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On the Local Isometric Embedding in R^3 of Surfaces with Gaussian Curvature of Mixed Sign

We study the old problem of isometrically embedding a 2-dimensional Riemannian manifold into Euclidean 3-space. It is shown that if the Gaussian curvature vanishes to finite order and its zero set consists of two Lipschitz curves intersecting transversely at a point, then local sufficiently smooth isometric embeddings exist.


Introduction
Does every smooth 2-dimensional Riemannian manifold admit a smooth local isometric embedding into R 3 , or heuristically, can every abstract surface be visualized at least locally? This natural question was first posed in 1873 by Schlaefli [16], and remarkably has remained to a large extent unanswered. It is the purpose of this paper to provide a general sufficient condition under which local embeddings exist.
The local isometric embedding problem for surfaces is equivalent to finding local solutions of a particular Monge-Ampère equation, usually referred to as the Darboux equation. The primary difficulty in analyzing this equation arises from the fact that it changes from elliptic to hyperbolic type, whenever the Gaussian curvature of the given metric passes from positive to negative curvature. Consequently the hypotheses of any result must take into account the manner in which the Gaussian curvature, K , vanishes. The classical results deal with the cases in which the curvature does not vanish, or the metric is analytic. It was not until 1985/86 that the first degenerate cases (when K vanishes) were treated, by Lin. He showed the existence of sufficiently smooth embeddings if the metric is sufficiently smooth and K ≥ 0 [11], or K(0) = 0, |∇K(0)| = 0 [12]. Smooth embeddings of smooth surfaces were obtained by Han, Hong, and Lin [5] when K ≤ 0 and ∇K possesses a certain nondegeneracy, and by Han [3] when K vanishes across a single smooth curve (see also [1], [2], and [8] for related results). Lastly if K = |∇K(0)| = 0, |∇ 2 K(0)| = 0 then Khuri [9] has proven the existence of sufficiently smooth embeddings for sufficiently smooth surfaces. For more details on this problem and other related topics the reader is referred to [4]. Here we will show Theorem 1.1. Let g ∈ C m * , m * ≥ 36(N + 10), be a Riemannian metric defined on a neighborhood of the origin in the plane, with Gaussian curvature K vanishing there to finite order N . If the zero set K −1 (0) consists of two C m * curves intersecting transversely at the origin, then g admits a C m , m ≤ 1 12 m * − N − 24, isometric embedding into R 3 on some neighborhood of the origin. Remark 1.1. Our methods actually treat a slightly more general situation in that the zero set K −1 (0) may consist of more than two curves intersecting transversely at the origin. However, in this setting there should not be more than two regions on which K ≥ 0.
The embeddings produced by this theorem are referred to as sufficiently smooth, since higher regularity of the metric implies higher regularity for the embedding. However, this theorem does not guarantee that C ∞ metrics give rise to C ∞ embeddings, as the methods used here require the domain of existence to shrink whenever higher regularity is demanded of the solution. On the other hand, it is likely that techniques similar to those found in [3] and [6] may lead to a C ∞ version of Theorem 1.1. We also point out that counterexamples to the existence of local isometric embeddings have been found for metrics of low regularity, by Pogorelov [15] when g ∈ C 2,1 , and by Nadirashvili and Yuan [13] who recently generalized this result. Moreover counterexamples to the local solvability of smooth Monge-Ampère equations have been found in [10]. Yet it is still very much an open question, whether or not there are any smooth (or sufficiently smooth) counterexamples to the local isometric embedding problem.
As mentioned above this problem is equivalent to finding local solutions of a particular Monge-Ampère equation. To see this we employ a standard method originally introduced by Weingarten [20]. That is, we search for a function z(u, v) defined in a neighborhood of the origin such that the new metric g − dz 2 is flat. Note that g − dz 2 will be Riemannian as long as |∇ g z| < 1. Since flat metrics are locally isometric to Euclidean space, there exist two C m functions x(u, v), y(u, v) (if g ∈ C m and z ∈ C m+1 , see [7]) such that g − dz 2 = dx 2 + dy 2 . The map (u, v) → (x(u, v), y(u, v), z(u, v)) then provides the desired embedding. Furthermore a straightforward calculation shows that g − dz 2 is flat if and only if z satisfies det Hess g z = K(det g)(1 − |∇ g z| 2 ). (1.1) This Monge-Ampère equation is the so called Darboux equation, and the question of its local solvability is equivalent to the local isometric embedding problem. It is trivial to construct approximate local solutions of (1.1), and thus it is natural to employ an implicit function theorem to prove existence. Under the hypotheses of Theorem 1.1 the linearization will be of mixed type, and thus we will necessarily lose derivatives when it is inverted. This suggests that we use a version of the Nash-Moser implicit function theorem, which essentially reduces the problem to a study of the linearized equation. We will show that the linearization has a particularly nice canonical form, when an appropriate coordinate system is chosen and certain perturbation terms which behave like quadratic error in the Nash-Moser iteration are removed. This was first observed by Han in [3]. More precisely, the canonical form is given by: Lu = (aKu x ) x + bu yy + cKu x + du y (1.2) where a, b > 0. The significance of this particular structure is that it explicitly illustrates how the Gaussian curvature affects the type of the linearization. Moreover it is also important that the first order coefficient cK vanishes whenever the principal symbol changes type, as this leads to the so called Levi conditions [14] in the hyperbolic regions, which facilitate the making of estimates. Under the assumptions of Theorem 1.1 on the Gauss curvature, there are four separate regions of elliptic or hyperbolic type for (1.2), each having a Lipschitz smooth boundary. We will develop the appropriate existence and regularity theory for (1.2) in each of these regions, and show that combined with a Nash-Moser iteration this leads to a corresponding solution of the nonlinear equation (1.1) in each region. These separate solutions will then be patched together to form a solution on a full neighborhood of the origin. This paper is organized as follows. In section §2 we obtain the canonical form (1.2), and in sections §3 and §4 the linear existence theory is established in the elliptic and hyperbolic regions, respectively. Lastly in section §5 we employ a version of the Nash-Moser iteration to solve (1.1) in the elliptic and hyperbolic regions separately, and also show how the solutions obtained can be patched together to yield the desired solution.

