You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: '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.'
---

**On the Local Isometric Embedding in $\mathbb{R}^{3}$ of Surfaces with Gaussian Curvature of Mixed Sign**

QING HAN & MARCUS KHURI

**1. Introduction**

Does every smooth 2-dimensional Riemannian manifold admit a smooth local isometric embedding into $\mathbb{R}^{3}$, 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 $K\geq 0$ \[11\], or $K(0)=0$, $|\nabla K(0)|\neq 0$ \[12\]. Smooth embeddings of smooth surfaces were obtained by Han, Hong, and Lin \[5\] when $K\leq 0$ and $\nabla K$ 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 $K=|\nabla K(0)|=0$, $|\nabla^{2}K(0)|\neq 0$ 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 $g\in C^{m_{*}}$, $m_{*}\geq
36(N+10)$, 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 $C^{m}$, $m\leq \frac{1}{12}m_{*}-N-24$, isometric embedding into $\mathbb{R}^{3}$ 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 $K\geq 0$.*

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 $C^{\infty}$ metrics give rise to $C^{\infty}$ 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 $C^{\infty}$ 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 $g\in C^{2,1}$, 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 $|\nabla_{g}z|<1$. Since flat metrics are locally isometric to Euclidean space, there exist two $C^{m}$ functions $x(u,v)$, $y(u,v)$ (if $g\in C^{m}$ and $z\in C^{m+1}$, see \[7\]) such that $g-dz^{2}=dx^{2}+dy^{2}$. The map $(u,v)\mapsto(x(u,v),y(u,v),z(u,v))$ then provides the desired embedding. Furthermore a straightforward calculation shows that $g-dz^{2}$ is flat if and only if $z$ satisfies $$\det\mathrm{Hess}_{g}z=K(\det g)(1-|\nabla_{g}z|^{2}).$$ 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: $$Lu=(aKu_{x})_{x}+bu_{yy}+cKu_{x}+du_{y}$$ 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 $\S 2$ we obtain the canonical form (1.2), and in sections $\S 3$ and $\S 4$ the linear existence theory is established in the elliptic and hyperbolic regions, respectively. Lastly in section $\S 5$ 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.

**2. 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 $$z=z_{0}+\varepsilon^{5}w,$$ where $\varepsilon>0$ 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 $$\Phi(w):=\det\nabla_{ij}z-K|g|(1-|\nabla_{g}z|^{2})=0$$ where $|g|=\det g_{ij}$ and $\nabla_{ij}$ are covariant derivatives with respect to these coordinates. We choose $$z_{0}=\frac{1}{2}(y^{1})^{2}+\sum_{n=3}^{m_{*}}p_{n}(y)$$ where each $p_{n}$ is a homogeneous polynomial of degree $n$, chosen so that $$\partial^{\alpha}\Phi(0)=0,\text{ }\text{ }\text{ }\text{ }\text{
}|\alpha|\leq m_{*}-2.$$ Here $\alpha=(\alpha_{1},\alpha_{2})$ 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 $\nabla_{11}z_{0}(0)=1$.

Upon rescaling coordinates by $y^{i}=\varepsilon^{2} x^{i}$, the linearization of $\Phi$ at a function $w$, $$\mathcal{L}(w)u=\frac{d}{dt}\Phi(w+tu)|_{t=0},$$ is given by $$\varepsilon^{-1}\mathcal{L}(w)u
=a^{ij}u_{;ij}-2\varepsilon^{4}K|g|\langle\nabla_{g}z,\nabla_{g}u\rangle$$ where $u_{;ij}$ denote covariant derivatives in $x^{i}$ coordinates, $\langle\cdot,\cdot\rangle$ is the inner product associated with $g$, and $$(a^{ij})=\left(\begin{array}{cc}
\nabla_{22}z & -\nabla_{12}z \\
-\nabla_{12}z & \nabla_{11}z
\end{array}\right)$$ is the cofactor matrix of $\mathrm{Hess}_{g}z$. Note that the quantity $|g|^{-1}a^{ij}$ transforms like a contravariant 2-tensor. According to the assumptions of Theorem 1.1, $K^{-1}(0)$ divides a small neighborhood of the origin into domains $\{\Omega_{\kappa}^{+}\}_{\kappa=1}^{\kappa_{0}}$ on which $K>0$, and $\{\Omega_{\varrho}^{-}\}_{\varrho=1}^{\varrho_{0}}$ on which $K<0$ (obviously $\kappa_{0}+\varrho_{0}=4$). The following lemma gives the desired canonical form. We will denote the Sobolev space of square integrable derivatives by $H^{m}$, with norm $\parallel\cdot\parallel_{H^{m}}$.

**Lemma 2.1.** *Let $g\in C^{m_{*}}$ and $w\in
C^{\infty}$ with $|w|_{C^{3}}<1$. Given a domain $\Omega_{\kappa}^{+}$, or $\Omega_{\varrho}^{-}$, $\varrho=1,2$, or $\Omega_{\varrho}^{-}$, $\varrho=3,4$, and given small $\sigma,\delta>0$, there exists a local $C^{m_{*}-2}$ change of coordinates $\xi^{i}=\xi^{i}(x)$ such that* $$\Omega_{\kappa}^{+}\cap B_{\sigma}(0)=\{(\xi^{1},\xi^{2})\mid
0<\xi^{2}<(\tan\delta)\xi^{1},\text{ }|\xi|<\sigma\},$$ *or* $$\Omega_{\varrho}^{-}\cap B_{\sigma}(0)=\{(\xi^{1},\xi^{2})\mid
h(\xi^{1})<\xi^{2},\text{ }|\xi|<\sigma\},\text{ }\text{ }\text{
}\text{ }\varrho=1,2,$$ *or* $$\Omega_{\varrho}^{-}\cap B_{\sigma}(0)=\{(\xi^{1},\xi^{2})\mid
h(\xi^{2})<\xi^{1},\text{ }|\xi|<\sigma\},\text{ }\text{ }\text{
}\text{ }\varrho=3,4,$$ *for some Lipschitz function $h(\xi^{i})$ (not necessarily the same for different regions) satisfying $h(0)=0$ and $|h(\xi^{i})-|\xi^{i}||_{C^{1}}=O(\sigma)$. In this new coordinate system the linearization takes the form* $$\varepsilon^{-1}\mathcal{L}(w)u=a^{22}L(w)u+(a^{22})^{-1}
\Phi(w)[\partial_{x^{1}}^{2}u-\partial_{x^{1}}\log
(a^{22}\sqrt{|g|})\partial_{x^{1}}u],$$ *where* $$L(w)u=\partial_{\xi^{1}}(k\partial_{\xi^{1}}u)+\partial_{\xi^{2}}^{2}u+
c\partial_{\xi^{1}}u +d\partial_{\xi^{2}}u$$ *with* $$\begin{aligned}
k&=&K\overline{k}(x,w,\nabla w,\nabla^{2}w,\nabla\xi),\\
c&=& K\overline{c}(x,w,\nabla
w,\nabla^{2}w,\nabla^{3}w,\nabla\xi,\nabla^{2}\xi)
+(a^{22})^{-2}\partial_{x^{1}}\Phi(w)\partial_{x^{1}}\xi^{1},\\
d&=&\varepsilon^{2}\overline{d}(x,w,\nabla w,\nabla^{2}w),\end{aligned}$$ *for some $\overline{k},\overline{c},\overline{d}\in
C^{m_{*}-4}$ such that $\overline{k}>1/2$ if $\varepsilon=\varepsilon(m)$ is chosen sufficiently small. Moreover there exists a constant $C_{m}$ independent of $\varepsilon$, $\delta$ such that* $$\parallel\xi\parallel_{H^{m}}\leq\delta^{-1}C_{m}(1+\parallel
w\parallel_{H^{m+4}}),\text{ }\text{ }\text{ }\text{ }\text{
}m\leq m_{*}-2.$$

**Remark 2.1.** *In the estimate (2.4), $\delta$ is only relevant for the regions $\{\Omega_{\kappa}^{+}\}_{\kappa=1}^{\kappa_{0}}$. Furthermore, since the curvature $K$ vanishes at least to second order, it may be possible to eliminate the role of $\delta$ 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 $\Omega_{\kappa}^{+}$, $\Omega_{\varrho}^{-}$ are sector domains, that is, each occupies the region between two lines passing through the origin. Furthermore we may assume that $\Omega_{1}^{-}$ ($\Omega_{2}^{-}$) contains the positive (negative) $x^{2}$-axis. Note that according to the hypotheses of Theorem 1.1, $\partial\Omega_{\kappa}^{+}-\{(0,0)\}$ and $\partial\Omega_{\varrho}^{-}-\{(0,0)\}$ 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 $y^{i}=\varepsilon^{2}x^{i}$), and therefore $$a^{22}>0,\text{ }\text{ }\text{ }\text{
}a^{12}=O(\varepsilon^{2}).$$ It is now an easy exercise in linear algebra to show that for each domain $\Omega_{\kappa}^{+}$, or $\Omega_{\varrho}^{-}$, $\varrho=1,2$, or $\Omega_{\varrho}^{-}$, $\varrho=3,4$, there exists a linear change of coordinates $\overline{x}=(\overline{x}^{1},\overline{x}^{2})$ such that $$\Omega_{\kappa}^{+}=\{(\overline{x}^{1},\overline{x}^{2})\mid
0<\overline{x}^{2}<\overline{x}^{1},\text{ }|\overline{x}|<1\},$$ or $$\Omega_{\varrho}^{-}=\{(\overline{x}^{1},\overline{x}^{2})\mid
|\overline{x}^{1}|<\overline{x}^{2}<1\},\text{ }\text{ }\text{
}\text{ }\varrho=1,2,$$ or $$\Omega_{\varrho}^{-}=\{(\overline{x}^{1},\overline{x}^{2})\mid
|\overline{x}^{2}|<\overline{x}^{1}<1\},\text{ }\text{ }\text{
}\text{ }\varrho=3,4,$$ and such that $$\partial_{\overline{y}^{1}}^{2}(y^{1})^{2}>0,\text{ }\text{
}\text{ }\text{ }
\partial_{\overline{y}^{1}}\partial_{\overline{y}^{2}}(y^{1})^{2}=0.$$ Here $\overline{y}^{i}=\varepsilon^{2}\overline{x}^{i}$. It follows that $a^{22}>0$ and $a^{12}=O(\varepsilon^{2})$ are preserved under this linear change of coordinates. For convenience we will still denote $\overline{y}^{i}$ by $y^{i}$ and $\overline{x}^{i}$ by $x^{i}$.

We may write (2.3) as $$L_{1}(w)u=a_{1}^{ij}u_{x^{i}x^{j}}
+a_{1}^{i}u_{x^{i}}:=\varepsilon^{-1}\mathcal{L}(w)u,$$ where $a_{1}^{ij}=a^{ij}$, $$a_{1}^{l}=-\varepsilon^{2}(a^{ij}\Gamma_{ij}^{l}+2K|g|z^{l})$$ with $z^{l}=g^{li}z_{y^{i}}$, and $\Gamma_{ij}^{l}$ are Christoffel symbols in $y^{i}$ coordinates. According to (2.1) $$a_{1}^{11}=\nabla_{22}z=(a^{22})^{-1}[K|g|(1-|\nabla_{g}z|^{2})+(\nabla_{12}z)^{2}
+\Phi(w)].$$ We then set $$L_{2}(w)u=a_{2}^{ij}u_{x^{i}x^{j}}+a_{2}^{i}u_{x^{i}}:=
L_{1}(w)u-(a^{22})^{-1}\Phi(w)u_{x^{1}x^{1}},$$ that is $a_{1}^{ij}=a_{2}^{ij}$ and $a_{1}^{i}=a_{2}^{i}$ except for $$a_{2}^{11}=(a^{22})^{-1}[K|g|(1-|\nabla_{g}z|^{2})+(\nabla_{12}z)^{2}].$$ Also let $$L_{3}(w)u=a_{3}^{ij}u_{x^{i}x^{j}}+a_{3}^{i}u_{x^{i}}:=(a^{22})^{-1}L_{2}(w)u.$$

