of the paper. Furthermore, we have that

cf. (33). Our rescaling of the inner stream function is valid only if the inner vorticity distribution is non-zero, \( a\not=0\) . If, however, \( a=0\) , we may simply choose \( \varphi_{\text{in}}=\psi_{\text{in}}=0\) .

The resulting problem thus depends only on two non-dimensionalized physical control quantities: the mass ratio \( \rho\) and the surface tension parameter \( \sigma\) . The speed of the ring \( W\) , the flux constant \( \gamma\) and the Bernoulli constant \( \nu\) will all depend on these quantities and on the ring geometry. Furthermore, the assumptions on the surface tension (14), (15), and (16) become

Our main result can now be stated in the following precise form:

Reversing the above rescaling, we recover Theorem 2.

We will find our solution to this overdetermined boundary problem by applying the implicit function theorem to the jump condition (47). Apparently, in the limiting case \( (\theta,\epsilon)=(0,0)\) , the cross-section domain \( \Omega\) becomes the unit disk and the left-hand side of (47) reads

which is indeed constant because the limit potentials \( \varphi_{\text{in}}\) and \( \varphi_{\text{out}}\) defined now on \( B\) and \( \mathbb{R}^2\) , respectively, are radially symmetric functions. We will give more insights into the strategy of the proof in the subsequent subsection. At this point, however, we already remark that in the case of positive surface tensions, the linear operator associated with the jump condition (47) has the Fourier symbol

which may actually degenerate unless assumption (49) is satisfied. The limiting problem for a hollow vortex ring was previously studied by Wegmann and Crowdy [45], who discovered bifurcations from a circular vortex sheet. These bifurcations bear some resemblance to the well-known Crapper waves, which are non-trivial, steady irrotational capillary water waves. After fixing the mass density, the bifurcations occur at precisely the same critical values that we need to exclude in (49). It would be interesting to investigate such bifurcations in the context of our thin vortex rings as well. We plan to address this problem in future work.

Our goal is to apply the implicit function theorem to the jump condition (47). We thus have to establish that the geometric quantities and the solution given through (39)–(48) depend continuously differentiable on the pair \( (\epsilon,\theta)\) . With this, after pulling back the jump condition (47) to the unit circle, we have that

where now

where \( \kappa_\theta\) is the mean curvature of \( \Omega_\theta\) and \( n_\theta\) its normal. The circulation condition (48) becomes

Our goal is to find a unique shape function \( \theta=\theta(\epsilon)\) such that the jump condition (57) is additionally satisfied. To this end, we use the abstract notation \( \text{const}\) to represent the Bernoulli constant \( \nu\) , indicating that the jump condition will be solved in a quotient space. As previously stated, we will apply the implicit function theorem to solve this equation. In the base case \( (\theta,\epsilon)=(0,0)\) , the equation is trivially satisfied because, on the ball \( \Omega_0=B_1(0)\) , the curvature \( h_{0,0}\) is constant. The solutions \( \varphi_{\text{in}}\) and \( \varphi_{\text{out}}\) to the inner and outer problems, respectively, are radially symmetric, which ensures that their normal (and thus radial) derivatives are constant on the interface \( \partial\Omega_0\) . Furthermore, the coefficient \( \epsilon W\) vanishes in this case.

Conceptually, the limiting problem for \( \epsilon=0\) should correspond to a similar reformulation for a stationary 2D vortex.

We can not expect the implicit function theorem alone to prove the statement, as the limiting problem is clearly invariant under translation, and hence the derivative of (57) will degenerate in at least one direction. To deal with this degeneracy we will introduce an additional parameter \( S\) , which, roughly speaking, corresponds to the additional degree of freedom that the speed \( W\) gives us and which is not present in the limiting problem of a 2D vortex. This extra parameter will then allow us to perform a Lyapunov–Schmidt reduction. We will see that in terms of Fourier series the subspace that is missing in the image of the derivative is the span of \( \cos\alpha=x_1\) (up to lower order terms).

To solve the equation for positive \( \epsilon\) , we will show that all quantities \( h_{\theta,\epsilon}\) , \( \lambda_{\theta,\epsilon}\) and \( \mu_{\theta,\epsilon}\) are in a suitable sense continuously differentiable in \( (\theta,\epsilon)\) . This is comparably simple for the curvature \( h_{\theta,\epsilon}\) and the inner potential \( \lambda_{\theta,\epsilon}\) . We treat these terms in Sections 3 and 4 below.

For the outer problem, the analysis is more difficult for two reasons. The first one is that the (formal) limiting problem

does not have a good solution theory in \( H^1\) anymore. Indeed solutions to this are of the form \( p\log|x|+q\notin H^1\) if \( \Omega_\theta=B\) and only unique under additional assumptions (such as fixing the prefactor of the logarithm), while for \( \epsilon>0\) , the solution for \( \psi_{\text{out}}\) does not require a selection principle.