The Linearized Canonical Form
In this section we will bring the linearization of (1.1) into the canonical form (1.2). Before doing this, however, we must specify at which function the linearization will be evaluated. For this we need an appropriate approximate solution, z 0 . We will then search for a solution of (1.1) in the form where ε > 0 is a small parameter. Let y = (y 1 , y 2 ) be local coordinates in a neighborhood of the origin with g = g ij dy i dy j , then we are interested in solving where |g| = det g ij and ∇ ij are covariant derivatives with respect to these coordinates. We choose where each p n is a homogeneous polynomial of degree n, chosen so that Here α = (α 1 , α 2 ) is a multi-index, and m * is as in Theorem 1.1. Note that such a polynomial z 0 may be found in the usual way by following the proof of the Cauchy-Kowalevski Theorem, since the line y 2 = 0 is noncharacteristic for (2.1) as ∇ 11 z 0 (0) = 1. Upon rescaling coordinates by y i = ε 2 x i , the linearization of Φ at a function w , is given by where u ;ij denote covariant derivatives in x i coordinates, ·, · is the inner product associated with g , and is the cofactor matrix of Hess g z . Note that the quantity |g| −1 a ij transforms like a contravariant 2-tensor. According to the assumptions of Theorem 1.1, The following lemma gives the desired canonical form. We will denote the Sobolev space of square integrable derivatives by H m , with norm · H m .
, and given small σ, δ > 0, there exists a local C m * −2 change of coordinates ξ i = ξ i (x) such that , for some Lipschitz function h(ξ i ) (not necessarily the same for different regions) satisfying h(0) = 0 and |h(ξ i ) − |ξ i || C 1 = O(σ). In this new coordinate system the linearization takes the form is chosen sufficiently small. Moreover there exists a constant C m independent of ε, δ such that Remark 2.1. In the estimate (2.4), δ is only relevant for the regions {Ω + κ } κ 0 κ=1 . Furthermore, since the curvature K vanishes at least to second order, it may be possible to eliminate the role of δ in the arguments of the next section.
Proof. We may choose an initial coordinate system x = (x 1 , x 2 ) so that each of the elliptic and hyperbolic regions Ω + κ , Ω − ̺ are sector domains, that is, each occupies the region between two lines passing through the origin. Furthermore we may assume that Ω − 1 (Ω − 2 ) contains the positive (negative) x 2 -axis. Note that according to the hypotheses of Theorem 1.1, ∂Ω + κ −{(0, 0)} and ∂Ω − ̺ −{(0, 0)} are both C m * smooth, so that this initial transformation is also C m * . The approximate solution z 0 is chosen with respect to this initial coordinate system (recall that y i = ε 2 x i ), and therefore It is now an easy exercise in linear algebra to show that for each domain Ω + κ , or Ω − ̺ , ̺ = 1, 2, or Ω − ̺ , ̺ = 3, 4, there exists a linear change of coordinates x = (x 1 , x 2 ) such that Here y i = ε 2 x i . It follows that a 22 > 0 and a 12 = O(ε 2 ) are preserved under this linear change of coordinates. For convenience we will still denote y i by y i and x i by We may write (2.3) as with z l = g li z y i , and Γ l ij are Christoffel symbols in y i coordinates. According to (2.1) We then set that is a ij 1 = a ij 2 and a i 1 = a i 2 except for We now define the desired change of coordinates by In order to obtain the correct expression in these new coordinates for the domains Ω + κ , or Ω − ̺ , ̺ = 1, 2, or Ω − ̺ , ̺ = 3, 4, we impose the initial conditions respectively. Note that since the curves x 1 → (x 1 , x 1 ) and x 1 → (x 1 , 0) are noncharacteristic for (2.7), equation (2.7) with initial condition (2.8) has a unique C m * −2 solution on some neighborhood of the origin. Furthermore, standard methods for first order equations combined with the Gagliardo-Nirenberg inequalities (Lemma 5.2 below) yields (2.4). Note also that (ξ 1 , ξ 2 ) forms a new coordinate system near the origin since according to the respective initial conditions given by (2.8).
We shall now calculate the coefficients of L 4 (w). Observe that (2.6) yields Calculating the second term on the right-hand side yields, x 1 a 12 + a 11 x 1 a 22 + a 22 a 12 x 2 − a 12 a 22 Therefore (2.5), (2.7), and (2.9) imply that A computation shows However, we see that the coefficient of z y i is in fact a curvature term. More precisely, if we denote it by χ i then where R i jkl is the Riemann tensor for g in y i coordinates (recall that Γ i lj are Christoffel symbols in y i coordinates). A similar calculation shows that Therefore after solving for ξ 1 x 2 in (2.7), (2.10) becomes that we have Lastly, it is trivial to calculate the remaining coefficients: Then by defining and recalling that Γ j j1 = 1 2 ∂ y 1 log |g|, we obtain the desired result. Q.E.D.

Linear Theory in the Elliptic Regions
In light of Lemma 2.1, it will be sufficient for our purposes to study the question of existence and regularity for the operator L(w), instead of the pure linearization L(w). More precisely, in this section we will study the Dirichlet problem for a modified version of L(w) in an elliptic region. First note that by using polar coordinates ξ 1 = r cos θ , ξ 2 = r sin θ , we can transform the elliptic region Ω + κ ∩ B σ (0) of Lemma 2.1 into a rectangle Ω = {(r, θ) | 0 < r < σ, 0 < θ < δ}.