We now define the desired change of coordinates by $$\xi^{1}=\xi^{1}(x^{1},x^{2}),\text{ }\text{ }\text{ }\text{
}\xi^{2}=x^{2},$$ with $$a^{12}\xi^{1}_{x^{1}}+a^{22}\xi^{1}_{x^{2}}=0,$$ so that if $$L_{4}(w)u=a_{4}^{ij}u_{\xi^{i}\xi^{j}}+a_{4}^{i}u_{\xi^{i}}:=L_{3}(w)u$$ then $$a_{4}^{12}=a_{3}^{ij}\xi^{1}_{x^{i}}\xi^{2}_{x^{j}}=0.$$ In order to obtain the correct expression in these new coordinates for the domains $\Omega_{\kappa}^{+}$, or $\Omega_{\varrho}^{-}$, $\varrho=1,2$, or $\Omega_{\varrho}^{-}$, $\varrho=3,4$, we impose the initial conditions $$\xi^{1}(x^{1},x^{1})=(\tan\delta)^{-1}x^{1},\text{ }\text{ }\text{
or }\text{ }\text{ }\xi^{1}(x^{1},0)=x^{1},\text{ }\text{ }\text{
or }\text{ }\text{ }\xi^{1}(x^{1},0)=x^{1},$$ respectively. Note that since the curves $x^{1}\mapsto(x^{1},x^{1})$ and $x^{1}\mapsto(x^{1},0)$ 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 $(\xi^{1},\xi^{2})$ forms a new coordinate system near the origin since $$(\tan\delta)^{-1}=\xi^{1}_{x^{1}}(0,0)+\xi^{1}_{x^{2}}(0,0)=
\xi^{1}_{x^{1}}(0,0)\left(1-\frac{a^{12}}{a^{22}}(0,0)\right),$$ or $$\xi^{1}_{x^{1}}(0,0)=1,$$ 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 $$\begin{aligned}
a_{4}^{11}=a_{3}^{ij}\xi^{1}_{x^{i}}\xi^{1}_{x^{j}}\!\!\!
&=&\!\!\!(\nabla_{11}z)^{-2}[K|g|(1-|\nabla_{g}z|^{2})+(\nabla_{12}z)^{2}](\xi^{1}_{x^{1}})^{2}\\
& &\!\!\!
-2(\nabla_{11}z)^{-1}(\nabla_{12}z)\xi^{1}_{x^{1}}\xi^{1}_{x^{2}}+(\xi_{x^{2}}^{1})^{2},\end{aligned}$$ but (2.7) gives $$\xi^{1}_{x^{2}}=(\nabla_{11}z)^{-1}(\nabla_{12}z)\xi^{1}_{x^{1}}$$ so $$a_{4}^{11}=(a^{22})^{-2}K|g|(1-|\nabla_{g}z|^{2})(\xi^{1}_{x^{1}})^{2}.$$

Next we examine $a_{4}^{1}$. By (2.7) $$\xi^{1}_{x^{1}x^{2}}=-(a^{12}_{3})_{x^{1}}\xi^{1}_{x^{1}}-a^{12}_{3}\xi^{1}_{x^{1}x^{1}},\text{
}\text{ }\text{ }\text{
}\xi^{1}_{x^{2}x^{2}}=-(a^{12}_{3})_{x^{2}}\xi^{1}_{x^{1}}-a^{12}_{3}\xi^{1}_{x^{1}x^{2}},$$ so (2.6) produces $$\begin{aligned}
a^{1}_{4}\!\!&=&\!\!a^{ij}_{3}\xi^{1}_{x^{i}x^{j}}+
a^{i}_{3}\xi^{1}_{x^{i}}\\
&=&\!\!(a^{22})^{-2}K|g|(1-|\nabla_{g}z|^{2})\xi^{1}_{x^{1}x^{1}}
-\left[\left(\frac{a^{12}}{a^{22}}\right)\left(\frac{a^{12}}{a^{22}}\right)_{x^{1}}
+\left(\frac{a^{12}}{a^{22}}\right)_{x^{2}}\right]\xi^{1}_{x^{1}}+a^{i}_{3}
\xi^{1}_{x^{i}}.\nonumber\end{aligned}$$ Calculating the second term on the right-hand side yields, $$\begin{aligned}
(a^{22})^{2}\left[\left(\frac{a^{12}}{a^{22}}\right)\left(\frac{a^{12}}{a^{22}}\right)_{x^{1}}
+\left(\frac{a^{12}}{a^{22}}\right)_{x^{2}}\right]\!\!&=&\!\!
a^{12}a^{12}_{x^{1}}-(a^{22})^{-1}(a^{12})^{2}a^{22}_{x^{1}}+a^{22}a^{12}_{x^{2}}
-a^{12}a^{22}_{x^{2}}\\
&=&\!\!
a^{12}a^{12}_{x^{1}}-a^{11}a^{22}_{x^{1}}+a^{22}a^{12}_{x^{2}}-a^{12}a^{22}_{x^{2}}\\
& &\!\!
+(a^{22})^{-1}a^{22}_{x^{1}}(\det a^{ij})\\
&=&\!\!-a^{12}_{x^{1}}a^{12}+a^{11}_{x^{1}}a^{22}+a^{22}a^{12}_{x^{2}}
-a^{12}a^{22}_{x^{2}}\\
& &\!\!+(a^{22})^{-1}a^{22}_{x^{1}}(\det a^{ij})-(\det
a^{ij})_{x^{1}}.\end{aligned}$$ Therefore (2.5), (2.7), and (2.9) imply that $$\begin{aligned}
a^{22}a^{1}_{4}\!\!&=&\!\!-[a_{x^{j}}^{ij}+\varepsilon^{2}(a^{lj}\Gamma_{lj}^{i}
+2K|g|z^{i})]\xi^{1}_{x^{i}}+((a^{22})^{-1}\det
a^{ij})_{x^{1}}\xi^{1}_{x^{1}}\\
&
&+(a^{22})^{-1}K|g|(1-|\nabla_{g}z|^{2})\xi^{1}_{x^{1}x^{1}}.\nonumber\end{aligned}$$ A computation shows $$\begin{aligned}
&
&\!\!a^{11}_{y^{1}}+a^{12}_{y^{2}}+a^{lj}\Gamma_{lj}^{1}\nonumber\\
&=&\!\!
-\Gamma_{j2}^{j}z_{y^{1}y^{2}}+\Gamma_{j1}^{j}z_{y^{2}y^{2}}\nonumber\\
& &\!\!+(\Gamma_{12,y^{2}}^{i}
-\Gamma_{22,y^{1}}^{i}-\Gamma_{11}^{1}\Gamma_{22}^{i}+2\Gamma_{12}^{1}\Gamma_{12}^{i}
-\Gamma_{22}^{1}\Gamma_{11}^{i})z_{y^{i}}\\
&=&\!\! \Gamma_{j2}^{j}a^{12}+\Gamma_{j1}^{j}a^{11}\nonumber\\
& &\!\!+(\Gamma_{12,y^{2}}^{i}
-\Gamma_{22,y^{1}}^{i}-\Gamma_{11}^{1}\Gamma_{22}^{i}+2\Gamma_{12}^{1}\Gamma_{12}^{i}
-\Gamma_{22}^{1}\Gamma_{11}^{i}-\Gamma_{j2}^{j}\Gamma_{12}^{i}
+\Gamma_{j1}^{j}\Gamma_{22}^{i})z_{y^{i}}.\nonumber\end{aligned}$$ However, we see that the coefficient of $z_{y^{i}}$ is in fact a curvature term. More precisely, if we denote it by $\chi^{i}$ then $$\chi^{i}=\Gamma_{12,y^{2}}^{i}-\Gamma_{22,y^{1}}^{i}
+\Gamma_{12}^{j}\Gamma_{j2}^{i}-\Gamma_{22}^{j}\Gamma_{j1}^{i}
=-R^{i}_{212}=-g^{i1}|g|K$$ where $R^{i}_{jkl}$ is the Riemann tensor for $g$ in $y^{i}$ coordinates (recall that $\Gamma_{lj}^{i}$ are Christoffel symbols in $y^{i}$ coordinates). A similar calculation shows that $$a^{12}_{y^{1}}+a^{22}_{y^{2}}+a^{lj}\Gamma_{lj}^{2}
=\Gamma_{j1}^{j}a^{12}+\Gamma_{j2}^{j}a^{22}-g^{i2}|g|Kz_{y^{i}}.$$ Therefore after solving for $\xi^{1}_{x^{2}}$ in (2.7), (2.10) becomes $$\begin{aligned}
a^{22}a_{4}^{1}&=&(a^{22})^{-1}K|g|(1-|\nabla_{g}z|^{2})\xi^{1}_{x^{1}x^{1}}
-\varepsilon^{2}K|g|z^{i}\xi^{1}_{x^{i}}\\
& &-[\varepsilon^{2}\Gamma_{j1}^{j}(a^{22})^{-1}\det
a^{ij}-((a^{22})^{-1}\det a^{ij})_{x^{1}}]\xi^{1}_{x^{1}}.\end{aligned}$$ It now follows from $$\det a^{ij}=\Phi(w)+K|g|(1-|\nabla_{g}z|^{2})$$ that we have $$\begin{aligned}
a^{22}a_{4}^{1}&=&-\varepsilon^{2}K|g|z^{i}\xi^{1}_{x^{i}}
+\partial_{x^{1}}[(a^{22})^{-1}K|g|(1-|\nabla_{g}z|^{2})\xi^{1}_{x^{1}}]\\
&
&-[\varepsilon^{2}\Gamma_{j1}^{j}(a^{22})^{-1}K|g|(1-|\nabla_{g}z|^{2})
+\varepsilon^{2}\Gamma_{j1}^{j}(a^{22})^{-1}\Phi(w)
-((a^{22})^{-1}\Phi(w))_{x^{1}}]\xi^{1}_{x^{1}}.\end{aligned}$$

Lastly, it is trivial to calculate the remaining coefficients: $$a_{4}^{22}=1,\text{ }\text{ }\text{ }\text{ }a_{4}^{2}
=-\varepsilon^{2}(a^{22})^{-1}(a^{ij}\Gamma_{ij}^{2}+2K|g|z^{2}).$$ Then by defining $$L(w)u:=L_{4}(w)u+(\varepsilon^{2}\Gamma_{j1}^{j}+\partial_{x^{1}}\log
a^{22})(a^{22})^{-2}\Phi(w)\xi^{1}_{x^{1}}u_{\xi^{1}}$$ and recalling that $\Gamma_{j1}^{j}=\frac{1}{2}\partial_{y^{1}}\log|g|$, we obtain the desired result. Q.E.D.

**3. 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 $\mathcal{L}(w)$. 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 $\xi^{1}=r\cos\theta$, $\xi^{2}=r\sin\theta$, we can transform the elliptic region $\Omega_{\kappa}^{+}\cap B_{\sigma}(0)$ of Lemma 2.1 into a rectangle $$\Omega=\{(r,\theta)\mid 0<r<\sigma,\text{ }0<\theta<\delta\}.$$ Under these coordinates we find that $$L(w)u=\mathcal{K}u_{rr}+\mathcal{A}u_{r\theta}+\mathcal{B}u_{\theta\theta}
+\mathcal{C}u_{r}+\mathcal{D}u_{\theta}$$ where $$\begin{aligned}
\mathcal{K}&=&k\cos^{2}\theta+\sin^{2}\theta,\\
\mathcal{A}&=&2(1-k)\frac{\sin\theta\cos\theta}{r},\\
\mathcal{B}&=&k\frac{\sin^{2}\theta}{r^{2}}+\frac{\cos^{2}\theta}{r^{2}},\\
\mathcal{C}&=&k\frac{\sin^{2}\theta}{r}+\frac{\cos^{2}\theta}{r}+
(c+\partial_{\xi^{1}}k)\cos\theta+d\sin\theta,\\
\mathcal{D}&=&2(k-1)\frac{\sin\theta\cos\theta}{r^{2}}
-(c+\partial_{\xi^{1}}k)\frac{\sin\theta}{r}
+d\frac{\cos\theta}{r}.\end{aligned}$$ It will be convenient to cut-off these coefficients away from the origin. So let $\varphi\in C^{\infty}([0,\infty))$ be a nonnegative cut-off function with $$\varphi(r)=\begin{cases}
1 & \text{if $0<r<\frac{1}{2}\sigma$,}\\
0 & \text{if $\sigma<r$},
\end{cases}$$ and define $$Lu=\overline{K}u_{rr}+\overline{A}u_{r\theta}+\overline{B}u_{\theta\theta}
+\overline{C}u_{r}+\overline{D}u_{\theta}$$ where $$\overline{K}=\varphi^{2}\mathcal{K},\text{ }\text{ }
\overline{A}=\varphi\mathcal{A},\text{ }\text{ }
\overline{B}=\mathcal{B},\text{ }\text{
}\overline{C}=\varphi\mathcal{C},\text{ }\text{
}\overline{D}=\varphi\mathcal{D}.$$

We will study the boundary value problem: $$Lu=f\text{ }\text{ }\text{ in }\text{ }\text{ }\Omega,\text{
}\text{ }\text{ }u(r,0)=u(r,\delta)=0,\text{ }\text{ }\text{
}\partial_{r}^{s}u(0,\theta)=0,\text{ }\text{ }\text{ }0\leq s\leq
s_{0},$$ 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 $\S 5$). 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 $$\parallel u\parallel_{(m,l,\gamma)}^{2}=\int_{\Omega}\sum_{0\leq s\leq m,\text{
} 0\leq t\leq l \atop s+t\leq\mathrm{max}(m,l)}\lambda^{-s}
r^{-\gamma+2s}(\partial_{r}^{s}\partial_{\theta}^{t}u)^{2}$$ where $\lambda,\gamma>0$ are large parameters, and let $H^{(m,l,\gamma)}(\Omega)$ be the closure of $\overline{C}^{\infty}(\Omega)$ with respect to this norm, where $\overline{C}^{\infty}(\Omega)$ 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 $H^{m}(\Omega)$, with norm $\parallel\cdot\parallel_{m}$. The following simple lemma describes the boundary behavior exhibited by elements of the weighted spaces.