Instead of directly working with the PDE for the limiting problem (59), (60), we will use a more indirect approach and reformulate this problem as

which does indeed yield a unique solution \( \phi=\frac{1}{2\pi}\log|x-y|*(\partial_n\phi\mathcal{H}^1\text\sf{{L}}\partial\Omega_\theta)\) by classical potential theory, see e.g. [46] for further reading.

This formulation is also available for \( \epsilon>0\) . If one lets \( K(x,y)\) denote the fundamental solution of \( -\nabla\cdot(\frac{1}{r}\nabla\cdot)\) , then we can write

See Section 5.1 for a detailed derivation.

There is also a natural analog for (62), which is the circulation condition (33), that is,

The basic strategy is then to study the convergence of (61), (62) to (63), (64) (pulled back to \( \partial B\) ).

The constants \( W\) and \( \gamma\) are still undetermined at this point. One of their degrees of freedom should be spent by fixing the circulation, the other by choosing the reduction parameter \( S\) .

The second difficulty is that \( W\) and \( \gamma\) apparently blow up logarithmically as \( \epsilon\rightarrow 0\) , as their asymptotics in Theorem 3 show. As the main term in \( K\) is a logarithm \( \log|x-y|\) (see (103) and (104)), it will also naturally contain \( |\log\epsilon|\) -terms after pulling back too. Therefore one needs to be very careful to not pick up any terms with an asymptotic \( \epsilon|\log\epsilon|\) or similar, which would not be continuously differentiable. In particular, for a general \( W\) and \( \gamma\) , the function \( \mu_{\theta,\epsilon}\) will in general not be differentiable in \( \epsilon\) .

The approach for extracting a continuously differentiable \( \mu_{\theta,\epsilon}\) , with a well-behaved dependence on the reduction parameter \( S\) , is to compute the asymptotic of the pullback of \( K\) , and then adjust \( W\) and \( \gamma\) to match the desired asymptotic of \( \mu_{\theta,\epsilon}\) . The desired asymptotic of \( \mu\) is

where \( \tilde{\mu}_{\theta,0}\) is the solution of the limiting problem given through \( -\partial_n \phi\circ \chi_\theta\) in (61), (62).

The crucial step is that both \( \gamma +\frac{1}{2}W(1+\epsilon(\chi_\theta)_1)^2\) and the image of this ansatz under \( K\) lie in the span of \( 1\) and \( y_1\) up to terms of order \( \epsilon^2|\log\epsilon|\) . This then allows to construct \( \mu,W,\gamma\) so that (65) holds with higher order terms of vanishing derivative at \( 0\) .

In Section 6, we finally solve the identity (57) for small values of \( \epsilon\) . We will see that the derivative in \( \theta\) at \( 0\) is the Fourier multiplier

which is invertible for \( l\neq 1\) (the mode \( l=1\) corresponds to translations and is handled by the reduction) under the condition (49).

Our paper is rather technical. To facilitate reading, we provide an overview of the notation:

• We typically write \( x\) for the space variable on \( \Omega_{\theta}\) or \( \partial \Omega_\theta\) and \( y\) for the variable on \( \partial B\) or \( B\) .

• \( \theta(\alpha)\) denotes the (signed) distance between \( \Omega_\theta\) and \( \partial B\) at angle \( \alpha\) .

• \( \Theta\) denotes a (fixed) extension of \( \theta\) to \( B_1(0)\) .

• \( \chi_\theta(y) = (1+\theta(y))y\) is the canonical map from \( \partial B_1(0)\) to \( \partial \Omega_{\theta}\) . Occasionally we will write only \( \chi\) .

• \( \mathcal{X}_\theta(y) = (1+\Theta(y))y\) is an extension of \( \chi_\theta\) which maps \( B_1(0)\) to \( \Omega_\theta\) .

• The Jacobian of the map \( \chi_\theta\) between the boundaries equals

  \[ \begin{align} m_\theta:=\sqrt{(\partial_\alpha\theta)^2+(\theta+1)^2}. \end{align} \]

  (67)

• Let \( X(\alpha) = (\cos \alpha , \sin \alpha)^T\) be the radial vector in polar coordinates as a function of the angle \( \alpha\in \mathbb{T}\simeq \partial B\) . Then the pullback of the tangent \( \tau_\theta\) of \( \partial\Omega_\theta\) is given by

  \[ \tau_{\theta}\circ \eta_\theta= \frac1{m_{\theta}} \left( (1+\theta) X^{\perp} +\partial_{\alpha}\theta X\right), \]

  and the pullback of the tangent \( n_\theta\) of \( \partial\Omega_\theta\) is given by

  \[ \begin{equation} n_{\theta}\circ \chi_\theta=\frac1m_{\theta} \left((1+\theta) X - \partial_\alpha \theta X^{\perp}\right). \end{equation} \]

  (68)

  We will occasionally just write \( n\) instead of \( n_\theta\) .

We would like to collect some general facts about \( H^s\) spaces and Fréchet differentiability in them that we will frequently use.