Under these coordinates we find that L(w)u = Ku rr + Au rθ + Bu θθ + Cu r + Du θ where K = k cos 2 θ + sin 2 θ, It will be convenient to cut-off these coefficients away from the origin. So let ϕ ∈ C ∞ ([0, ∞)) be a nonnegative cut-off function with We will study the boundary value problem: for some large integer s 0 . The motivation for considering this problem stems from our method for constructing solutions to the nonlinear problem (2.1) (see section §5). Namely, we shall construct solutions in the elliptic and hyperbolic regions separately, and then show that they can be patched together. This requires certain compatibility conditions at the origin, and the boundary conditions of (3.1) guarantee that they will be satisfied. Of course, a necessary condition for solving (3.1) is that f must also vanish to a corresponding high order at the origin. It is therefore convenient to introduce the following weighted Sobolev spaces which control the amount of vanishing. Define norms where λ, γ > 0 are large parameters, and let H (m,l,γ) (Ω) be the closure of C ∞ (Ω) with respect to this norm, where C ∞ (Ω) is the space of smooth functions which vanish in a neighborhood of r = 0. We will always denote the traditional Sobolev spaces having square integrable derivatives up to and including order m by H m (Ω), with norm · m . The following simple lemma describes the boundary behavior exhibited by elements of the weighted spaces.
In analogy with the theory of strictly elliptic equations in a sector domain such as Ω, the regularity of a solution to (3.1) will depend on the size of the angle forming the domain. More precisely, smaller angles yield higher regularity. According to Lemma 2.1 we are free to choose the angle δ arbitrarily small, with the only price being paid with the blow-up of estimate (2.4). This blow-up, however, can be controlled in the context of equation (3.1) by taking ε = ε(δ) to be sufficiently small, since whenever ξ(x) or its derivatives appear in the coefficients of the operator L, they are always multiplied by ε. These considerations lead to existence and regularity for (3.1), and the first step needed to establish such a result is the following basic estimate. Define a λ,γ (r, θ) = λθ 2 − 1 r γ , and as in Lemma 2.1 let w ∈ C ∞ throughout this section.
Proof. Let 0 < r 0 < σ and set Ω r 0 = Ω ∩ {(r, θ) | r 0 < r < σ}. Then for any a ∈ C ∞ (Ω r 0 ), integration by parts yields where (ν 1 , ν 2 ) denotes the unit outer normal to ∂Ω r 0 . By choosing a = a λ,γ−2 and observing that the desired result follows since all boundary terms vanish according to Lemma 3.1 (after letting r 0 → 0). Note that ε is chosen small depending on δ , in order to control the blow-up (implied by the estimate (2.4)) found in the coefficients of L. Q.E.D.
This lemma is the main tool used to establish the basic existence result of the next where (·, ·) is the L 2 (Ω) inner product and L * is the formal adjoint of L.
Here r 0 > 0 is assumed to be sufficiently small, and C > 0 is a constant depending on m, λ, γ , and v . The proof that such a solution exists may be found in appendix A, section §6. Our first goal is to establish an estimate of the form The boundary conditions of (3.3) allow us to integrate by parts in a manner similar to the proof of Lemma 3.2 to find, Note that since all the coefficients of L vanish at r = σ , except B , no boundary terms at r = σ appear in the above formula. Furthermore since if δ = δ(λ) and ε = ε(δ) are sufficiently small. Next we calculate and observe that integrating by parts, again with the help of the boundary conditions in (3.3), produces Also observe that Then the combination of (3.5), (3.6), (3.7), and (3.8) produces (3.4) for λ = λ(m) sufficiently large and δ = δ(λ), ε = ε(δ) sufficiently small. We need one last estimate before proving existence. Since r 4 η ∈ H (m,0,γ+2) (Ω) whenever η ∈ H (m,1,γ) (Ω), we have Here (·, ·) (m,0,γ+2) denotes the inner product on H (m,0,γ+2) (Ω), and the norm · (−m,−1,γ) comes from the dual space H (−m,−1,γ) (Ω) of H (m,1,γ) (Ω), which may be obtained as the completion of L 2 (Ω) in this negative norm.

Now apply (3.4) to obtain
which when combined with (3.9) yields Consider the linear functional F : X → R where X = L * (r 4 C ∞ (Ω)), given by Note that we use f ∈ H (m,1,γ) (Ω) here (ie. f has to have this particular boundary behavior at r = 0). Thus we can apply the Hahn-Banach theorem to get a bounded extension of F (still denoted F ) defined on all of H (−m,−1,γ) (Ω). It follows that there exists a unique u ∈ H (m,1,γ) (Ω) such that Since every v ∈ C ∞ (Ω) can be written as v = r 4 v for some v ∈ C ∞ (Ω), u is a weak solution of (3.1). Q.E.D.
, then the strict ellipticity of L on the interior of Ω shows that the solution given by Theorem 3.1 satisfies u ∈ H (m,1,γ) (Ω) ∩ C m * −3 (Ω), since the coefficients of L will be in C m * −4 when g ∈ C m * . In particular Lu = f pointwise on the interior of Ω, so that Let H m γ (Ω) := H (m,m,γ) (Ω) with norm · (m,γ) . Then if f ∈ H m γ (Ω) we may continue to differentiate the expression for u θθ to eventually obtain (by induction) that u ∈ H m γ (Ω). In order to determine the boundary values for u, integrate by parts in expression (3.2) to obtain for all v ∈ C ∞ (Ω). This implies that u(r, 0) = u(r, δ) = 0, as B > 0. It also shows that we cannot arbitrarily prescribe boundary values for u at r = σ , since all the coefficients of L (except B ) vanish at r = σ . However it is clear that the boundary values at r = σ are given explicitly in terms of f (σ, θ) according to (3.11); although we will not have need of this fact. Moreover, the boundary behavior at r = 0 is completely determined by the fact that u ∈ H m γ (Ω), so that if m ≥ s 0 + 1 and γ > 2m then u will vanish to the desired order s 0 at r = 0 by Lemma 3.1. We summarize all that we have found with the following theorem, and also give an a priori estimate needed for the Nash-Moser iteration of section §5.