**Lemma 3.1.** *Suppose that $u\in
H^{(m,l,\gamma)}(\Omega)\cap C^{\max(m,l)-1}(\Omega)$, then for any $0<r_{0}<\sigma$ we have* $$\int_{r=r_{0}}(\partial_{r}^{s}\partial_{\theta}^{t}u)^{2}\leq
r_{0}^{\gamma-2(s+1)}C\parallel
u\parallel_{(m,l,\gamma)}^{2},\text{ }\text{ }s\leq m-1,\text{
}\text{ }t\leq l-1,\text{ }\text{ }s+t\leq\max(m,l)-1,$$ *where the constant $C$ depends only on $\sigma-r_{0}$.*

*Proof.* When $s\leq m-1$, $t\leq l-1$, $s+t\leq\max(m,l)-1$ we have $r^{-\gamma/2+s+1}\partial_{r}^{s}\partial_{\theta}^{t}u\in
H^{1}(\Omega)\cap C^{0}(\Omega)$. The desired result now follows from the standard trace theorem for Sobolev spaces. Q.E.D.

In analogy with the theory of strictly elliptic equations in a sector domain such as $\Omega$, 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 $\delta$ 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 $\varepsilon=\varepsilon(\delta)$ to be sufficiently small, since whenever $\xi(x)$ or its derivatives appear in the coefficients of the operator $L$, they are always multiplied by $\varepsilon$. 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 $$a_{\lambda,\gamma}(r,\theta)=\frac{\lambda\theta^{2}-1}{r^{\gamma}},$$ and as in Lemma 2.1 let $w\in C^{\infty}$ throughout this section.

**Lemma 3.2.** *Suppose that $|w|_{C^{4}}<1$ and let $u\in H^{(2,1,\gamma+2)}(\Omega)\cap C^{2}(\Omega)$ with $u(r,0)=u(r,\delta)=0$. If $\delta=\delta(\lambda)$ and $\varepsilon=\varepsilon(\delta)$ are sufficiently small, then* $$\int_{\Omega}a_{\lambda,\gamma-2}uLu\geq C\int_{\Omega}\lambda
r^{-\gamma}u^{2}+r^{-\gamma+2}(\varphi\sin\theta
u_{r}+r^{-1}\cos\theta u_{\theta})^{2}$$ *for some constant $C>0$ independent of $\lambda$, $\delta$, $\varepsilon$, and $w$.*

*Proof.* Let $0<r_{0}<\sigma$ and set $\Omega_{r_{0}}=\Omega\cap\{(r,\theta)\mid r_{0}<r<\sigma\}$. Then for any $a\in C^{\infty}(\Omega_{r_{0}})$, integration by parts yields $$\begin{aligned}
\int_{\Omega_{r_{0}}}auLu
&=&\int_{\Omega_{r_{0}}}-a(\overline{K}u_{r}^{2}+\overline{A}u_{r}u_{\theta}
+\overline{B}u_{\theta}^{2})\\
&
&+\frac{1}{2}\int_{\Omega_{r_{0}}}
[(a\overline{K})_{rr}+(a\overline{A})_{r\theta}+(a\overline{B})_{\theta\theta}
-(a\overline{C})_{r}-(a\overline{D})_{\theta}]u^{2}\\
&
&+\int_{\partial\Omega_{r_{0}}}a(\overline{K}uu_{r}\nu_{1}+\overline{A}uu_{r}\nu_{2}
+\overline{B}uu_{\theta}\nu_{2})\\
& &+\frac{1}{2}\int_{\partial\Omega_{r_{0}}}
[-(a\overline{K})_{r}\nu_{1}-(a\overline{A})_{\theta}\nu_{1}
-(a\overline{B})_{\theta}\nu_{2}+a\overline{C}\nu_{1}+a\overline{D}\nu_{2}]u^{2},\end{aligned}$$ where $(\nu_{1},\nu_{2})$ denotes the unit outer normal to $\partial\Omega_{r_{0}}$. By choosing $a=a_{\lambda,\gamma-2}$ and observing that $$|(a_{\lambda,\gamma-2}\overline{K})_{rr}|+|(a_{\lambda,\gamma-2}\overline{A})_{r\theta}|
+|(a_{\lambda,\gamma-2}\overline{C})_{r}|+
|(a_{\lambda,\gamma-2}\overline{D})_{\theta}|=O(r^{-\gamma}),$$ $$(a_{\lambda,\gamma-2}\overline{B})_{\theta\theta}=2\lambda
r^{-\gamma}\cos^{2}\theta(1+O(\theta+\lambda^{-1})),$$ the desired result follows since all boundary terms vanish according to Lemma 3.1 (after letting $r_{0}\rightarrow 0$). Note that $\varepsilon$ is chosen small depending on $\delta$, 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 theorem. Let $\widehat{C}^{\infty}(\Omega)$ denote the space of $\overline{C}^{\infty}(\Omega)$ functions $v$ satisfying $v(r,0)=v(r,\delta)=0$. Given $f\in H^{(m,1,\gamma)}(\Omega)$, we will refer to a function $u\in H^{(m,1,\gamma)}(\Omega)$ as a weak solution of (3.1) if $$(u,L^{*}v)=(f,v)\text{ }\text{ }\text{ all }\text{ }\text{
}v\in\widehat{C}^{\infty}(\Omega),$$ where $(\cdot,\cdot)$ is the $L^{2}(\Omega)$ inner product and $L^{*}$ is the formal adjoint of $L$.

**Theorem 3.1.** *Suppose that $g\in C^{m_{*}}$, $|w|_{C^{4}}<1$, and $f\in H^{(m,1,\gamma)}(\Omega)$. If $m\leq
m_{*}-4$ and $\delta=\delta(m)$, $\varepsilon=\varepsilon(m,\delta)$ are sufficiently small, then there exists a weak solution $u\in H^{(m,1,\gamma)}(\Omega)$ of (3.1).*

*Proof.* Given $v\in\widehat{C}^{\infty}(\Omega)$, let $\zeta\in H^{(m,\infty,\gamma+2)}(\Omega)\cap C^{\infty}(\Omega)$ be the unique solution of the ODE: $$\begin{aligned}
&
&\sum_{s=0}^{m}\lambda^{-s}(-1)^{s}\partial_{r}^{s}(a_{\lambda,\gamma-2(s-1)}
\partial_{r}^{s}\zeta)=v,\\
& &\zeta(r,0)=\zeta(r,\delta)=0,\text{ }\text{ }\text{
}\partial_{r}^{s}\zeta(\sigma,\theta)=0,\text{ }\text{ }\text{
}0\leq s\leq m-1,\nonumber\\
&
&\int_{r=r_{0}}(\partial_{r}^{s}\partial_{\theta}^{l}\zeta)^{2}\leq
r_{0}^{\gamma-2s}C,\text{ }\text{ }\text{ }0\leq s\leq 2m-1,\text{
}\text{ }\text{ }0\leq l<\infty.\nonumber\end{aligned}$$ Here $r_{0}>0$ is assumed to be sufficiently small, and $C>0$ is a constant depending on $m$, $\lambda$, $\gamma$, and $v$. The proof that such a solution exists may be found in appendix A, section $\S 6$.