On the boundary \( \partial B\) , we can write \( L^2\) -functions as Fourier series, since \( \partial B\simeq \mathbb{T}\) . We then set

This is the same as the fractional Sobolev space \( W^{2,s}\) defined via the Gagliardo seminorm. For \( s\notin \mathbb{Z}+\frac{1}{2}\) and \( s> \frac{1}{2}\) there is a Sobolev embedding \( H^s(\partial B)\hookrightarrow C^{s-\frac{1}{2}}(\partial B)\) .

There is a continuous linear trace extension operator from \( H^s(\partial B)\) to \( H^{s+\frac{1}{2}}(B)\) for \( s>0\) , that is, for every \( \theta \in H^s(\partial B)\) there is a \( \Theta\in H^{s+\frac{1}2}(B)\) such that

Here the norm on \( H^{l}(B)\) is defined as usual via

There is a Sobolev embedding \( H^s(B)\hookrightarrow C^{s-1}(B)\) for \( s\notin \mathbb{Z}\) . There is a bounded linear trace \( H^s(B)\rightarrow H^{s-\frac{1}{2}}(\partial B)\) for \( s>\frac{1}{2}\) . We refer the reader e.g. to the monograph [47] for further reading.

One important property is that if \( u\in H^s(\Omega', \mathbb{R}^l)\) for some bounded \( \Omega'\subset \mathbb{R}^d\) with sufficiently smooth boundary and \( s>\frac{d}{2}\) , then for every \( F\in C^{\lceil s\rceil}\) , the composition \( F(u)\) is in \( H^s\) as well and composition with \( F\) is continuous in \( u\) , for a proof see e.g. [48]. In particular, this implies that these spaces are closed under products.

One important application is that:

We remark that we are not going to make use of function spaces on any domains different from \( B\) and \( \partial B\) in the sequel.

Here and occasionally in the following, we allow for negative values of the cross-section parameter \( \epsilon\) for mathematical convenience.

With the previous proposition, we have already all the information at hand we need to deal with the surface tension term in (57). In the next lemma, we establish the regularity of two more geometric quantities that will be needed to analyze the pressure contributions in (57).

In the statement of the proposition, the brackets \( \langle\cdot,\cdot \rangle\) denote the \( L^2(\partial B)\) inner product.

As a trivial consequence, because \( V^k\) is a smooth manifold the geometric identities characterizing the tangent plane also hold at \( \theta\neq 0\) up to higher order terms. For instance, the volume condition implies the following.

Here we are using the natural identification of the tangent space with a subspace of \( H^k\) .

We have noticed in our reformulation (57) of the jump condition that we only need to understand the normal derivative of the outer homogeneous part \( \varphi_{\text{out}}\) along the boundary of the domain \( \Omega_{\theta}\) . It holds that

and hence

Recall that in this formulation, the normal \( n\) , including the one in the normal derivative, points in the direction outside of \( \Omega_{\theta}\) . Moreover, writing the elliptic equation in (42) globally in \( \mathbb{H}_{1/\epsilon}\) , we obtain

in the sense of distributions.

The Dirichlet condition in (43) can thus, using a single layer potential, be rewritten in the form

for any \( x\in \partial \Omega_{\theta}\) , where \( K_{0,\epsilon}\) denotes the fundamental solution of the elliptic operator \( -\nabla\cdot(\frac1{1+\epsilon x_1}\nabla\cdot )\) in \( \mathbb{H}_{1/\epsilon}\) . (For a rigorous proof that this is compatible with the conditions at the axis and \( \infty\) , see, e.g. Lemma 2.1.4 in [51].)

We will pull back this equation onto \( \partial B\) in order to analyze the asymptotic of the boundary values. Then it reads as

where we set

The Jacobian \( m_\theta\) was defined in (67).

In order to derive the precise form of the kernel \( K_{\theta,\epsilon}\) , we go back to the original operator \( -\nabla\cdot(\frac{1}{r}\nabla\cdot)\) in (22) with the boundary and decay conditions (19), (20), (21). Its fundamental solution \( K_{\mathbb{H}}\) takes the form

where the function \( F:(0,\infty)\to \mathbb{R}\) is defined by

see, for instance, [52] or Chapter 19 of the lecture notes [53]. For further reference, we notice the following expansion:

Changing now variables from \( (r,z)\) on \( \partial \Omega\) to \( y\in \partial B\) yields

We shall occasionally write \( \mathcal{K}_{\theta,\epsilon}\) for the associated linear operator,

With \( \mu_{\theta,\epsilon}\) introduced in Section 1.4,

the boundary equation (100) now takes the short form

We have to show that \( \mu_{\epsilon,\theta}\) depends continuous differentiably on \( (\epsilon,\theta)\) . This is nontrivial as the kernel \( K_{\theta, \epsilon}\) is (logarithmically) diverging as \( \epsilon\to 0\) , see Lemma 12 below, which is, in fact, a behavior that is inherited by the velocity \( W\) and the flux constant \( \gamma\) . The limit \( \epsilon\rightarrow 0\) of (107) is thus singular, so we can not expect \( \mu_{\theta,\epsilon}\) to depend smoothly on \( \epsilon\) . In order to make sure that at least a first derivative exists, we will need to carefully analyze the behavior of the integral kernel \( K_{\theta,\epsilon}\) and we will have to construct appropriate boundary values.