. Furthermore, there exists a constant C m independent of δ and ε such that Proof. The first half of this theorem follows from the discussion directly above, and thus it only remains to establish the estimate. From Lemma 3.2 we have that Solving for u θθ as in (3.11) then yields Applying the proof of Lemma 3.2 to the above equation gives from which we find that if λ is sufficiently large. Eventually, with the help of |w| C 6 < 1 and (3.11), we obtain By repeatedly differentiating with respect to r , we can continue this procedure and apply the Gagliardo-Nirenberg inequalities (Lemma 5.2 below) in the usual way to obtain To see how this works, we will show the calculation for one representative term that appears on the right-hand side of the equation after differentiating; the remaining terms may be treated similarly. Differentiate (3.1) s-times to obtain Since ∂ s r u ∈ H (2,1,γ−2s) (Ω) when s ≤ m * − 6, the basic estimate yields Furthermore observe that for some constants C t 1 and C t 2 . Therefore we may apply Lemma 5.2 (i), the Sobolev embedding theorem, and |w| C 6 < 1 to find where | · | ∞ denotes the L ∞ (Ω) norm. Note that the first term of the last line above, may be absorbed into the left-hand side of (3.12) for large λ(s).
The remaining derivatives of u involving higher orders of ∂ θ may be estimated by differentiating (3.11) and using the above estimates. Since the coefficients of L depend on the derivatives of w up to and including order 3 and the derivatives of ξ (the coordinates of Lemma 2.1) up to and including order 2, with the help of (2.4) we obtain .
Lastly since f vanishes to all orders at r = 0, a little calculation shows that from which the desired result follows. Q.E.D.

Linear Theory in the Hyperbolic Regions
Here we shall study the question of existence and regularity for the Cauchy problem associated with the operator L(w) of Lemma 2.1, in the hyperbolic regions. In the previous section concerning the elliptic regions, after cutting-off some of the coefficients away from the origin, we were able to invert L(w) in the appropriate function spaces. However in the hyperbolic case, we will not necessarily be able to make such an inversion, and as a result we must consider a regularized version of L(w) as we explain below. For convenience let (x, y) denote the coordinates (ξ 1 , ξ 2 ) of Lemma 2.1, so that a portion of the given hyperbolic region Ω − ̺ ∩ B σ (0), ̺ = 1, 2, may be written as . Then in these coordinates L(w) is given by in the notation of Lemma 2.1. Consider the Cauchy problem for some given data φ, ψ and a large integer α 0 , where ∂Ω 1 denotes the "bottom" portion of the boundary given by y = h(x). The "top" portion of the boundary will be denoted ∂Ω 2 , and is given by y = σ . As in problem (3.1) the solution is required to vanish to high order at the origin, in order to satisfy certain compatibility conditions which arise when constructing solutions to the nonlinear problem (2.1) (see section §5). Of course a necessary condition to have such behavior is that φ, ψ , and f must also vanish to a corresponding high order at the origin. Equation (4.1) is degenerate hyperbolic, and as such, solvability of the Cauchy problem depends on the so called Levi conditions. These are relations between the coefficients K and C , which if satisfies would guarantee existence for the Cauchy problem (when y = h(x) is smooth and noncharacteristic). A simple example of such a relation is the condition (see [14]) for some constant M > 0. Unfortunately the quantity ∂ x 1 Φ(w) present in C prevents the validity of this inequality. However in the Nash iteration of the next section, ∂ x 1 Φ(w) will be uniformly small. Therefore it is natural to consider the regularized Cauchy problem There then exists a constant M > 0 such that However we cannot simply apply the results of [14] to obtain the desired solution of (4.2), since the Cauchy surface y = h(x) is not smooth. We will therefore prove existence by hand in what follows. Note that with regards to the Nash iteration θ is of quadratic error (see section §5), so that the small perturbation (4.2) will not affect convergence of this procedure. It will be convenient to first establish an existence result for (4.2) with homogeneous Cauchy data and with f vanishing to high order on all of ∂Ω 1 . To this end we define H (m,l) (Ω) (H (m,l) 0 (Ω)) to be the closure of all C ∞ (Ω) functions (which vanish to all orders at ∂Ω 1 ) in the norm where λ > 0 is a large parameter. The L 2 (Ω) inner product will as usual be denoted by (·, ·), and the formal adjoint of L θ by L * θ . Also as in Lemma 2.1 w ∈ C ∞ throughout this section. (Ω) of (4.2) with φ, ψ = 0, for each θ > 0. That is θ (x, y)e −λy and let ζ be the unique solution of where v is as stated in the theorem. Note that for each y ∈ (0, σ), this equation may be interpreted as an ODE in ζ y , and therefore the theory of such equations guarantees the existence of a unique solution to (4.5). Furthermore if the metric g ∈ C m * as in Lemma 2.1, then the coefficients of (4.5) are in C m * −m−4 . Thus as long as m ≤ m * − 6, we have ∂ s x ζ ∈ C 2 (Ω) for 0 ≤ s ≤ 2m − 1. We first note that the solution ζ of (4.5) satisfies extra boundary conditions, namely ∂ s To see this let ∂ T ζ be differentiation along the right portion of ∂Ω 1 (that is, the curve y = h(x), x > 0), which we denote by ∂Ω + 1 . Then since ζ| ∂Ω + 1 = ζ y | ∂Ω + 1 = 0 we have where h ′ = dh/dx. It follows that this shows (4.6) for m = 2, and continuing this procedure establishes the full result on ∂Ω + 1 . Also the same arguments hold to yield (4.6) on ∂Ω − 1 , the left portion of ∂Ω 1 (that is, the curve y = h(x), x < 0).