Our first goal is to establish an estimate of the form $$(L\zeta,r^{4}\sum_{s=0}^{m}\lambda^{-s}(-1)^{s}\partial_{r}^{s}(
a_{\lambda,\gamma-2(s-1)}\partial_{r}^{s}\zeta))\geq
C\parallel\zeta\parallel_{(m,1,\gamma)}^{2}.$$ 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, $$\begin{aligned}
& &(L\zeta,r^{4}\sum_{s=0}^{m}\lambda^{-s}(-1)^{s}\partial_{r}^{s}
(a_{\lambda,\gamma-2(s-1)}\partial_{r}^{s}\zeta))\\
&=&
\sum_{s=0}^{m}\lambda^{-s} ([\partial_{r}^{s},L]\zeta
+L\partial_{r}^{s}\zeta,r^{4}a_{\lambda,\gamma-2(s-1)}\partial_{r}^{s}\zeta)\nonumber\\
& &+\sum_{s=0}^{m}\lambda^{-s}\left(\sum_{l<s}
\left(\begin{array}{c}
s \\
l
\end{array}\right)\partial_{r}^{s-l}r^{4}\partial_{r}^{l}L\zeta,
a_{\lambda,\gamma-2(s-1)}\partial_{r}^{s}\zeta\right).\nonumber\end{aligned}$$ Note that since all the coefficients of $L$ vanish at $r=\sigma$, except $\overline{B}$, no boundary terms at $r=\sigma$ appear in the above formula. Furthermore since $r^{4}a_{\lambda,\gamma-2(s-1)}=a_{\lambda,\gamma-2(s+1)}$ and $\partial_{r}^{s}\zeta\in H^{(2,1,\gamma-2(s-1))}(\Omega)$, $0\leq
s\leq m-2$ (for $m-1\leq s\leq m$ the boundary behavior of $\partial_{r}^{s}\zeta$ given by (3.3) is also adequate), Lemma 3.2 implies that $$\begin{aligned}
&
&\sum_{s=0}^{m}(L\partial_{r}^{s}\zeta,\lambda^{-s}r^{4}a_{\lambda,\gamma-2(s-1)}
\partial_{r}^{s}\zeta)\\
&\geq& C\int_{\Omega}\sum_{s=0}^{m}\lambda^{-s}[
r^{-\gamma+2(s+1)}(\varphi\sin\theta(\partial_{r}^{s}\zeta)_{r}+r^{-1}\cos\theta
(\partial_{r}^{s}\zeta)_{\theta})^{2}+\lambda
r^{-\gamma+2s}(\partial_{r}^{s}\zeta)^{2}]\nonumber\\
&\geq& C\int_{\Omega}\left[\sum_{s=0}^{m-1}\lambda^{-s}
r^{-\gamma+2s}
(\partial_{r}^{s}\zeta_{\theta})^{2}+\sum_{s=0}^{m}\lambda^{1-s}r^{-\gamma+2s}
(\partial_{r}^{s}\zeta)^{2}\right]\nonumber\\
&\geq& C\parallel\zeta\parallel_{(m,1,\gamma)}^{2}\nonumber\end{aligned}$$ if $\delta=\delta(\lambda)$ and $\varepsilon=\varepsilon(\delta)$ are sufficiently small. Next we calculate $$\begin{aligned}
[\partial_{r}^{s},L]\zeta&=&\sum_{l<s}\left(\begin{array}{c}
s \\
l
\end{array}\right)[\partial_{r}^{s-l}\overline{K}(\partial_{r}^{l}\zeta)_{rr}
+\partial_{r}^{s-l}\overline{A}(\partial_{r}^{l}\zeta)_{r\theta}\\
&
&+\partial_{r}^{s-l}\overline{B}(\partial_{r}^{l}\zeta)_{\theta\theta}+
\partial_{r}^{s-l}\overline{C}(\partial_{r}^{l}\zeta)_{r}+
\partial_{r}^{s-l}\overline{D}(\partial_{r}^{l}\zeta)_{\theta}],\end{aligned}$$ and observe that integrating by parts, again with the help of the boundary conditions in (3.3), produces $$|(\sum_{l<s}\left(\begin{array}{c}
s \\
l
\end{array}\right)\partial_{r}^{s-l}\overline{K}(\partial_{r}^{l}\zeta)_{rr},
r^{4}a_{\lambda,\gamma-2(s-1)}\partial_{r}^{s}\zeta)|\leq
C_{s}\int_{\Omega} \sum_{l\leq s}r^{-\gamma+2l}
(\partial_{r}^{l}\zeta)^{2},$$ $$|(\sum_{l<s}\left(\begin{array}{c}
s \\
l
\end{array}\right)\partial_{r}^{s-l}\overline{A}(\partial_{r}^{l}\zeta)_{r\theta},
r^{4}a_{\lambda,\gamma-2(s-1)}\partial_{r}^{s}\zeta)|\leq
C_{s}\int_{\Omega}[r^{-\gamma+2s}(\partial_{r}^{s}\zeta)^{2}+
\sum_{l<s}r^{-\gamma+2l} (\partial_{r}^{l}\zeta_{\theta})^{2}],$$ $$|(\sum_{l<s}\left(\begin{array}{c}
s \\
l
\end{array}\right)\partial_{r}^{s-l}\overline{B}(\partial_{r}^{l}\zeta)_{\theta\theta},
r^{4}a_{\lambda,\gamma-2(s-1)}\partial_{r}^{s}\zeta)|\leq
C_{s}\int_{\Omega}[r^{-\gamma+2s}(\partial_{r}^{s}\zeta)^{2}+
\sum_{l<s}r^{-\gamma+2l} (\partial_{r}^{l}\zeta_{\theta})^{2}],$$ $$|(\sum_{l<s}\left(\begin{array}{c}
s \\
l
\end{array}\right)\partial_{r}^{s-l}\overline{C}(\partial_{r}^{l}\zeta)_{r},
r^{4}a_{\lambda,\gamma-2(s-1)}\partial_{r}^{s}\zeta)|\leq
C_{s}\int_{\Omega} \sum_{l\leq s}r^{-\gamma+2l}
(\partial_{r}^{l}\zeta)^{2},$$ $$|(\sum_{l<s}\left(\begin{array}{c}
s \\
l
\end{array}\right)\partial_{r}^{s-l}\overline{D}(\partial_{r}^{l}\zeta)_{\theta},
r^{4}a_{\lambda,\gamma-2(s-1)}\partial_{r}^{s}\zeta)|\leq
C_{s}\int_{\Omega}[r^{-\gamma+2s}(\partial_{r}^{s}\zeta)^{2}+
\sum_{l<s}r^{-\gamma+2l} (\partial_{r}^{l}\zeta_{\theta})^{2}].$$ Also observe that $$\begin{aligned}
& &\sum_{l<s}\left(\begin{array}{c}
s \\
l
\end{array}\right)\partial_{r}^{s-l}r^{4}\partial_{r}^{l}L\zeta\\
&=& \sum_{s-4\leq l<s}\left(\begin{array}{c}
s \\
l
\end{array}\right)\frac{4!}{(4-s+l)!}r^{4-s+l}[\sum_{t\leq l}\left(\begin{array}{c}
l \\
t
\end{array}\right)(\partial_{r}^{l-t}\overline{K}(\partial_{r}^{t}\zeta)_{rr}\\
&
+&\!\!\partial_{r}^{l-t}\overline{A}(\partial_{r}^{t}\zeta)_{r\theta}+
\partial_{r}^{l-t}\overline{B}(\partial_{r}^{t}\zeta)_{\theta\theta}
+\partial_{r}^{l-t}\overline{C}(\partial_{r}^{t}\zeta)_{r}
+\partial_{r}^{l-t}\overline{D}(\partial_{r}^{t}\zeta)_{\theta})],\end{aligned}$$ so similar calculations yield $$\begin{aligned}
& &|(\sum_{l<s}\left(\begin{array}{c}
s \\
l
\end{array}\right)\partial_{r}^{s-l}r^{4}\partial_{r}^{l}L\zeta,
a_{\lambda,\gamma-2(s-1)}\partial_{r}^{s}\zeta)|\\
&\leq& C_{s}\int_{\Omega}\left[\sum_{l\leq
s}r^{-\gamma+2l}(\partial_{r}^{l}\zeta)^{2}+
\sum_{l<s}r^{-\gamma+2l}
(\partial_{r}^{l}\zeta_{\theta})^{2}\right].\nonumber\end{aligned}$$ Then the combination of (3.5), (3.6), (3.7), and (3.8) produces (3.4) for $\lambda=\lambda(m)$ sufficiently large and $\delta=\delta(\lambda)$, $\varepsilon=\varepsilon(\delta)$ sufficiently small.

We need one last estimate before proving existence. Since $r^{4}\eta\in H^{(m,0,\gamma+2)}(\Omega)$ whenever $\eta\in
H^{(m,1,\gamma)}(\Omega)$, we have $$\begin{aligned}
\parallel r^{4}v\parallel_{(-m,-1,\gamma)}&:=&\sup_{\eta\in H^{(m,1,\gamma)}
(\Omega)}\frac{|(\eta,r^{4}v)|}{\parallel\eta\parallel_{(m,1,\gamma)}}\\
&=&\sup_{\eta\in H^{(m,1,\gamma)}
(\Omega)}\frac{|(r^{4}(\lambda\theta^{2}-1)\eta,\zeta)_{(m,0,\gamma+2)}|}{\parallel
\eta\parallel_{(m,1,\gamma)}}\nonumber\\
&\leq&C\sup_{\eta\in H^{(m,1,\gamma)}
(\Omega)}\frac{\parallel\eta\parallel_{(m,0,\gamma)}\parallel\zeta\parallel_{(m,0,\gamma)}}{\parallel
\eta\parallel_{(m,1,\gamma)}}\nonumber\\
&\leq&C\parallel\zeta\parallel_{(m,1,\gamma)}.\nonumber\end{aligned}$$ Here $(\cdot,\cdot)_{(m,0,\gamma+2)}$ denotes the inner product on $H^{(m,0,\gamma+2)}(\Omega)$, and the norm $\parallel\cdot\parallel_{(-m,-1,\gamma)}$ comes from the dual space $H^{(-m,-1,\gamma)}(\Omega)$ of $H^{(m,1,\gamma)}(\Omega)$, which may be obtained as the completion of $L^{2}(\Omega)$ in this negative norm.

Now apply (3.4) to obtain $$\begin{aligned}
\parallel \zeta\parallel_{(m,1,\gamma)}\parallel L^{*}r^{4}v
\parallel_{(-m,-1,\gamma)}&\geq&
(\zeta,L^{*}r^{4}v)\\
&=&(L\zeta,r^{4}v)\\
&=&(L\zeta,r^{4}\sum_{s=0}^{m}\lambda^{-s}(-1)^{s}\partial_{r}^{s}
(a_{\lambda,\gamma-2(s-1)}\partial_{r}^{s}\zeta))\\
&\geq& C\parallel \zeta\parallel_{(m,1,\gamma)}^{2},\end{aligned}$$ which when combined with (3.9) yields $$\parallel r^{4}v\parallel_{(-m,-1,\gamma)}\leq C\parallel
L^{*}r^{4}v\parallel_{(-m,-1,\gamma)}.$$ Consider the linear functional $F:X\rightarrow\mathbb{R}$ where $X=L^{*}(r^{4}\widehat{C}^{\infty}(\Omega))$, given by $$F(L^{*}r^{4}v)=(f,r^{4}v).$$ According to (3.10) we have that $F$ is bounded on the subspace $X$ of $H^{(-m,-1,\gamma)}(\Omega)$ since $$\begin{aligned}
|F(L^{*}r^{4}v)|=|(f,r^{4}v)|&\leq& \parallel
f\parallel_{(m,1,\gamma)}\parallel
r^{4}v\parallel_{(-m,-1,\gamma)}\\
&\leq& C\parallel f\parallel_{(m,1,\gamma)}\parallel
L^{*}r^{4}v\parallel_{(-m,-1,\gamma)}.\end{aligned}$$ Note that we use $f\in H^{(m,1,\gamma)}(\Omega)$ 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 $H^{(-m,-1,\gamma)}(\Omega)$. It follows that there exists a unique $u\in H^{(m,1,\gamma)}(\Omega)$ such that $$F(\eta)=(u,\eta)\text{ }\text{ }\text{ all }\text{ }\text{
}\eta\in H^{(-m,-1,\gamma)}(\Omega).$$ Now restrict $\eta$ back to $X$ to obtain $$(u,L^{*}r^{4}v)=(f,r^{4}v)\text{ }\text{ }\text{ all }\text{
}\text{ }v\in\widehat{C}^{\infty}(\Omega).$$ Since every $\overline{v}\in\widehat{C}^{\infty}(\Omega)$ can be written as $\overline{v}=r^{4}v$ for some $v\in\widehat{C}^{\infty}(\Omega)$, $u$ is a weak solution of (3.1). Q.E.D.

If $f\in H^{(m,1,\gamma)}(\Omega)\cap
C^{m_{*}-4}(\Omega)$, then the strict ellipticity of $L$ on the interior of $\Omega$ shows that the solution given by Theorem 3.1 satisfies $u\in H^{(m,1,\gamma)}(\Omega)\cap C^{m_{*}-3}(\Omega)$, since the coefficients of $L$ will be in $C^{m_{*}-4}$ when $g\in
C^{m_{*}}$. In particular $Lu=f$ pointwise on the interior of $\Omega$, so that $$-u_{\theta\theta}=\overline{B}^{-1}[\overline{K}u_{rr}+\overline{A}u_{r\theta}
+\overline{C}u_{r}+\overline{D}u_{\theta}-f]\text{ }\text{ }\text{
in }\text{ }\text{ }\Omega.$$ Since $\overline{B}^{-1}=O(r^{2})$ as $r\rightarrow 0$ and $u\in
H^{(m,1,\gamma)}(\Omega)$ it follows that $u_{\theta\theta}\in
H^{(m-2,0,\gamma)}(\Omega)$. Let $H_{\gamma}^{m}(\Omega):=H^{(m,m,\gamma)}(\Omega)$ with norm $\parallel\cdot\parallel_{(m,\gamma)}$. Then if $f\in
H_{\gamma}^{m}(\Omega)$ we may continue to differentiate the expression for $u_{\theta\theta}$ to eventually obtain (by induction) that $u\in H_{\gamma}^{m}(\Omega)$.

In order to determine the boundary values for $u$, integrate by parts in expression (3.2) to obtain $$\begin{aligned}
0&=&\int_{\partial\Omega}(\overline{K}vu_{r}\nu_{1}-(\overline{K}
v)_{r}u\nu_{1}+\overline{A}vu_{\theta}\nu_{1}-(\overline{A}v)_{r}u\nu_{2}\\
&
&+\overline{B}vu_{\theta}\nu_{2}-(\overline{B}v)_{\theta}u\nu_{2}
+\overline{C}vu\nu_{1}+\overline{D}vu\nu_{2}),\end{aligned}$$ for all $v\in\widehat{C}^{\infty}(\Omega)$. This implies that $u(r,0)=u(r,\delta)=0$, as $\overline{B}>0$. It also shows that we cannot arbitrarily prescribe boundary values for $u$ at $r=\sigma$, since all the coefficients of $L$ (except $\overline{B}$) vanish at $r=\sigma$. However it is clear that the boundary values at $r=\sigma$ are given explicitly in terms of $f(\sigma,\theta)$ 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 $u\in
H^{m}_{\gamma}(\Omega)$, so that if $m\geq s_{0}+1$ and $\gamma>2m$ 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 $\S 5$.