We start our step-by-step analysis with a study of the renormalized limiting operator: Considering the expansion of \( F\) in (104), it is not difficult to observe formally that

and it is thus natural to study the kernel

as an auxiliary object, which is simply the pullback of the Newtonian potential \( -\frac{1}{2\pi}\log|x-y|\) to \( \partial B\) . We denote by \( \widetilde{\mathcal{K}}_{\theta,0}\) the associated linear maps, cf. (106).

Picking (formally) \( \epsilon=0\) in (107), we are led to the auxiliary problem

which corresponds to solving the Laplace equation in \( \mathbb{R}^2\setminus \partial \Omega_\theta\) with constant Dirichlet boundary data,

It is clear that \( \phi\) is constant inside of \( \Omega_{\theta}\) , and one can select a unique solution by requiring that

We are thus concerned with the classical capacity potential problem for the set \( \Omega_{\theta}\) .

Going back to the notation introduced earlier, the normal derivative jump \( \tilde \mu_{\theta,0} = [\partial_n \phi\circ \chi_{\theta}]\) solves the problem

As mentioned earlier, we will use the solution \( \tilde{\mu}_{\theta,0}\) as the formal interpretation of \( \mu_{\theta,\epsilon}\) at \( \epsilon=0\) . Differentiability properties will be studied below in Lemma 5.

The approximate map \( \widetilde{K}_{\theta,0}\) is in fact smooth in \( \theta\) and its derivative can be understood quite easily. In preparation for proving this, we first need to show that these kinds of kernels behave well on \( H^k(\partial B)\) -spaces.

Our first goal will be to show that \( \mathcal{K}_{\theta,0}\) is smooth in \( \theta\) as a map from \( H^{k-1}(\partial B)\) to \( H^k(\partial B)\) . This requires us to develop some machinery about the general mapping properties of such kernels.

The key result of this subsection is Proposition 4.

We first want to understand the kernel at \( \theta=0\) , for this it will be convenient to understand the action of the Newtonian potential operator \( \tilde{\mathcal{K}}_{0,0}\) in terms of Fourier series. For this we will occasionally identify \( \mathbb{R}^2\) with \( \mathbb{C}\) , so that \( e^{i\alpha}\) parametrizes \( \partial B\) with \( \alpha\in \mathbb{T}\) .

In order to see what kind of terms we will need to study for \( \theta\neq 0\) , we first expand the kernel.

This motivates that we should study kernels involving difference quotients of \( \theta\) .

Here comes our key result of this subsection.

We first want to understand the behavior of the limiting kernel \( \widetilde{\mathcal{K}}_{\theta,0}\) , which is smooth (Lemma 8), easy to understand in terms of Fourier series at \( \theta=0\) (Lemma 5) and invertible modulo constants (Lemma 10).

For the convenience of the reader, we recall that in the statement of the lemma, \( \tilde{\mathcal{K}}_{0,0}\) is the operator associated with the Newtonian potential \( - \frac1{2\pi}\log |y-\tilde y|\) . We remark that the derivative \( \left. \mathrm{D}_{\theta}\right|_{\theta=0} \widetilde{\mathcal{K}}_{\theta,0}\) has a nontrivial kernel which reflects the translation invariance of the limiting problem. Indeed, using (110) below we immediately see that the derivative vanishes for \( f=\text{const} \) and \( \theta(x)=y_1\) .

The following Lemma follows directly from Lemma 5.

The invertibility of \( \widetilde{\mathcal{K}}_{\theta,0}\) near \( \widetilde{\mathcal{K}}_{0,0}\) follows now via a perturbation argument:

By choosing \( g=0\) in the above statement, we derive the regularity of \( \tilde \mu_{\theta,0}\) as defined in (108).

It will also be convenient for later to estimate the unknown constant in (108).

In this subsection, we will analyze the asymptotic of \( \mathcal{K}_{\theta,\epsilon}\) and show that this kernel is still invertible modulo constants (Lemma 14).

The linear maps associated with the explicit terms in this expansion are all also continuously Fréchet differentiable (except for the singularity of \( \log\frac{1}{\epsilon}\) at \( \epsilon=0\) ) with operator norms and derivatives of the obvious order as a straightforward application of Lemma 7 and Proposition 4 shows. This implies the following bounds, which we state without a proof.

In this section, we finally conclude the existence of a continuously differentiable \( \mu_{\theta,\epsilon}\) in Proposition 6, where \( \mu_{\theta,\epsilon}\) is the solution of the boundary equation

satisfying the integral constraint

We recall that, at this point, \( W\) and \( \gamma\) are not defined yet. Instead, identifying them appropriately will be part of the problem.