We will now establish the basic estimate on which the existence proof is based. More precisely we will show that for some constant C > 0. Observe that the following calculations hold for 0 ≤ s ≤ m according to the boundary conditions (4.6): where (ν 1 , ν 2 ) denotes the unit outer normal to ∂Ω. Next observe that if λ > 0 is sufficiently large then and in light of the Levi condition (4.3) as well as D = O(ε) we have as λK y + O(K 2 x ) ≤ 0 near ∂Ω 1 for large λ. It follows that for large λ depending on m, where we have used a Poincaré type inequality to estimate ζ L 2 (Ω) . Note that the boundary integral has the correct sign on ∂Ω 2 . We claim that it also has the correct sign on ∂Ω 1 . To see this we use the boundary condition (4.6). That is, by (4.6) It follows that the boundary integral on ∂Ω 1 is nonnegative, and hence (4.7) holds.
We will need one last estimate before proving existence. Namely To prove existence apply (4.7) to obtain where C ∞ (Ω) consists of all C ∞ (Ω) functions with v| ∂Ω 2 = v y | ∂Ω 2 = 0. Consider the linear functional F : X → R, where X = L * θ C ∞ (Ω), given by According to (4.9) we have that F is bounded on the subspace X of H . Note that the generalized Schwarz inequality (the first inequality in the above sequence) holds because f ∈ H (Ω) such that Q.E.D.
In order to obtain higher regularity for the solution given by Theorem 4.1, we will utilize the following standard lemma concerning the difference quotient u q (x, y) := u(x, y + q) − u(x, y) q .
The Sobolev space of square integrable derivatives up to and including order m will be denoted by H m (Ω) with norm · m , and the completion of C ∞ (Ω) functions which vanish to all orders at ∂Ω 1 in the norm · m shall be denoted by H m 0 (Ω). (Ω) be the weak solution given by Theorem 4.1, so that (4.4) holds. If m ≤ 1 then this corollary follows directly from Theorem 4.1, so assume that m ≥ 2. We may integrate by parts to obtain for all v ∈ C ∞ (Ω), where (ν 1 , ν 2 ) is the unit outer normal to ∂Ω. Note that since u θ , ∂ x u θ ∈ H 1 (Ω) both u θ | ∂Ω and ∂ x u θ | ∂Ω are meaningful in L 2 (∂Ω), and in particular as u θ , ∂ x u θ ∈ H 1 0 (Ω) we have u θ | ∂Ω 1 = ∂ x u θ | ∂Ω 1 = 0 in the L 2 (∂Ω 1 ) sense. Thus we may write Then since u θ , f ∈ H (0,1) (Ω) and |∇η| 2 ≤ Cη , Lemma 4.1 (i) yields √ ηu q θ ≤ C for some constant C independent of q , if |q| is sufficiently small. Now Lemma 4.1 (ii) shows that u θ ∈ H (0,1) loc (Ω), as η was arbitrary. Hence ∂ 2 y u θ ∈ L 2 loc (Ω). It follows that the equation L θ u θ = f holds in L 2 loc (Ω), and since we can solve for ∂ 2 y u θ , we can boot-strap in the usual way to obtain u θ ∈ H m loc (Ω). Next observe that the above restrictions on η may be relaxed if q > 0, that is in this case η is only required to vanish in a neighborhood of ∂Ω 2 . Then the same procedure yields ηu θ ∈ H m (Ω) for all such η . Furthermore since L θ is strictly hyperbolic, we can use the regularity theory for such operators to obtain estimates for u θ near ∂Ω 2 . It follows that u θ ∈ H m (Ω).
To show that u θ ∈ H m 0 (Ω), it is enough to observe that where the equality is interpreted in the L 2 (∂Ω 1 ) sense when s+t = m−1. This follows from the fact that f satisfies (4.10), in the following way. First note that as in the arguments used to establish (4.6), u θ | ∂Ω 1 = ∂ x u θ | ∂Ω 1 = 0 implies that ∂ y u θ | ∂Ω 1 = 0. Furthermore using the notation of those arguments we have

However from equation (4.2) we find that
The same arguments also apply to ∂Ω − 1 , so by differentiating equation (4.2) we can continue this procedure to obtain (4.10).
This corollary yields the existence of a regular solution to (4.2) when φ = ψ = 0 and f vanishes to high order on ∂Ω 1 . However, we are interested in solving (4.2) in the general case when φ, ψ , and f vanish to high order at the origin but are otherwise arbitrary. The next lemma will enable us to obtain the general case from Corollary 4.1. Here and below H m 0 (Ω) will denote the completion of C ∞ (Ω) in the norm · m , and H m 0 (∂Ω 1 ) will be defined similarly with respect to the · m,∂Ω 1 norm. Recall that C ∞ (Ω) consists of C ∞ (Ω) functions which vanish in a neighborhood of the origin.
We remark that an equivalent and perhaps more concrete way to obtain the extension is as the unique weak solution η θ ∈ H m+2 0 (Ω) of the boundary value problem: Q.E.D.
Now in order to solve (4.2) with φ ∈ H . Observe that in order for u θ to have the desired vanishing at the origin we require m ≥ α 0 + 2. Our next task shall be to estimate u θ independent of θ , in a manner suitable for application to the Nash iteration of the next section. A significant difference between the following theorem and its analogue for the elliptic regions (Theorem 3.2), is that the loss of derivatives here depends on the degree to which the Gaussian curvature vanishes at the origin.