**Theorem 3.2.** *Suppose that $g\in C^{m_{*}}$ and $f\in \overline{C}^{\infty}(\Omega)$. If $s_{0}+1\leq m\leq
m_{*}-4$, $\gamma> 2m$, $|w|_{C^{6}}<1$, and $\delta=\delta(m)$, $\varepsilon=\varepsilon(m,\delta)$ are sufficiently small then there exists a unique solution $u\in H^{m}_{\gamma}(\Omega)\cap
C^{m_{*}-3}(\Omega)$ of (3.1). Furthermore, there exists a constant $C_{m}$ independent of $\delta$ and $\varepsilon$ such that* $$\parallel u\parallel_{(m,\gamma)}\leq C_{m}(\parallel
f\parallel_{m+2+\gamma}+\parallel w\parallel_{m+6}\parallel
f\parallel_{5+\gamma}),$$ *for each $m\leq m_{*}-6$.*

*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 $$\int_{\Omega}r^{-\gamma+2}(\lambda u^{2}+u_{\theta}^{2})\leq C
\int_{\Omega}r^{-\gamma+6}(f^{2}+r^{-2}u_{r}^{2}).$$ Now differentiate equation (3.1) to obtain, $$Lu_{r}=f_{r}-\overline{K}_{r}u_{rr}-\overline{A}_{r}u_{r\theta}
-\overline{B}_{r}u_{\theta\theta}-\overline{C}_{r}u_{r}
-\overline{D}_{r}u_{\theta}.$$ Solving for $u_{\theta\theta}$ as in (3.11) then yields $$\begin{aligned}
L_{1}u_{r}&:=&\overline{K}(u_{r})_{rr}+\overline{A}(u_{r\theta})_{r}+
\overline{B}(u_{r})_{\theta\theta}\\
& &+(\overline{C}+\overline{K}_{r}-\overline{B}^{-1}
\overline{B}_{r}\overline{K})(u_{r})_{r}+(\overline{D}+\overline{A}_{r}
-\overline{B}^{-1} \overline{B}_{r}\overline{A})(u_{r})_{\theta}
\\
&=&f_{r}-(\overline{C}_{r}-\overline{B}^{-1}\overline{B}_{r}
\overline{C})u_{r}-(\overline{D}_{r}-\overline{B}^{-1}
\overline{B}_{r}\overline{D})u_{\theta}.\end{aligned}$$ Applying the proof of Lemma 3.2 to the above equation gives $$C\int_{\Omega}[\lambda r^{-\gamma+4}u_{r}^{2}+r^{-\gamma+6}
(\varphi\sin\theta u_{rr}+r^{-1}\cos\theta u_{r\theta})^{2}]\leq
(a_{\lambda,\gamma-6}u_{r},L_{1}u_{r}),$$ from which we find that $$\int_{\Omega}r^{-\gamma+4}(\lambda u_{r}^{2}+u_{r\theta}^{2})\leq
C\int_{\Omega}r^{-\gamma+6}(f^{2}+r^{2}f_{r}^{2}+u_{rr}^{2})$$ if $\lambda$ is sufficiently large. Eventually, with the help of $|w|_{C^{6}}<1$ and (3.11), we obtain $$\parallel u\parallel_{(3,3,\gamma-2)}\leq C\parallel
f\parallel_{(3,3,\gamma-6)}.$$ 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 $$\begin{aligned}
& &\int_{\Omega}\left(\lambda\sum_{s\leq
m}r^{-\gamma+2s+2}(\partial_{r}^{s}u)^{2}+\sum_{s<
m}r^{-\gamma+2s+2}(\partial_{r}^{s}u_{\theta})^{2}\right)\\
&\leq& C_{m}(\parallel
f\parallel^{2}_{(m,0,\gamma-6)}+\Lambda_{m}^{2}\parallel
u\parallel^{2}_{(3,3,\gamma-2)})\nonumber\\
&\leq&\!\!C_{m}(\parallel
f\parallel_{(m,\gamma)}^{2}+\Lambda_{m}^{2}\parallel
f\parallel_{(3,\gamma)}^{2}),\nonumber\end{aligned}$$ where $$\Lambda_{m}=\parallel r^{2}\overline{K}\parallel_{m}+
\parallel r^{2}\overline{A}\parallel_{m}+\parallel r^{2}\overline{B}\parallel_{m}
+\parallel r^{2}\overline{C}\parallel_{m}+
\parallel r^{2}\overline{D}\parallel_{m}+1.$$

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 $$\begin{aligned}
L_{s}\partial_{r}^{s}u&:=&\overline{K}(\partial_{r}^{s}u)_{rr}+\overline{A}(\partial_{r}^{s}u)_{r\theta}
+\overline{B}(\partial_{r}^{s}u)_{\theta\theta}\\
& &+(\overline{C}+s\overline{K}_{r}-s\overline{B}^{-1}
\overline{B}_{r}\overline{K})(\partial_{r}^{s}u)_{r}+(\overline{D}+s\overline{A}_{r}
-s\overline{B}^{-1}
\overline{B}_{r}\overline{A})(\partial_{r}^{s}u)_{\theta}
\\
&=&\partial_{r}^{s}f-\partial_{r}^{s-1}(\overline{C}_{r}u_{r})+\cdots.\end{aligned}$$ Since $\partial_{r}^{s}u\in H^{(2,1,\gamma-2s)}(\Omega)$ when $s\leq m_{*}-6$, the basic estimate yields $$\int_{\Omega}r^{-\gamma+2s+2}[\lambda(\partial_{r}^{s}u)^{2}
+(\partial_{r}^{s}u_{\theta})^{2}]\leq
C\int_{\Omega}r^{-\gamma+2s+6}[(\partial_{r}^{s}f)^{2}
+(\partial_{r}^{s-1}(\overline{C}_{r}u_{r}))^{2}]+\cdots.$$ Furthermore observe that $$\partial_{r}^{s-1}(\overline{C}_{r}u_{r})=\sum_{l\leq s-1}\left(\begin{array}{c}
s-1 \\
l
\end{array}\right)\partial_{r}^{s-1-l}\overline{C}_{r}\partial_{r}^{l}u_{r},$$ and $$\partial_{r}^{s-1-l}\overline{C}_{r}=\partial_{r}^{s-1-l}(r^{-2}r^{2}\overline{C}_{r})=\sum_{t_{1}\leq
s-1-l}C_{t_{1}}r^{-2-(s-1-l-t_{1})}\partial_{r}^{t_{1}}(r^{2}\overline{C}_{r}),$$ $$r^{-\gamma/2+s+3+[-2-(s-1-l-t_{1})]}\partial_{r}^{l}u_{r}=\sum_{t_{2}\leq
l}C_{t_{2}}\partial_{r}^{t_{2}}(r^{-\gamma/2+t_{1}+t_{2}+2}u_{r}),$$ 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 $$\begin{aligned}
& &\parallel
r^{-\gamma/2+s+3}\partial_{r}^{s-1}(\overline{C}_{r}u_{r})\parallel\\
&\leq& \sum_{2\leq t_{1}+t_{2}\leq s-1}C_{t_{1}t_{2}}\parallel
\partial_{r}^{t_{1}}(r^{2}\overline{C}_{r})\partial_{r}^{t_{2}}(r^{-\gamma/2+
t_{1}+t_{2}+2}u_{r})\parallel\\
& &+\sum_{t_{1}+t_{2}<2}C_{t_{1}t_{2}}\parallel
\partial_{r}^{t_{1}}(r^{2}\overline{C}_{r})\partial_{r}^{t_{2}}(r^{-\gamma/2+
t_{1}+t_{2}+2}u_{r})\parallel\\
&\leq&C_{s}(|r^{2}\overline{C}_{r}|_{\infty}\parallel
u_{r}\parallel_{(s-1,0,\gamma-4)}+\parallel
r^{2}\overline{C}_{r}\parallel_{s-1}|r^{-\gamma/2+4}u_{r}|_{\infty})\\
& &+C_{s}\parallel u\parallel_{(2,2,\gamma-2)}\\
&\leq& C_{s}(\parallel
u\parallel_{(s,0,\gamma-2)}+\Lambda_{s}\parallel
u\parallel_{(3,3,\gamma-2)}),\end{aligned}$$ where $|\cdot|_{\infty}$ denotes the $L^{\infty}(\Omega)$ norm. Note that the first term of the last line above, may be absorbed into the left-hand side of (3.12) for large $\lambda(s)$.

The remaining derivatives of $u$ involving higher orders of $\partial_{\theta}$ 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 $\xi$ (the coordinates of Lemma 2.1) up to and including order 2, with the help of (2.4) we obtain $$\parallel u\parallel_{(m,\gamma-2)}\leq C_{m}(\parallel
f\parallel_{(m,\gamma)}+\parallel w\parallel_{m+6}\parallel
f\parallel_{(3,\gamma)}).$$ Lastly since $f$ vanishes to all orders at $r=0$, a little calculation shows that $$\int_{\Omega}r^{-\gamma+2s}(\partial_{r}^{s}\partial_{\theta}^{t}f)^{2}\leq
C_{s}\int_{\Omega}(\partial_{r}^{s+\gamma/2+1}\partial_{\theta}^{t}f)^{2},$$ from which the desired result follows. Q.E.D.

**4. 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 $(\xi^{1},\xi^{2})$ of Lemma 2.1, so that a portion of the given hyperbolic region $\Omega^{-}_{\varrho}\cap B_{\sigma}(0)$, $\varrho=1,2$, may be written as $$\Omega=\{(x,y)\mid h(x)<y<\sigma\}$$ for some Lipschitz function $h(x)$ satisfying $h(0)=0$ and $|h(x)-|x||_{C^{1}}=O(\sigma)$. Then in these coordinates $L(w)$ is given by $$Lu=(\overline{K}u_{x})_{x}+u_{yy}+\overline{C}u_{x}+\overline{D}u_{y}$$ with $$\overline{K}=\overline{k}K,\text{ }\text{ }\text{ }\text{ }\text{
}\overline{C}=\overline{c}K+(a^{22})^{-2}\partial_{x^{1}}\xi^{1}
\partial_{x^{1}}\Phi(w),\text{ }\text{ }\text{ }\text{ }\text{
}\overline{D}=\varepsilon^{2}\overline{d},$$ in the notation of Lemma 2.1. Consider the Cauchy problem $$Lu=f\text{ }\text{ in }\text{ }\Omega,\text{ }\text{
}u|_{\partial\Omega_{1}}=\phi,\text{ }\text{
}u_{y}|_{\partial\Omega_{1}}=\psi,\text{ }\text{
}\partial^{\alpha}u(0,0)=0,\text{ }\text{ }0\leq|\alpha|\leq
\alpha_{0},$$ for some given data $\phi$, $\psi$ and a large integer $\alpha_{0}$, where $\partial\Omega_{1}$ denotes the “bottom" portion of the boundary given by $y=h(x)$. The “top" portion of the boundary will be denoted $\partial\Omega_{2}$, and is given by $y=\sigma$. 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 $\S 5$). Of course a necessary condition to have such behavior is that $\phi$, $\psi$, 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 $\overline{K}$ and $\overline{C}$, 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 \[14\]) $$\overline{C}\leq M\sqrt{|\overline{K}|}\text{ }\text{ }\text{ in
}\text{ }\text{ }\Omega,$$ for some constant $M>0$. Unfortunately the quantity $\partial_{x^{1}}\Phi(w)$ present in $\overline{C}$ prevents the validity of this inequality. However in the Nash iteration of the next section, $\partial_{x^{1}}\Phi(w)$ will be uniformly small. Therefore it is natural to consider the regularized Cauchy problem $$L_{\theta}u=f\text{ }\text{ in }\text{ }\Omega,\text{ }\text{
}u|_{\partial\Omega_{1}}=\phi,\text{ }\text{
}u_{y}|_{\partial\Omega_{1}}=\psi,\text{ }\text{
}\partial^{\alpha}u(0,0)=0,\text{ }\text{
}0\leq|\alpha|\leq\alpha_{0},$$ where $L_{\theta}$ differs from $L$ in that $\overline{K}$ is replaced by $\overline{K}_{\theta}:=\overline{K}-\theta$ with $\theta=|\Phi(w)|_{C^{1}}$. There then exists a constant $M>0$ such that $$\overline{C}\leq M|\overline{K}_{\theta}|\text{ }\text{ }\text{ in
}\text{ }\text{ }\Omega.$$ 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 $\theta$ is of quadratic error (see section $\S 5$), 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 $\partial\Omega_{1}$. To this end we define $H^{(m,l)}(\Omega)$ ($H_{0}^{(m,l)}(\Omega)$) to be the closure of all $C^{\infty}(\Omega)$ functions (which vanish to all orders at $\partial\Omega_{1}$) in the norm $$\parallel u\parallel^{2}_{(m,l)}=\int_{\Omega}\sum_{0\leq s\leq
m\atop 0\leq t\leq
l}\lambda^{-s}(\partial_{x}^{s}\partial_{y}^{t}u)^{2},$$ where $\lambda>0$ is a large parameter. The $L^{2}(\Omega)$ inner product will as usual be denoted by $(\cdot,\cdot)$, and the formal adjoint of $L_{\theta}$ by $L_{\theta}^{*}$. Also as in Lemma 2.1 $w\in C^{\infty}$ throughout this section.