Anticipating that the leading order contribution of \( \mu_{\theta,\epsilon}\) is given by the solution \( \tilde \mu_{\theta,0}\) to the approximate problem studied in Subsection 5.3 above, we make for any \( S\in\mathbb{R}\) the ansatz

where \( \mathcal{E}_{\theta,\epsilon}(S)\) shall be an error term that is (more or less) of second order. The precise form of the first order in \( \epsilon\) contribution is motivated by the corresponding term on the right-hand side of (149).

We will construct \( W\) and \( \gamma\) as functions of \( S\) by matching their asymptotics with that of the image of this ansatz under \( \mathcal{K}_{\theta,\epsilon}\) . The speed \( W\) is determined through the following lemma.

Our proof actually reveals that

is an admissible choice.

We postpone the proof of this lemma for a moment.

Inserting the expansions (151) and (152) into the boundary equation (149), we obtain the relation

Moreover, as we have fixed the circulations in (58) and (108) and because

we also have the integral condition

Using this insight, we will obtain smallness and continuous differentiability of \( \mathcal{E}_{\theta,\epsilon}\) via Lemma 14.

In fact, given \( \widetilde{\mathcal{E}}_{\theta,\epsilon}\) from Lemma 15 and determining \( \mathcal{E}_{\theta,\epsilon}\) via (157) and (159), the formula in (151) can be actually considered as a definition of \( \mu_{\theta,\epsilon}\) . The desired boundary equation (149) will automatically hold then for some implicitly given \( \gamma\) , whose asymptotics we calculate below in Lemma 16.

We stress that \( \mu\) , \( W\) , \( \gamma\) and \( \mathcal{E}_{\theta,\epsilon}\) are all affine functions in the reduction parameter \( S\) .

We collect some trivial consequences.

We finally provide the proof of Lemma 15.

It remains to compute \( \gamma\) :

For the uniqueness of the solution, it will also be crucial that the values of \( W\) and \( \gamma\) determine \( S\) , if the circulation is given.

We shall split \( \mathcal{F}=\mathcal{F}_1+\mathcal{F}_2\) , where \( \mathcal{F}_1\) is the projection of \( \mathcal{F}\) onto the span of \( x_1\) (with respect to the standard scalar product in \( L^2(\partial B)/\mathbb{R}\) ), that is,

and \( \mathcal{F}_2\) is the projection onto the orthogonal complement of this. As explained in Subsection 1.4, \( \mathcal{F}_1\) is the part that we cannot control with the implicit function theorem due to the translation invariance of the limiting problem and which hence requires a Lyapunov–Schmidt reduction.

In this first step, our goal is to ensure that we can pick \( S=S_{\theta,\epsilon}\) as a function of \( \epsilon\) and \( \theta\) uniquely such that \( \mathcal{F}_1(S_{\theta,\epsilon})=0\) for small enough \( (\epsilon,\theta)\in \mathcal{M}\) , under the constraint

In the next subsection, we will show that the contribution to \( \mathcal{F}_2\) through \( S\) as a function of \( (\theta,\epsilon)\) is continuously Fréchet differentiable in \( (\epsilon,\theta)\in \mathcal{M}\) with derivative \( 0\) at \( 0\) in \( H^{k-1}(\partial B)\) .

Because \( \mu\) is affine in \( S\) and the other terms are independent of \( S\) , the functional \( \mathcal{F}_1\) is a quadratic polynomial in \( S\) , of which we first compute the asymptotics of the coefficients.

We first note that it holds that \( \mathcal{F}(0,0,0)=\text{const}\) , because \( \lambda_{0,0}=-2\) by (87), \( \mu_{0,0}(0)=\frac1{2\pi}\) by Proposition 5 and (151) and \( h_{0,0}=1\) , and because \( (\epsilon \sigma)^{-1}\) is bounded in the case of positive surface tension by our assumption in (185). In particular, it must hold that \( \mathcal{F}_1(0,0,0)=0\) .

We now expand \( \mathcal{F}_1\) into a first-order Taylor series. Using the smoothness \( \lambda_{\theta,\epsilon}\) established in Proposition 3 and recalling Remark 1, we see that

because \( \langle 1,y_1\rangle =0\) . Similarly, using the smoothness of \( h_{\theta,\epsilon}\) by Proposition 1, we observe that

because via two integrations by parts, it holds that \( \langle \partial_{\tau}^2 \theta,y_1\rangle = -\langle \theta,y_1\rangle\) . The term \( \mu_{\theta,\epsilon}(0)\) is not smooth, but the second-order error in its expansion is of order \( |\log\epsilon|(\epsilon^2+\left\lVert \theta \right\rVert_{H^k}^2)\) by Proposition 6. Using this proposition and Proposition 5 together with Remark 2, we find that

if we expand along a geodesic (in the sense that we take the image of a line under the diffeomorphism between a neighborhood of \( 0\) in \( \mathcal{M}\) and an open set of \( \mathrm{span}\{1,x_1\}^\perp\subset H_{\text{sym}}^k(\partial B)\) as described in Subsection 2) so that \( \delta \theta\in T_0V^k\) satisfies \( \|\delta \theta\|_{H^k}\lesssim \|\theta\|_{H^k}\) .