, and |w| C 2N+4 < 1 where N ≤ m * − 2 is the largest integer such that ∂ α K(0, 0) = 0 for all |α| ≤ N . If α 0 + 2 ≤ m ≤ m * − 6 and ε = ε(m) is sufficiently small, then there exists a unique solution u θ ∈ H m 0 (Ω) of (4.2) for each θ > 0. Furthermore there exists a constant C m independent of ε and θ such that Proof. The existence of a solution u θ ∈ H m * −6 0 (Ω) for each θ > 0 follows directly from the discussion preceding the statement of this theorem. In order to make estimates it will be advantageous to have a zeroth order term for L θ . Therefore we set v θ = e − 1 2 y 2 u θ and observe that With the aim of treating the x-derivatives first, we differentiate the above equation to find We first assume that m ≤ N + 1. In this case let η θ be given by Lemma 4 Note that this implies that η θ ∈ H m+N +3 0 (Ω), ∂ α η θ | ∂Ω 1 = ∂ α v θ | ∂Ω 1 for all |α| ≤ m + N + 2, and we have the estimate As in the proof of Theorem 4.1, we set b = K −1 θ e −λy and integrate by parts: The boundary integral along ∂Ω 2 is nonnegative, and according to the choice of η θ it vanishes along ∂Ω 1 . Moreover, the same calculations as in the proof of Theorem 4.1 apply to the interior integral to yield We proceed to estimate each term on the right-hand side separately. First note that since m ≤ N + 1 and |w| C N+4 < 1, we have Next observe that since K vanishes (at most) to order N at the origin, there exists a constant C 0 > 0 such that |K| ≥ C −1 0 (y − h(x)) N +1 in Ω. Then in light of (4.11), a little calculation shows that It follows that applying (4.12) and summing from 0 to m produces if λ is sufficiently large. From this inequality we may obtain an estimate for the xderivatives of v θ (and hence for u θ ), with the help of (4.12). In addition, by solving for ∂ 2 y u θ in equation (4.2) all remaining derivatives up to and including order m may also be estimated.
We now assume that m ≥ N + 2. In order to isolate terms involving high order derivatives of w we break f (m) into two parts. Let v (s) Note that since f (Ω) and ∂ s to be valid. In this case the following estimate holds where we have used |w| C N+7 < 1 and the proof of Lemma 4.2 to estimate ∂ t y ∂ s . Therefore (4.13) applies to yield . We now proceed to estimate each term on the right-hand side of (4.17) separately. First note that since |w| C N+4 < 1, we have Next observe again that since |K| ≥ C −1 0 (y − h(x)) N +1 in Ω, (4.15) implies that Furthermore applying the Gagliardo-Nirenberg inequalities (Lemma 5.2), (4.12), (4.14), (4.16), and using |w| C 2N+4 < 1 produces It follows that we may combine (4.17)-(4.20) and utilize (4.12), (4.14), as well as (4.16) to obtain Since v θ , with the help of (4.16) the above inequality yields an estimate for the x-derivatives of u θ . The remaining derivatives of u θ may be estimated in the usual way, by solving for ∂ 2 y u θ from equation (4.2). Lastly taking ε = ε(m) sufficiently small yields the desired result. Q.E.D. However analogous existence and regularity results, requiring only slight modifications of the arguments above, hold for the Cauchy problem in the domains Ω − ̺ ∩ B σ (0), ̺ = 3, 4, when the Cauchy data are prescribed on either the "upper" or "lower" parts of these domains (that is, on one of the two differentiable components of the curve x = h(y)).

The Nash-Moser Iteration
In this section we will carry out Nash-Moser type iteration procedures to obtain solutions of (2.1) in each of the elliptic, hyperbolic, and mixed type regions separately. The solutions will then be patched together by choosing appropriate boundary values, to yield a solution on a full neighborhood of the origin. As a consequence of Lemma 2.1, we can assume (by a judicious choice of coordinates) that each elliptic region is given by , each hyperbolic region is given by or Ω − ̺ = {(ξ 1 , ξ 2 ) | h(ξ 2 ) < ξ 1 < σ}, ̺ = 3, 4. If ∂Ω + κ denotes the portion of the boundary consisting of the curves ξ 2 = 0 and ξ 2 = (tan δ)ξ 1 , ∂Ω − ̺ , ̺ = 1, 2, denotes the portion of the boundary given by ξ 2 = h(ξ 1 ), and ∂Ω − ̺ , ̺ = 3, 4, denotes either the upper or lower part of the boundary curve ξ 1 = h(ξ 2 ), then we aim to solve

2)
∂ α w − ̺ (0, 0) = 0, |α| ≤ α 0 , for each ̺ = 1, . . . , ̺ 0 , where ∂ ν denotes the outward normal derivative, α 0 is a large integer, and φ − ̺ , ψ − ̺ will be specified below. Both of the problems (5.1) and (5.2) will require slight modifications of the standard Nash-Moser procedure. This arises from the fact that instead of solving the linearized equation at each iteration, the theories developed in sections §3 and §4 require us to solve modified versions of the linearized equation. However the error incurred at each step by these modifications is of quadratic type and therefore does not prevent the procedure from converging to a solution. Below we shall carry out the proof in full detail for the hyperbolic regions, problem (5.2). Moreover, since the corresponding iteration for problem (5.1) differs only slightly from that of (5.2), we shall merely indicate the necessary changes required in this case.

Hyperbolic Regions
In the hyperbolic regions, the linearization L(w) and the operator L θ (w) that we invert in section §4 differ by perturbation terms coming from Lemma 2.1 as well as a regularizing term involving θ = |Φ(w)| C 1 . More precisely according to Lemma 2.1 where ξ i (x 1 , x 2 ) are the coordinates constructed in Lemma 2.1. Furthermore, as with all Nash-Moser iterative schemes we will need smoothing operators. Since the theory of section §4 is based on the Sobolev spaces H m 0 , the smoothing operators we employ should respect these spaces. For convenience, in the remainder of this section the hyperbolic region in question Ω − ̺ will be denoted by Ω, and the H m 0 (Ω) norm will be denoted by · m . Lemma 5.1. Given µ ≥ 1 there exists a linear smoothing operator S µ : L 2 (Ω) → H ∞ 0 (Ω) such that for all l, m ∈ Z ≥0 and u ∈ H l 0 (Ω), Furthermore there exists a linear smoothing operator S ′ µ : L 2 (Ω) → H ∞ (Ω) such that (i), (ii), and (iii) hold whenever u ∈ H l (Ω).