**Theorem 4.1.** *Suppose that $g\in C^{m_{*}}$, $|w|_{C^{4}}<1$, and $f\in H_{0}^{(m,0)}(\Omega)$. If $m\leq
m_{*}-6$ and $\varepsilon$ is sufficiently small, then there exists a weak solution $u_{\theta}\in H_{0}^{(m,1)}(\Omega)$ of (4.2) with $\phi,\psi=0$, for each $\theta>0$. That is* $$(u_{\theta},L_{\theta}^{*}v)=(f,v)\text{ }\text{ }\textit{ all
}\text{ }\text{ }v\in C^{\infty}(\Omega)$$ *with $v|_{\partial\Omega_{2}}=v_{y}|_{\partial\Omega_{2}}=0$.*

*Proof.* Set $b(x,y)=\overline{K}_{\theta}^{-1}(x,y)e^{-\lambda y}$ and let $\zeta$ be the unique solution of $$\sum_{s=0}^{m}(-1)^{s+1}\lambda^{-s}\partial_{x}^{s}(b\partial_{x}^{s}\zeta_{y})=v\text{
}\text{ }\text{ in }\text{ }\text{ }\Omega,\text{ }\text{
}\zeta|_{\partial\Omega_{1}}=\partial_{x}^{s}\zeta_{y}|_{\partial\Omega_{1}}=0,\text{
}\text{ }0\leq s\leq m-1,$$ where $v$ is as stated in the theorem. Note that for each $y\in(0,\sigma)$, this equation may be interpreted as an ODE in $\zeta_{y}$, and therefore the theory of such equations guarantees the existence of a unique solution to (4.5). Furthermore if the metric $g\in C^{m_{*}}$ as in Lemma 2.1, then the coefficients of (4.5) are in $C^{m_{*}-m-4}$. Thus as long as $m\leq m_{*}-6$, we have $\partial_{x}^{s}\zeta\in C^{2}(\Omega)$ for $0\leq s\leq
2m-1$.

We first note that the solution $\zeta$ of (4.5) satisfies extra boundary conditions, namely $$\partial_{x}^{s}\partial_{y}^{t}\zeta|_{\partial\Omega_{1}}=0,\text{
}\text{ }\text{ }\text{ }s+t\leq m,\text{ }\text{ }\text{ }\text{
}0\leq t\leq 2.$$ To see this let $\partial_{T}\zeta$ be differentiation along the right portion of $\partial\Omega_{1}$ (that is, the curve $y=h(x)$, $x>0$), which we denote by $\partial\Omega_{1}^{+}$. Then since $\zeta|_{\partial\Omega_{1}^{+}}=\zeta_{y}|_{\partial\Omega_{1}^{+}}=0$ we have $$0=\partial_{T}\zeta=\frac{1}{\sqrt{1+h'^{\!\text{ }2}}}
(\zeta_{x}+h'\zeta_{y})|_{\partial\Omega_{1}^{+}}
=\frac{\zeta_{x}}{\sqrt{1+h'^{\!\text{
}2}}}|_{\partial\Omega_{1}^{+}},$$ where $h'=dh/dx$. It follows that $$0=\partial_{T}\zeta_{x}=\frac{1}{\sqrt{1+h'^{\!\text{
}2}}}(\zeta_{xx}+h'\zeta_{xy})|_{\partial\Omega_{1}^{+}}
=\frac{\zeta_{xx}}{\sqrt{1+h'^{\!\text{
}2}}}|_{\partial\Omega_{1}^{+}},$$ $$0=\partial_{T}\zeta_{y}=\frac{1}{\sqrt{1+h'^{\!\text{
}2}}}(\zeta_{xy}+h'\zeta_{yy})|_{\partial\Omega_{1}^{+}}
=\frac{h'\zeta_{yy}}{\sqrt{1+h'^{\!\text{
}2}}}|_{\partial\Omega_{1}^{+}}.$$ Since $|h(x)-|x||_{C^{1}}=O(\sigma)$ this shows (4.6) for $m=2$, and continuing this procedure establishes the full result on $\partial\Omega_{1}^{+}$. Also the same arguments hold to yield (4.6) on $\partial\Omega_{1}^{-}$, the left portion of $\partial\Omega_{1}$ (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 $$(L_{\theta}\zeta,\sum_{s=0}^{m}(-1)^{s+1}\lambda^{-s}\partial_{x}^{s}(b\partial_{x}^{s}\zeta_{y}))
\geq C\parallel \zeta\parallel_{(m,1)}^{2},$$ for some constant $C>0$. Observe that the following calculations hold for $0\leq s\leq m$ according to the boundary conditions (4.6): $$\begin{aligned}
((\overline{K}_{\theta}\zeta_{x})_{x},(-1)^{s+1}\partial_{x}^{s}(b\partial_{x}^{s}\zeta_{y}))&=&
\int_{\Omega}\partial_{x}(b\partial_{x}^{s}\zeta_{y})\partial_{x}^{s}(\overline{K}_{\theta}
\zeta_{x})\\
&=&-\int_{\Omega}\left[\frac{1}{2}(b\overline{K}_{\theta})_{y}
(\partial_{x}^{s+1}\zeta)^{2}-b_{x}\overline{K}_{\theta}\partial_{x}^{s}\zeta_{y}\partial_{x}^{s+1}\zeta\right]\\
& &-\int_{\Omega}b\partial_{x}^{s}\zeta_{y}\partial_{x}\left[
\sum_{l=1}^{s}\left(\begin{array}{c}
s \\
l
\end{array}\right)\partial_{x}^{l}\overline{K}_{\theta}
\partial_{x}^{s+1-l}\zeta\right]\\
&
&+\int_{\partial\Omega}\frac{1}{2}b\overline{K}_{\theta}(\partial_{x}^{s+1}\zeta)^{2}
\nu_{2},\end{aligned}$$ $$\begin{aligned}
(\zeta_{yy},(-1)^{s+1}\partial_{x}^{s}(b\partial_{x}^{s}\zeta_{y}))&=&
-\int_{\Omega}b\partial_{x}^{s}\zeta_{yy}\partial_{x}^{s}\zeta_{y}
+\int_{\partial\Omega}b\partial_{x}^{s-1}\zeta_{yy}\partial_{x}^{s}\zeta_{y}\nu_{1}\\
&=&\int_{\Omega}\frac{1}{2}b_{y}(\partial_{x}^{s}\zeta_{y})^{2}
+\int_{\partial\Omega}
\left[b\partial_{x}^{s-1}\zeta_{yy}\partial_{x}^{s}\zeta_{y}\nu_{1}
-\frac{1}{2}b(\partial_{x}^{s}\zeta_{y})^{2}\nu_{2}\right],\end{aligned}$$ $$\begin{aligned}
(\overline{C}\zeta_{x},(-1)^{s+1}\partial_{x}^{s}(b\partial_{x}^{s}\zeta_{y}))&=&
-\int_{\Omega}b\partial_{x}^{s}\zeta_{y}\partial_{x}^{s}(\overline{C}\zeta_{x})\\
&=&-\int_{\Omega}\left[b\overline{C}\partial_{x}^{s}\zeta_{y}\partial_{x}^{s+1}\zeta
+b\partial_{x}^{s}\zeta_{y}\sum_{l=1}^{s}\left(\begin{array}{c}
s \\
l
\end{array}\right)\partial_{x}^{l}\overline{C}
\partial_{x}^{s+1-l}\zeta\right],\end{aligned}$$ $$\begin{aligned}
(\overline{D}\zeta_{y},(-1)^{s+1}\partial_{x}^{s}(b\partial_{x}^{s}\zeta_{y}))&=&
-\int_{\Omega}b\partial_{x}^{s}\zeta_{y}\partial_{x}^{s}(\overline{D}\zeta_{y})\\
&=&-\int_{\Omega}\left[b\overline{D}(\partial_{x}^{s}\zeta_{y})^{2}
+b\partial_{x}^{s}\zeta_{y}\sum_{l=1}^{s}\left(\begin{array}{c}
s \\
l
\end{array}\right)\partial_{x}^{l}\overline{D}
\partial_{x}^{s-l}\zeta_{y}\right],\end{aligned}$$ where $(\nu_{1},\nu_{2})$ denotes the unit outer normal to $\partial\Omega$. Next observe that if $\lambda>0$ is sufficiently large then $$b_{y}=-\frac{1}{\overline{K}_{\theta}}\left(\lambda+\frac{\partial_{y}\overline{K}_{\theta}}
{\overline{K}_{\theta}}\right)e^{-\lambda y}>0,\text{ }\text{
}\text{ }-(b\overline{K}_{\theta})_{y}=\lambda e^{-\lambda y}>0,$$ and in light of the Levi condition (4.3) as well as $\overline{D}=O(\varepsilon)$ we have $$\begin{aligned}
&
&-(b_{y}-2b\overline{D})(b\overline{K}_{\theta})_{y}-(b_{x}\overline{K}_{\theta}-b\overline{C})^{2}\\
&\geq& -\frac{e^{-2\lambda y}}{\overline{K}_{\theta}^{2}}\left[
\left(\lambda+\frac{\partial_{y}\overline{K}_{\theta}}
{\overline{K}_{\theta}}+O(\varepsilon)\right)\lambda\overline{K}_{\theta}
+O(\overline{K}_{x}^{2}+\overline{K}_{\theta}^{2})\right]\\
&\geq&-\frac{e^{-2\lambda
y}}{\overline{K}_{\theta}^{2}}\left[\frac{\lambda^{2}}{2}\overline{K}_{\theta}
+\lambda\overline{K}_{y}+O(\overline{K}_{x}^{2})\right]\\
&>& e^{-2\lambda y},\end{aligned}$$ as $\lambda\overline{K}_{y}+O(\overline{K}_{x}^{2})\leq 0$ near $\partial\Omega_{1}$ for large $\lambda$. It follows that for large $\lambda$ depending on $m$, $$\begin{aligned}
& &(L_{\theta}\zeta,\sum_{s=0}^{m}(-1)^{s+1}\lambda^{-s}\partial_{x}^{s}(b\partial_{x}^{s}\zeta_{y}))\\
&\geq& C\parallel
\zeta\parallel_{(m,1)}^{2}+\frac{\lambda^{-m}}{2}\int_{\partial\Omega}(b\overline{K}_{\theta}
(\partial_{x}^{m+1}\zeta)^{2}\nu_{2}+2b\partial_{x}^{m-1}\zeta_{yy}\partial_{x}^{m}\zeta_{y}\nu_{1}
-b(\partial_{x}^{m}\zeta_{y})^{2}\nu_{2}),\end{aligned}$$ where we have used a Poincaré type inequality to estimate $\parallel\zeta\parallel_{L^{2}(\Omega)}$. Note that the boundary integral has the correct sign on $\partial\Omega_{2}$. We claim that it also has the correct sign on $\partial\Omega_{1}$. To see this we use the boundary condition (4.6). That is, by (4.6) $\partial_{x}^{m-1}\zeta_{y}|_{\partial\Omega_{1}}=0$ so if $\partial_{T}$ is differentiation along $\partial\Omega_{1}-\{(0,0)\}$ then $$0=\partial_{T}\partial_{x}^{m-1}\zeta_{y}=-\nu_{2}\partial_{x}^{m}\zeta_{y}+\nu_{1}
\partial_{x}^{m-1}\zeta_{yy},$$ which yields $$b\partial_{x}^{m-1}\zeta_{yy}\partial_{x}^{m}\zeta_{y}\nu_{1}=b(\partial_{x}^{m}\zeta_{y})^{2}
\nu_{2}\geq 0.$$ Moreover since $\partial_{x}^{m}\zeta|_{\partial\Omega_{1}}=0$ we have $$0=\partial_{T}\partial_{x}^{m}\zeta|_{\partial\Omega_{1}}=
-\nu_{2}\partial_{x}^{m+1}\zeta+\nu_{1}\partial_{x}^{m}\zeta_{y},$$ which yields $$b\overline{K}_{\theta}(\partial_{x}^{m+1}\zeta)^{2}\nu_{2}
=b\overline{K}_{\theta}(\partial_{x}^{m}\zeta_{y})^{2}\frac{\nu_{1}^{2}}{\nu_{2}}.$$ It follows that the boundary integral on $\partial\Omega_{1}$ is nonnegative, and hence (4.7) holds.