Combining these expansions, in sum we obtain that

Moreover, similar calculations reveal that

Similarly, we can compute the asymptotic of the terms depending on \( S\) , using that only the contribution of \( \mu\) depends on \( S\) . We write

By construction and (166), we have that \( \mu(S)-\mu(0)=\epsilon S(e_1\cdot n_\theta)\circ \chi_\theta\) up to a term which is at most of the order \( |S||\log\epsilon|(\epsilon^2+\left\lVert \theta \right\rVert_{H^k}^2)\) . Also we have

as this is smooth and vanishing at \( \theta=0\) , with a corresponding bound on the derivative. Hence, we have for the first term on the right-hand side of (192) that

and the error term has a derivative one order lower. For the second term, we may use a similar computation via (166) and noticing that \( \langle y_1,y_1^2 \rangle=0\) , to see that

Together, this gives us that

For the derivatives, we perform a similar calculation, using the assumption (51) on the derivative of \( \sigma_\epsilon\) , and find that

and

All the error terms are precisely linear resp. quadratic in \( S\) .

Hence we see from (189) and (195), after dividing by \( \epsilon\) that \( \mathcal{F}_1(S)=0\) is a quadratic equation in \( S\) , namely

It will be more convenient to consider this as an equation in \( t=\frac{S}{\sigma\epsilon}\) . We thus are concerned with a quadratic equation of the form

where

According to our assumption that \( \left\lVert \theta \right\rVert_{H^k}\lesssim \epsilon^\ell\) (see (27)), the equation is certainly solvable in \( \mathbb{R}\) under the assumption (14), which was that

The two unique solutions are then (approximately) given by

and thus, if \( \theta\) and \( \epsilon\) are small enough, \( t\approx a/b\) is the unique solution satisfying \( |ct|=o(1)\) .

On the level of \( S=S_{\theta,\epsilon}\) , the argument shows that there is precisely one solution satisfying

for \( (\theta,\epsilon)\in \mathcal{M}\) small. It has the leading order asymptotics

where we used the bound (50) on \( \sigma\) , and thus

Because all error terms occurring in (198) are continuously differentiable in \( (\theta,\epsilon)\) , the (real-valued) implicit function theorem ensures that \( S_{\theta,\epsilon}\) depends in a neighborhood of \( (0,0)\) continuous differentiably on \( (\theta,\epsilon)\) . Differentiating the defining equation \( \langle \mathcal{F}(S),x_1\rangle=0\) and using the assumption (51) (which says \( |\partial_\epsilon\sigma_\epsilon|\lesssim \sigma_\epsilon \epsilon^{-1}\) ) as well as the identities (190), (192), (196), and (202) or, respectively, (191), (192), (197), and (201) we find on the one hand that

and thus,

for \( (\theta,\epsilon)\in \mathcal{M}\) sufficiently small, where we have used (200). On the other hand, it holds that

for \( (\theta,\epsilon)\in \mathcal{M}\) sufficiently small, and thus,

assuming that (200) and (185) are satisfied.

The same calculation can also be made for the functional without surface tension, where, writing \( S^{\sigma=0}_{\theta,\epsilon}\) instead of \( S_{\theta,\epsilon}\) for clarity, one gets

for the unique solution satisfying

for \( (\theta,\epsilon)\in \mathcal{M}\) small. The derivatives are controlled as follows:

Next, we show that the contribution of \( S_{\theta,\epsilon}\) to \( \mathcal{F}_2\) is neglectable, i.e. that \( \mathcal{F}_2(S_{0,0},0,0)=0\) in \( H^{k-2}(\partial B)/\mathbb{R}\) (or \( H^{k-1}(\partial B)/\mathbb{R}\) in the case \( \sigma=0\) ) and that \( \mathcal{F}_2(S_{\theta,\epsilon},\theta,\epsilon)\) is continuously Fréchet differentiable near \( 0\) with \( \mathrm{D}_{\theta,\epsilon}\big|_{\theta,\epsilon=0}\mathcal{F}_2(S_{\theta,\epsilon},\theta,\epsilon)=\mathrm{D}_{\theta,\epsilon}\big|_{\theta,\epsilon=0}\mathcal{F}_2(0,\theta,\epsilon)\) .