The next lemma contains the so called Gagliardo-Nirenberg inequalities, which will be used frequently throughout this section.
iii) Let D ⊂ R l be compact and contain the origin, and let G ∈ C ∞ (D). If u ∈ H m (Ω, D) ∩ L ∞ (Ω, D), then there exists a constant C 3 depending on m such that Proof. These estimates are standard consequences of the interpolation inequalities, and may be found in, for instance, [19]. Q.E.D. (Ω) such that Now suppose that in addition to w 0 , functions w 1 , w 2 , . . . , w n have been defined on Ω, and put v i = S i w i , 0 ≤ i ≤ n, where S i = S µ i . Then define w n+1 = w n + u n where u n is the unique solution of L θn (v n )u n = f n in Ω, u n | ∂Ω = ∂ ν u n | ∂Ω = 0, (5.5) given by Theorem 4.2, where θ n = |Φ(v n )| C 1 and f n will be specified below. Let Q n (w n , u n ) denote the quadratic error in the Taylor expansion of Φ at w n . Then according to (5.3) we have Φ(w n+1 ) = Φ(w n ) + L(w n )u n + Q n (w n , u n ) and ξ i n are the coordinates of Lemma 2.1 with respect to v n . We now define f n . In order to solve (5.5) with the theory of section §4, we require f n ∈ C ∞ (Ω). Furthermore, we need the right-hand side of (5.6) to tend to zero sufficiently fast, to make up for the error incurred at each step by solving (5.5) instead of solving the unmodified linearized equation. Therefore we set E 0 = 0, E n = n−1 i=0 e i , and define The following theorem contains the Moser estimate for solutions of (5.5), upon which the whole iteration scheme is based.
Theorem 5.1. Suppose that g ∈ C m * and N is as in Theorem 4.2. If m ≤ m * − N − 8, |v n | C 2N+4 < 1, and ε = ε(m) is sufficiently small then there exists a unique solution u n ∈ H m 0 (Ω) of (5.5) which satisfies the estimate u n m ≤ C m ( f n m+N + v n m+N +6 f n N +2 ), (5.9) for some constant C m independent of ε and θ n .
Proof. This will follow from Theorem 4.2 with φ = ψ ≡ 0. The only difference is that the Sobolev norms appearing in Theorem 4.2 are with respect to the coordinates ξ i n (x 1 , x 2 ) of Lemma 2.1. In order to obtain the current estimate from that of Theorem 4.2 we may utilize (2.4). Note that δ (of Lemma 2.1) is not chosen arbitrarily small in the hyperbolic regions, and so it does not appear in the above estimate. Q.E.D.
In what follows, we will show that the right-hand side of (5.8) tends to zero sufficiently fast to guarantee convergence of {w n } ∞ n=0 to a solution of (5.2). Let ρ be a positive number that will be chosen as large as possible, and set µ = ε − 1 2ρ , µ n = µ n . Furthermore, note that Φ(w 0 ) ∈ H m * −m 0 0 (Ω). The convergence of {w n } ∞ n=0 will follow from the following eight statements, valid for 0 ≤ m ≤ m * − N − 8 unless specified otherwise. These statements shall be proven by induction on n, for some constants C 1 , . . . , C 5 independent of n and ε, but dependent on m.
Assume that the above eight statements hold for all nonnegative integers less than or equal to n. The next four propositions will show that they also hold for n + 1. The case n = 0 will be proven shortly thereafter.
Proof. I n+1 : First note that by III n , for small ε. Therefore if m ≤ m * − N − 8 we may apply Theorem 5.1 to obtain a solution u n ∈ H m 0 (Ω) of (5.5) which satisfies the estimate (5.9). When m + 2N + 8 − ρ ≥ 1/2 this may be combined with V n , VII n , and ρ ≥ 2N + 8 to obtain for small ε. When m + 2N + 8 − ρ ≤ −1/2, the estimate v n m+N +6 ≤ C 3 ε placed in the above calculation gives the desired result.
II n+1 : Since w n+1 = n i=0 u i , we have III n+1 : By the largeness assumption on ρ we have 3N + 8 − ρ ≤ −1/2. Therefore II n+1 and V n+1 (proven below) imply that , e ′′′ n = Q n (w n , u n ). Proof. We will estimate e ′ n , e ′′ n , and e ′′′ n separately. According to (2.3) we may write Then Lemma 5.2 (i) and (iii), I n+1 , and IV n show that Note that we have used ρ ≥ 2N + 10, as well as m ≤ m * − N − 10 which allows us to apply I n+1 and IV n . We now estimate e ′′ n . By Lemma 5.2 (i) and (iii), I n+1 , V n , and VIII n , if µ is large and m + N + 4 − ρ ≥ 1/2. If m + N + 4 − ρ ≤ −1/2 then we may use the estimate v n m+2 ≤ C 3 ε to obtain the same outcome. Another application of Lemma 5.2 gives after noting that m ≤ min(m * − N − 10, m * − m 0 ) is required for VIII n to be valid, and similar methods yield Moreover, if l = ρ + 2 ≤ m * − 2 and n > 0 then we may apply Lemma 5.1 and recall that µ = ε − 1 2ρ to obtain ∇ αβ Φ(w n + tu n )∂ α u n ∂ β u n m dt )dt, where α = ∂ α (w n + tu n ) and β = ∂ β (w n + tu n ). The notation ∇ 2 Φ represents the collection of second partial derivatives with respect to the variables α and β . Furthermore it is easy to see that |∇ 2 Φ(0)| = O(ε). Therefore using Lemma 5.2 (iii), I n+1 , and II n , we have e ′′′ n m ≤ C m,16 [(ε+ w n 4 + u n 4 ) u n 4 u n m+2 +(ε+ w n m+2 + u n m+2 ) u n According to the above proposition if ρ + 1 ≤ m * − m 0 (in addition to the other required restrictions) then E n ∈ H ρ+1 (Ω) and the following estimate holds, which will be utilized in the next proposition: Proposition 5.3. If the hypotheses of Proposition 5.2 hold and ρ + 1 ≤ m * − m 0 , then VII n+1 holds for all 0 ≤ m < ∞.