We will need one last estimate before proving existence. Namely $$\begin{aligned}
\parallel v\parallel_{(-m,0)}&:=&\sup_{\eta\in
H_{0}^{(m,0)}(\Omega)}\frac{|(\eta,v)|}{\parallel
\eta\parallel_{(m,0)}}\\
&=&\sup_{\eta\in
H_{0}^{(m,0)}(\Omega)}\frac{|(\eta,
\sum_{s=0}^{m}(-1)^{s+1}\lambda^{-s}\partial_{x}^{s}(b\partial_{x}^{s}\zeta_{y}))|}{\parallel
\eta\parallel_{(m,0)}}\nonumber\\
&\leq&\theta^{-1}C\parallel \zeta\parallel_{(m,1)},\nonumber\end{aligned}$$ which follows after integration by parts. Here $\parallel\cdot\parallel_{(-m,0)}$ is the norm on the dual space $H_{0}^{(-m,0)}(\Omega)$ of $H_{0}^{(m,0)}(\Omega)$. This dual space may be obtained as the completion of $L^{2}(\Omega)$ in the norm $\parallel\cdot\parallel_{(-m,0)}$.

To prove existence apply (4.7) to obtain $$\begin{aligned}
\parallel \zeta\parallel_{(m,1)}\parallel L_{\theta}^{*}v
\parallel_{(-m,-1)}&\geq&
(\zeta,L_{\theta}^{*}v)=(L_{\theta}\zeta,v)\\
&=&(L_{\theta}\zeta,\sum_{s=0}^{m}(-1)^{s+1}\lambda^{-s}\partial_{x}^{s}(b\partial_{x}^{s}\zeta_{y}))\geq
C\parallel \zeta\parallel_{(m,1)}^{2},\end{aligned}$$ which together with (4.8) implies that $$\parallel
v\parallel_{(-m,0)}\leq\theta^{-1}C
\parallel L_{\theta}^{*}v\parallel_{(-m,-1)}\text{
}\text{ }\text{ all }\text{ }\text{
}v\in\widehat{C}^{\infty}(\Omega),$$ where $\widehat{C}^{\infty}(\Omega)$ consists of all $C^{\infty}(\Omega)$ functions with $v|_{\partial\Omega_{2}}=v_{y}|_{\partial\Omega_{2}}=0$. Consider the linear functional $F:X\rightarrow\mathbb{R}$, where $X=L_{\theta}^{*}\widehat{C}^{\infty}(\Omega)$, given by $$F(L_{\theta}^{*}v)=(f,v).$$ According to (4.9) we have that $F$ is bounded on the subspace $X$ of $H_{0}^{(-m,-1)}(\Omega)$ since $$|F(L_{\theta}^{*}v)|\leq\parallel f\parallel_{(m,0)}\parallel
v\parallel_{(-m,0)}\leq\theta^{-1}C\parallel f\parallel_{(m,0)}
\parallel L_{\theta}^{*}v\parallel_{(-m,-1)}.$$ Note that the generalized Schwarz inequality (the first inequality in the above sequence) holds because $f\in H_{0}^{(m,0)}(\Omega)$, that is, $f$ vanishes appropriately on $\partial\Omega_{1}$. Thus we can apply the Hahn-Banach theorem to obtain a bounded extension of $F$ defined on all of $H_{0}^{(-m,-1)}(\Omega)$. It follows that there exists $u_{\theta}\in H^{(m,1)}_{0}(\Omega)$ such that $$F(\eta)=(u_{\theta},\eta)\text{ }\text{ }\text{ all }\text{
}\text{ }\eta\in H_{0}^{(-m,-1)}(\Omega).$$ Now restrict $\eta$ back to $X$ to obtain $$(u_{\theta},L_{\theta}^{*}v)=(f,v)\text{ }\text{ }\text{ all
}\text{ }\text{ }v\in\widehat{C}^{\infty}(\Omega).$$ 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 $$u^{q}(x,y):=\frac{u(x,y+q)-u(x,y)}{q}.$$

**Lemma 4.1.** $i)$ *Let $u\in H^{(0,1)}(\Omega)$ and $\Omega'\subset\subset\Omega$ (that is, $\Omega'$ is compactly contained in $\Omega$). Then* $$\parallel u^{q}\parallel_{L^{2}(\Omega')}\leq\parallel
u_{y}\parallel_{L^{2}(\Omega)}$$ *for all $0<|q|<\frac{1}{2}\mathrm{dist}(\Omega',\partial\Omega)$.*

$ii)$ *If $u\in L^{2}(\Omega)$ and $\parallel
u^{q}\parallel_{L^{2}(\Omega')}\leq C$ for all $0<|q|<\frac{1}{2}\mathrm{dist}(\Omega',\partial\Omega)$, then $u\in H^{(0,1)}(\Omega')$*.

The Sobolev space of square integrable derivatives up to and including order $m$ will be denoted by $H^{m}(\Omega)$ with norm $\parallel\cdot\parallel_{m}$, and the completion of $C^{\infty}(\Omega)$ functions which vanish to all orders at $\partial\Omega_{1}$ in the norm $\parallel\cdot\parallel_{m}$ shall be denoted by $H_{0}^{m}(\Omega)$.

**Corollary 4.1.** *Under the hypotheses of Theorem 4.1, if $f\in H_{0}^{m}(\Omega)$ there exists a unique solution $u_{\theta}\in H_{0}^{m}(\Omega)$ of (4.2) with $\phi,\psi=0$, for each $\theta>0$.*

*Proof.* Let $u_{\theta}\in H_{0}^{(m,1)}(\Omega)$ be the weak solution given by Theorem 4.1, so that (4.4) holds. If $m\leq 1$ then this corollary follows directly from Theorem 4.1, so assume that $m\geq 2$. We may integrate by parts to obtain $$\begin{aligned}
-(\partial_{y}u_{\theta}+\overline{D}u_{\theta},v_{y})&=&
(f-\partial_{x}(\overline{K}_{\theta}\partial_{x}u_{\theta})-\overline{C}\partial_{x}
u_{\theta}+\overline{D}_{y}u_{\theta},v)\\
&
&+\int_{\partial\Omega}(\overline{C}vu_{\theta}\nu_{1}-v_{y}u_{\theta}
\nu_{2}-\overline{K}_{\theta}v_{x}u_{\theta}\nu_{1}+\overline{K}_{\theta}v\partial_{x}
u_{\theta}\nu_{1})\end{aligned}$$ for all $v\in\widehat{C}^{\infty}(\Omega)$, where $(\nu_{1},\nu_{2})$ is the unit outer normal to $\partial\Omega$. Note that since $u_{\theta},\partial_{x}u_{\theta}\in
H^{1}(\Omega)$ both $u_{\theta}|_{\partial\Omega}$ and $\partial_{x}u_{\theta}|_{\partial\Omega}$ are meaningful in $L^{2}(\partial\Omega)$, and in particular as $u_{\theta},\partial_{x}u_{\theta}\in H_{0}^{1}(\Omega)$ we have $u_{\theta}|_{\partial\Omega_{1}}=\partial_{x}u_{\theta}|_{\partial\Omega_{1}}=0$ in the $L^{2}(\partial\Omega_{1})$ sense. Thus we may write $$(\overline{u}_{\theta},v_{y})=(\overline{f},v)\text{ }\text{
}\text{ all }\text{ }\text{ }v\in\widehat{C}^{\infty}(\Omega),$$ where $$\overline{u}_{\theta}=-\partial_{y}u_{\theta}-\overline{D}u_{\theta},\text{
}\text{ }\text{ }\text{
}\overline{f}=f-\partial_{x}(\overline{K}_{\theta}\partial_{x}u_{\theta})
-\overline{C}\partial_{x}u_{\theta}+\overline{D}_{y}u_{\theta}.$$ Furthermore $$(\overline{u}_{\theta}^{q},v_{y})=(\overline{f}^{q},v)\text{
}\text{ }\text{ all }\text{ }\text{ }v\in C_{c}^{\infty}(\Omega),$$ so that choosing a sequence $v_{i}\in C^{\infty}_{c}(\Omega)$ with $v_{i}\rightarrow-\eta u_{\theta}^{q}$ in $H^{(0,1)}(\Omega)$ for some nonnegative $\eta\in C^{\infty}_{c}(\Omega)$, implies that $$\begin{aligned}
\parallel\sqrt{\eta}\overline{u}_{\theta}^{q}\parallel^{2}&\leq&
|(\overline{f}^{q},\eta
u_{\theta}^{q})|+|(\overline{u}_{\theta}^{q},\eta_{y}u_{\theta}^{q})|
+|(\overline{u}_{\theta}^{q},\eta(\overline{D}u_{\theta})^{q})|\\
&\leq&
\parallel\sqrt{\eta}\overline{f}^{q}\parallel\parallel\sqrt{\eta}u_{\theta}^{q}\parallel
+\parallel\sqrt{\eta}\overline{u}_{\theta}^{q}\parallel\parallel\frac{\eta_{y}}{\sqrt{\eta}}u_{\theta}^{q}\parallel
+\parallel\sqrt{\eta}\overline{u}_{\theta}^{q}\parallel\parallel\sqrt{\eta}(\overline{D}u_{\theta})^{q}\parallel.\end{aligned}$$ Then since $u_{\theta},\overline{f}\in H^{(0,1)}(\Omega)$ and $|\nabla\eta|^{2}\leq C\eta$, Lemma 4.1 $(i)$ yields $\parallel\sqrt{\eta}\overline{u}_{\theta}^{q}\parallel\leq C$ for some constant $C$ independent of $q$, if $|q|$ is sufficiently small. Now Lemma 4.1 $(ii)$ shows that $\overline{u}_{\theta}\in
H^{(0,1)}_{loc}(\Omega)$, as $\eta$ was arbitrary. Hence $\partial_{y}^{2}u_{\theta}\in L^{2}_{loc}(\Omega)$. It follows that the equation $L_{\theta}u_{\theta}=f$ holds in $L^{2}_{loc}(\Omega)$, and since we can solve for $\partial_{y}^{2}u_{\theta}$, we can boot-strap in the usual way to obtain $u_{\theta}\in H^{m}_{loc}(\Omega)$.

Next observe that the above restrictions on $\eta$ may be relaxed if $q>0$, that is in this case $\eta$ is only required to vanish in a neighborhood of $\partial\Omega_{2}$. Then the same procedure yields $\eta u_{\theta}\in H^{m}(\Omega)$ for all such $\eta$. Furthermore since $L_{\theta}$ is strictly hyperbolic, we can use the regularity theory for such operators to obtain estimates for $u_{\theta}$ near $\partial\Omega_{2}$. It follows that $u_{\theta}\in H^{m}(\Omega)$.

To show that $u_{\theta}\in H_{0}^{m}(\Omega)$, it is enough to observe that $$\partial_{x}^{s}\partial_{y}^{t}u_{\theta}|_{\partial\Omega_{1}}=0,\text{
}\text{ }\text{ }s+t\leq m-1,$$ where the equality is interpreted in the $L^{2}(\partial\Omega_{1})$ 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), $u_{\theta}|_{\partial\Omega_{1}}=\partial_{x}u_{\theta}|_{\partial\Omega_{1}}=0$ implies that $\partial_{y}u_{\theta}|_{\partial\Omega_{1}}=0$. Furthermore using the notation of those arguments we have $$0=\partial_{T}(\partial_{x}u_{\theta})=\frac{1}{\sqrt{1+h'^{\!\text{
}2}}}(\partial_{x}^{2}u_{\theta}
+h'\partial_{x}\partial_{y}u_{\theta})|_{\partial\Omega_{1}^{+}},$$ $$0=\partial_{T}(\partial_{y}u_{\theta})=\frac{1}{\sqrt{1+h'^{\!\text{
}2}}}(\partial_{x}\partial_{y}u_{\theta}
+h'\partial_{y}^{2}u_{\theta})|_{\partial\Omega_{1}^{+}}.$$ However from equation (4.2) we find that $$(-\theta\partial_{x}^{2}u_{\theta}+\partial_{y}^{2}u_{\theta})|_{\partial\Omega_{1}^{+}}=0,$$ hence $$\partial_{x}^{2}u_{\theta}|_{\partial\Omega_{1}^{+}}=\partial_{y}^{2}u_{\theta}|_{\partial\Omega_{1}^{+}}=
\partial_{x}\partial_{y}u_{\theta}|_{\partial\Omega_{1}^{+}}=0.$$ The same arguments also apply to $\partial\Omega_{1}^{-}$, so by differentiating equation (4.2) we can continue this procedure to obtain (4.10).

Lastly we note that since $u_{\theta}|_{\partial\Omega_{1}}=|\nabla
u_{\theta}|_{\partial\Omega_{1}}=0$, (4.7) with $m=1$ yields $$(L_{\theta}u_{\theta},\sum_{s=0}^{1}(-1)^{s+1}\lambda^{-s}\partial_{x}^{s}(b\partial_{x}^{s}\partial_{y}u_{\theta}))
\geq C\parallel u_{\theta}\parallel_{(1,1)}^{2},$$ from which uniqueness follows. Q.E.D.