Let \( J\) be the projection onto the orthogonal complement of \( x_1\) (with respect to the usual \( L^2\) scalar product). Using that only \( \mu\) depends on \( S\) , we may write

Most of the first summand disappears since we modded out \( x_1\) . Indeed, by Proposition 6 we know that

where the error term has a derivative \( \lesssim 1\) . On the other hand, by the same Proposition, we know that

The product of the main terms here belongs to \( \mathrm{span}\{x_1\}\) and disappears thus under the projection operator \( J\) . In particular, we have

By (167) it holds that

hence, using the bound (202) on \( S_{\theta,\epsilon}\) we see that

and thus, in particular,

for small \( (\theta,\epsilon)\in \mathcal{M}\) . The convergence of the last term is a consequence of (200).

In the case without surface tension, one has the same bound without the factor \( \frac{1}{\sigma\epsilon}\) , which still converges to \( 0\) .

In either case, the observation implies that

Next, we need to argue that \( \mathcal{F}_2(S_{\theta,\epsilon})\) is still continuously Fréchet differentiable and, in fact, has the same derivative as \( \mathcal{F}_2(0)\) at \( 0\) . As all involved terms are continuously differentiable away from \( 0\) , and \( \mathcal{F}_2(0,\theta,\epsilon)\) is continuously Fréchet differentiable by Propositions 1, 3 and 6, it suffices to show that \( \mathrm{D}_{\theta,\epsilon}(\mathcal{F}_2(S_{\theta,\epsilon})-\mathcal{F}_2(0))\rightarrow 0\) in \( H^{k-1}\) as \( (\theta,\epsilon)\to (0,0)\) .

To account for the terms that disappear due to the projection, we rewrite

where the additional summand \( \epsilon S x_1\) in the middle is annihilated by \( J\) .

Using the product rule, we then compute the derivative and estimate

where \( \mathrm{I}\) - \( \mathrm{IV}\) stand for the terms in each line. (In the case without surface tension, the term \( \mathrm{I}\) disappears, and the prefactor \( \frac{1}{\sigma\epsilon}\) needs to be replaced with \( 1\) .) Derivatives with respect to \( \theta\) are estimated analogously. We omit the details.

Using the assumption in (51) and our last estimate (214), we know that

as \( (\theta,\epsilon)\to (0,0)\) in \( \mathcal{M}\) .

To estimate \( \mathrm{II}\) , we use the product rule and the fact that \( H^{k-1}\) is closed under products to estimate

Invoking (166), the first summand is controlled by \( \frac{1}{\epsilon \sigma_\epsilon}|S_{\theta,\epsilon}||\log\epsilon|(\epsilon^{2}+\left\lVert \theta \right\rVert_{H^k}^2)\) , which is vanishing as a consequence of the bound (202) on \( S_{\theta,\epsilon}\) . For bounding the second summand, we use that \( \mu\) is affine in \( S\) and write

Applying the bounds (202) and (204) on \( S_{\theta,\epsilon}\) and its derivative, and the estimates (166) and (168), we find that

and thus, also the second summand vanishes as \( (\theta,\epsilon)\to (0,0)\) in \( \mathcal{M}\) .

We proceed analogously with the third term. Using the product rule and (202) and (204), we find

The right-hand side is asymptotically vanishing as a consequence of (164) and the asymptotics \( \mu(0) = \tilde \mu +\mathcal{E}(0) = \frac1{2\pi} +o(1)\) .

Finally, to bound the term \( \mathrm{IV}\) , we use again that \( \mu(S)\) is affine and estimate

Making use of (202), (204), (166) and (168), we find

which is vanishing as \( \epsilon \to 0\) by the virtue of (200).

The case \( \sigma=0\) proceeds the same way, the term \( \mathrm{I}\) drops out, and for the other terms, one can make the exact same calculations with the bounds (207) and (208) on \( S^{\sigma=0}\) .

It remains to apply the implicit function theorem to the functional \( (\theta,\epsilon)\mapsto \mathcal{F}_2(S_{\theta,\epsilon},\theta,\epsilon)\) defined in a small neighborhood of the origin \( (0,0)\) in \( \mathcal{M}\) . It takes values in the space

(resp. \( k-1\) , if \( \sigma=0\) ), which is isomorphic to \( T_0V^{k-2}\) (resp. \( T_0V^{k-1}\) ) by Proposition 2 via the canonical identification. To apply the implicit function theorem, we only need to show that the derivative (with respect to \( \theta\) ) of \( \mathcal{F}_2(S_{\theta,\epsilon},\theta,\epsilon)\) is invertible from \( T_0V^k\) to \( \mathcal{N}\) . By the previous step, we only need to consider the derivative of \( \mathcal{F}_2(0,\theta,\epsilon)\) at \( (0,0)\) , which in fact equals the derivative of \( \mathcal{F}(0,\theta,\epsilon)\) at \( (0,0)\) , projected onto the orthogonal complement of \( x_1\) .

By the product rule and the Propositions 1, 3, 5, and 6 this derivative, tested against \( \delta\theta\) , equals

where \( \omega\) was introduced in (185).