Proof. By (5.7) as well as Lemma 5.2 (i) and (iii), Next observe that (2.2) together with (5.14) below yields (5.12) Then (5.10) implies that for all m ≥ ρ + 1, If m < ρ + 1, then applying similar methods along with VI n+1 to produces the same estimate found in (5.13). Therefore plugging into (5.11) produces if m+N +4−ρ ≥ 1/2. If m+N +4−ρ ≤ −1/2 and m ≥ 2, then using v n+1 m+2 ≤ C 3 ε in the estimate above gives the desired result. Moreover if 0 ≤ m < 2, then in place of (5.11) we use the estimate Applying the estimate (5.10) along with (5.12) and εµ ρ−m ≤ ε 1/2 produces if ε is sufficiently small. Lastly a similar estimate may be obtained for Φ(v n+1 ) by writing To complete the proof by induction we will now prove the case n = 0. Here we will assume that the initial data are appropriately small: (5.14) Then according to (5.4), II 0 , III 0 , IV 0 , and V 0 are trivial as long as ε is small enough. Furthermore applying (5.12) and again taking ε to be sufficiently small yields VII 0 and VIII 0 . In addition by the proof of Proposition 5.1 we obtain the following stronger version of I 1 , Since ρ ≥ 3N + 9 we must then have m * ≥ max(3N + 16, 3N + m 0 + 10).

Elliptic Regions
Here we shall set up the iteration procedure for problem (5.1). For convenience we will denote the domain Ω + κ by Ω. Set w 0 = 0 and suppose that functions w 1 , . . . , w n have been defined on Ω. If S i = S µ i are smoothing operators given by Lemma 5.1, then we put v i = S i w i , 0 ≤ i ≤ n, and define w n+1 = w n + u n where u n is the solution of L(v n )u n = f n in Ω, u n | ∂Ω = 0, (5.15) given by Theorem 5.2 below, L(v n ) is the operator of Lemma 2.1, and f n will also be specified below. Let Q n (w n , u n ) again denote the quadratic error and L(w n ) the linearization of (5.1), then according to (5.3) (with θ n = 0) we have Lastly we set E 0 = 0, E n = n−1 i=0 e i , and define f n according to (5.7). It is clear that similar arguments as those used for the hyperbolic regions will show that {w n } ∞ n=0 converges to a solution of (5.1) if a Moser estimate (like that found in Theorem 5.1) holds for the solution of (5.15). In order to establish such an estimate using the theory of section §3, we need to extend the coefficients of L(v n ) outside of Ω and cut them off. For this purpose we will use the following extension lemma. Lemma 5.3 [18]. Let X be a bounded convex domain in R 2 , with Lipschitz smooth boundary. Then there exists a linear operator E X : Theorem 5.2. Suppose that g ∈ C m * . If m ≤ 1 3 (m * − 8), |v n | C 6 < 1, and δ = δ(m), ε = ε(m, δ) are sufficiently small, then there exists a solution u n ∈ H m 0 (Ω) of (5.15) which satisfies the estimate for some constant C m independent of δ and ε and where 2m < γ < m * − m − 6.
We may now apply arguments similar to those in the hyperbolic regions to obtain a solution of (5.1). More precisely, the proofs of Propositions 5.1-5.4 yield the following restrictions on ρ, γ , and m * in the elliptic regions: ρ ≥ 2γ + 54, ρ + 1 = 1 3 (m * − 14).
Lastly we observe that since v vanishes in a neighborhood of r = 0, the solution ζ satisfies a version of the so called Euler differential equation in this domain. All solutions of this equation may be written down explicitly. In particular, for r sufficiently small ζ must be a linear combination of 2m functions of the form: r α (log r) β where α ∈ C and β ∈ Z ≥0 . However according to Lemma 3.1 r=r 0 (∂ s r ∂ l θ ζ) 2 ≤ r γ−2s 0 C ζ 2 (m,l+1,γ+2) , s ≤ m − 1, 0 ≤ l < ∞. (6.2) Therefore each term in the linear combination must satisfy r α (log r) β = O(r γ/2 ) as r → 0. (6. 3) The desired boundary behavior at r = 0 now follows from (6.2) and (6.3).

Appendix B
The purpose of this section is to construct the smoothing operators S µ of Lemma 5.1. The construction will differ from the standard one for S ′ µ (see [16]), in that the smoothed functions are required to vanish identically at the origin. This of course is only possible if the function being smoothed already vanishes in an appropriate sense at the origin.
We first construct smoothing operators on the plane, and will later restrict them back to the bounded domain Ω. Fix χ ∈ C ∞ c (R 2 ) such that χ ≡ 1 on some neighborhood of the origin, and let χ(x) = R 2 e 2πiξ·x χ(ξ)dξ be its inverse Fourier Transform. Then χ is a Schwartz function and satisfies R 2 χ(x)dx = 1, R 2 x α χ(x)dx = 0, |α| > 0.
Furthermore since supp ∂ β (1 − η µ ) ⊂ {|x| < µ −1 } we may apply the same methods used to establish (7.2) to obtain It follows that (iii) holds. The desired smoothing operators on Ω may be obtained from S µ in the following way. If Ω is a bounded convex Lipschitz domain, then Lemma 5.3 yields an extension operator E : H m (Ω) → H m (R 2 ). We then define smoothing operators S µ : L 2 (Ω) → H ∞ 0 (Ω) by S µ u = (S µ Eu)| Ω . As E is bounded, it is clear that Lemma 5.1 will also hold for S µ .