Using the Remarks 1 and 2, we may express this derivative as the Fourier multiplier

Because our tangent vectors are symmetric, \( \widehat{\delta \theta}(l)=\widehat{\delta \theta}(-l)\) , see Proposition 2, it is enough to consider nonnegative wave numbers \( l\) . Furthermore, because they are mean-zero functions and orthogonal on \( x_1=\cos(\alpha)\) , it holds \( \widehat{\delta\theta}(0)=\widehat{\delta\theta}(1)=0\) , and we may hence restrict to \( l\ge 2\) .

The derivative is thus an invertible map from \( T_0V^{k}\) to \( \mathcal{N}\) if

for all \( l\in\mathbb{N}_{\geq 2}\) .

We remark that if \( \omega=0\) , this is always the case. In general, invertibility is thus ensured precisely if

In the case without surface tension, one gets instead the Fourier multiplier

which is always invertible because its modulus is bounded below by \( O(l)\) .

In either case, we obtain existence from the implicit function theorem.

Next, we compute the asymptotics. First, note that (a posteriori!) we have that \( \theta=\theta_{\epsilon}\) is differentiable with respect to \( \epsilon\) and, in particular, \( \left\lVert \theta_{\epsilon} \right\rVert_{H^k}\lesssim \epsilon\) .

By the definition of \( W\) in (156), and the formulas for \( S\) in (201) and (207) we have that

both for the case with and without surface tension.

Similarly, for the flux constant, we obtain via Lemma 16 that

Regarding the unknown constant in the equation \( \mathcal{F}_2(S_{\theta_{\epsilon},\epsilon},\theta_{\epsilon},\epsilon) = \text{const}\) , which we denote by \( \nu_{\epsilon}\) in what follows, we have that

by the virtue of (214). Averaging over \( \partial B\) , we may write

Making now use of the asymptotics implied by Propositions 1, 3, 5, and 6 and invoking orthogonality properties, we deduce that

The error is differentiable and vanishing at \( \epsilon=0\) .

To compute the derivative of \( \theta\) with respect to \( \epsilon\) , we first note

by our analysis in the previous subsection. In view of the before-mentioned propositions, the first term on the right-hand side is vanishing. Thanks to the invertibility of \( \mathrm{D}_{\theta}\mathcal{F}(0,0,0)\) , it follows that \( \partial_{\epsilon} \theta_{\epsilon}\) is zero at \( \epsilon=0\) , and thus

by Taylor’s theorem.

For a given \( S_{\epsilon,\theta}\) and \( \epsilon\) such that the bounds (202), (204) and (206) (resp.  (207) and (208) hold, the solution \( \theta_{\epsilon}\in V^k\) is unique if \( \epsilon\) is small enough by the implicit function theorem, provided that the extra condition \( \left\lVert \theta \right\rVert_{H^k}\leq\epsilon^\ell\) holds.

We have shown in the first step that \( S_{\epsilon,\theta}\) is uniquely determined through \( \mathcal{F}_1(\theta,\epsilon,S_{\epsilon,\theta})=0\) for small \( \epsilon\) , as long as \( S_{\theta,\epsilon} |\log \epsilon|\left(\epsilon^2 +\|\theta\|_{H^k}^2\right)=o(1)\) holds due to the bounds on the discarded second solution.

Now suppose that there is some other solution \( \tilde \theta,\tilde \varphi_{\text{in}},\tilde \varphi_{\text{out}}, \tilde W,\tilde \gamma\) fulfilling (39)-(48) that is not the one we have constructed. Then \( \tilde \varphi_{\text{in}}\) is uniquely determined by \( \tilde \theta\) as solutions to the Laplace equation are unique. Moreover, \( \tilde \varphi_{\text{out}}\) gives rise to some \( \tilde \mu\) , and by applying the fundamental solution \( \mathcal{K}_{\tilde \theta,\epsilon}\) as in Subsection 5.1, this \( \tilde \mu\) also uniquely determines \( \tilde \varphi_{\text{out}}\) . By Lemma 17, this \( \tilde \mu\) must be of the form \( \mu(\tilde S)\) for some \( \tilde S\) fulfilling \( |\tilde S|\lesssim |\tilde W| + |\log\epsilon|\) and thus

by our assumption on \( \tilde W\) in (55).

Since it must hold that \( \mathcal{F}_1(\tilde S,\tilde\theta,\epsilon)=0\) , we have \( \tilde S=S(\tilde\theta,\epsilon)\) by the first step, where \( S(\cdot,\cdot)\) is the same function for all solutions \( (\tilde\theta,\tilde \varphi_{in},\tilde \varphi_{out},\tilde W,\tilde \gamma)\) .

But this implies by the implicit function theorem that \( S'\) and \( \tilde\theta\) must be the solution we constructed, contradicting non-uniqueness.

The smoothness of \( \theta\) follows from the fact that \( k\geq 5\) was arbitrary, and hence \( \Omega_\theta\) is smooth. The smoothness of \( \varphi_{\text{in}}\) and \( \varphi_{\text{out}}\) follows from classical elliptic regularity, see e.g. [57].\(\square\)

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