This corollary yields the existence of a regular solution to (4.2) when $\phi=\psi=0$ and $f$ vanishes to high order on $\partial\Omega_{1}$. However, we are interested in solving (4.2) in the general case when $\phi$, $\psi$, 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 $\overline{H}_{0}^{m}(\Omega)$ will denote the completion of $\overline{C}^{\infty}(\Omega)$ in the norm $\parallel\cdot\parallel_{m}$, and $\overline{H}_{0}^{m}(\partial\Omega_{1})$ will be defined similarly with respect to the $\parallel\cdot\parallel_{m,\partial\Omega_{1}}$ norm. Recall that $\overline{C}^{\infty}(\Omega)$ consists of $C^{\infty}(\Omega)$ functions which vanish in a neighborhood of the origin.

**Lemma 4.2.** *Suppose that $g\in C^{m_{*}}$, $|w|_{C^{6}}<1$, and $\phi\in\overline{H}_{0}^{m+1}(\partial\Omega_{1})$, $\psi\in\overline{H}_{0}^{m}(\partial\Omega_{1})$, $f\in\overline{H}_{0}^{m}(\Omega)$ with $m\leq m_{*}-6$. Then there exists a function $\eta_{\theta}\in\overline{H}_{0}^{m+2}(\Omega)$ such that $\eta_{\theta}|_{\partial\Omega_{1}}=\phi$, $\partial_{y}\eta_{\theta}|_{\partial\Omega_{1}}=\psi$, and $\partial_{y}^{t}(f-L_{\theta}\eta_{\theta})|_{\partial\Omega_{1}}=0$, $0\leq t\leq m-1$, with* $$\begin{aligned}
\parallel \eta_{\theta}\parallel_{m+2}&\leq& C_{m}(\parallel
f\parallel_{m}+\parallel\phi\parallel_{m+1,\partial\Omega_{1}}
+\parallel\psi\parallel_{m,\partial\Omega_{1}}\\
& &+\parallel w\parallel_{m+6}(\parallel
f\parallel_{2}+\parallel\phi\parallel_{2,\partial\Omega_{1}}
+\parallel\psi\parallel_{2,\partial\Omega_{1}})).\end{aligned}$$

*Proof.* We may assume that a unique solution $u_{\theta}\in\overline{H}_{0}^{m+2}(\Omega)$ of (4.2) exists, since here we shall only use its boundary values which can be explicitly determined in terms of $\phi$, $\psi$, and $f$, as $\partial\Omega_{1}$ is noncharacteristic for $L_{\theta}$. Then because $\Omega$ is a Lipschitz domain, the linear restriction map $H^{m+2}(\Omega)\rightarrow H^{m+1}(\partial\Omega_{1})$ is bounded and onto (see \[18\]). By quotienting with the kernel and applying the closed graph theorem, we obtain a bounded inverse $H^{m+1}(\partial\Omega_{1})\rightarrow
H^{m+2}(\Omega)/H^{m+2}_{0}(\Omega)$ with respect to the quotient norm. We may then use this map to obtain an extension $\eta_{\theta}$ of $u_{\theta}$ from $\partial\Omega_{1}$ to $\Omega$ with $\partial^{\alpha}\eta_{\theta}|_{\partial\Omega_{1}}=\partial^{\alpha}u_{\theta}|_{\partial\Omega_{1}}$ for all $|\alpha|\leq m+1$, and $$\parallel \eta_{\theta}\parallel_{m+2}\leq
C_{m}\sum_{|\alpha|\leq
m+1}\parallel\partial^{\alpha}u_{\theta}\parallel_{0,\partial\Omega_{1}}.$$ Applying the Gagliardo-Nirenberg inequalities (Lemma 5.2) to the expression for $\partial^{\alpha}u_{\theta}|_{\partial\Omega_{1}}$ in terms of $\phi$, $\psi$, and $f$ yields the desired estimate.

We remark that an equivalent and perhaps more concrete way to obtain the extension is as the unique weak solution $\eta_{\theta}\in\overline{H}_{0}^{m+2}(\Omega)$ of the boundary value problem: $$\sum_{s=0}^{m+2}(-1)^{s}\Delta^{s}\eta_{\theta}=0\text{ }\text{
}\text{ in }\text{ }\text{ }\Omega,\text{ }\text{ }\text{ }\text{
}\partial_{y}^{s}\eta_{\theta}|_{\partial\Omega_{1}}
=\partial_{y}^{s}u_{\theta}|_{\partial\Omega_{1}},\text{ }\text{
}\text{ }\text{ }0\leq s\leq m+1,$$ $$\left(\sum_{l=s}^{m+1-s}(-1)^{l}\partial_{y}\Delta^{l}\eta_{\theta}\right)
_{\partial\Omega_{2}}=0,\text{ }\text{ }\text{ }\text{ }0\leq
s\leq\left[\frac{m+1}{2}\right],$$ $$\left(\sum_{l=s+1}^{m+1-s}(-1)^{l}\Delta^{l}\eta_{\theta}\right)
_{\partial\Omega_{2}}=0,\text{ }\text{ }\text{ }\text{ }0\leq
s\leq\left[\frac{m+(-1)^{m}}{2}\right].$$ Q.E.D.

Now in order to solve (4.2) with $\phi\in\overline{H}_{0}^{m+1}(\partial\Omega_{1})$, $\psi\in\overline{H}_{0}^{m}(\partial\Omega_{1})$, and $f\in\overline{H}_{0}^{m}(\Omega)$, we note that if $\eta_{\theta}\in\overline{H}_{0}^{m+2}(\Omega)$ is as in Lemma 4.2 and $\overline{u}_{\theta}\in H_{0}^{m}(\Omega)$ is given by Corollary 4.1 with $\overline{f}_{\theta}=f-L_{\theta}\eta_{\theta}\in
H_{0}^{m}(\Omega)$, then $u_{\theta}=\overline{u}_{\theta}+\eta_{\theta}\in\overline{H}_{0}^{m}(\Omega)$ satisfies (4.2). Observe that in order for $u_{\theta}$ to have the desired vanishing at the origin we require $m\geq\alpha_{0}+2$.

Our next task shall be to estimate $u_{\theta}$ independent of $\theta$, 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.

**Theorem 4.2.** *Suppose that $g\in C^{m_{*}}$, $\phi,\psi\in\overline{C}^{\infty}(\partial\Omega_{1})$, $f\in\overline{C}^{\infty}(\Omega)$, and $|w|_{C^{2N+4}}<1$ where $N\leq m_{*}-2$ is the largest integer such that $\partial^{\alpha}K(0,0)=0$ for all $|\alpha|\leq N$. If $\alpha_{0}+2\leq m\leq m_{*}-6$ and $\varepsilon=\varepsilon(m)$ is sufficiently small, then there exists a unique solution $u_{\theta}\in\overline{H}_{0}^{m}(\Omega)$ of (4.2) for each $\theta>0$. Furthermore there exists a constant $C_{m}$ independent of $\varepsilon$ and $\theta$ such that* $$\begin{aligned}
\parallel u_{\theta}\parallel_{m}&\leq& C_{m}(\parallel
f\parallel_{m+N}+\parallel\phi\parallel_{m+N+1,\partial\Omega_{1}}
+\parallel\psi\parallel_{m+N,\partial\Omega_{1}}\\
& &+\parallel w\parallel_{m+N+6}(\parallel
f\parallel_{N+2}+\parallel\phi\parallel_{N+3,\partial\Omega_{1}}
+\parallel\psi\parallel_{N+2,\partial\Omega_{1}}))\end{aligned}$$ *for each $m\leq m_{*}-N-8$.*

*Proof.* The existence of a solution $u_{\theta}\in\overline{H}_{0}^{m_{*}-6}(\Omega)$ for each $\theta>0$ 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 $L_{\theta}$. Therefore we set $v_{\theta}=e^{-\frac{1}{2}y^{2}}u_{\theta}$ and observe that $$\overline{L}_{\theta}v_{\theta}:=\partial_{x}(\overline{K}_{\theta}\partial_{x}v_{\theta})
+\partial_{y}^{2}v_{\theta}+\overline{C}\partial_{x}v_{\theta}+(2y+\overline{D})
\partial_{y}v_{\theta}+(1+y^{2}+y\overline{D})v_{\theta}=e^{-\frac{1}{2}y^{2}}f:=\overline{f}.$$ With the aim of treating the $x$-derivatives first, we differentiate the above equation to find $$\begin{aligned}
\overline{L}_{\theta}^{(m)}\partial_{x}^{m}v_{\theta}&=&\partial_{x}^{m}\overline{f}
-\sum_{s=3}^{m}\left(\begin{array}{c}
m \\
s
\end{array}\right)\partial_{x}^{s}\overline{K}_{\theta}\partial_{x}^{m-s}
(\partial_{x}^{2}v_{\theta})\\
& &-\sum_{s=2}^{m}\left(\begin{array}{c}
m \\
s
\end{array}\right)\partial_{x}^{s}(\overline{C}+\partial_{x}\overline{K}_{\theta})
\partial_{x}^{m-s}(\partial_{x}v_{\theta})\\
& &-\sum_{s=1}^{m}\left(\begin{array}{c}
m \\
s
\end{array}\right)[\partial_{x}^{s}\overline{D}\partial_{x}^{m-s}(\partial_{y}v_{\theta})
+y\partial_{x}^{s}\overline{D}\partial_{x}^{m-s}v_{\theta}]\\
&:=&\partial_{x}^{m}\overline{f}+\overline{f}^{(m)}(v_{\theta}),\end{aligned}$$ where $$\begin{aligned}
\overline{L}_{\theta}^{(m)}v&:=&(\overline{K}_{\theta}v_{x})_{x}+v_{yy}
+(\overline{C}+m\partial_{x}\overline{K}_{\theta})v_{x}
+(2y+\overline{D})v_{y}\\
&
&+\left(1+y^{2}+y\overline{D}+m\partial_{x}\overline{C}+\frac{m(m+1)}{2}\partial_{x}^{2}
\overline{K}_{\theta}\right)v.\end{aligned}$$

We first assume that $m\leq N+1$. In this case let $\eta_{\theta}$ be given by Lemma 4.2 such that $\eta_{\theta}|_{\partial\Omega_{1}}=v_{\theta}|_{\partial\Omega_{1}}$, $\partial_{y}\eta_{\theta}|_{\partial\Omega_{1}}=\partial_{y}v_{\theta}|_{\partial\Omega_{1}}$ and $$\partial_{y}^{t}(\overline{f}-\overline{L}_{\theta}\eta_{\theta})|_{\partial\Omega_{1}}=0,\text{
}\text{ }\text{ }\text{ }0\leq t\leq m+N.$$ Note that this implies that $\eta_{\theta}\in\overline{H}_{0}^{m+N+3}(\Omega)$, $\partial^{\alpha}\eta_{\theta}|_{\partial\Omega_{1}}=\partial^{\alpha}v_{\theta}|_{\partial\Omega_{1}}$ for all $|\alpha|\leq m+N+2$, and we have the estimate $$\begin{aligned}
\parallel\eta_{\theta}\parallel_{m+N+3}&\leq& C(\parallel
f\parallel_{m+N+1}+\parallel\phi\parallel_{m+N+2,\partial\Omega_{1}}
+\parallel\psi\parallel_{m+N+1,\partial\Omega_{1}}\\
& &+\parallel w\parallel_{m+N+7}(\parallel
f\parallel_{2}+\parallel\phi\parallel_{2,\partial\Omega_{1}}
+\parallel\psi\parallel_{2,\partial\Omega_{1}})).\nonumber\end{aligned}$$ Furthermore the function $\overline{v}_{\theta}:=v_{\theta}-\eta_{\theta}$ satisfies $$\overline{L}_{\theta}^{(m)}\partial_{x}^{m}\overline{v}_{\theta}=
\partial_{x}^{m}(\overline{f}-\overline{L}_{\theta}\eta_{\theta})
+\overline{f}^{(m)}(\overline{v}_{\theta}).$$ As in the proof of Theorem 4.1, we set $b=\overline{K}_{\theta}^{-1}e^{-\lambda y}$ and integrate by parts: $$\begin{aligned}
& &\!\!\!(-b\partial_{y}\partial_{x}^{m}\overline{v}_{\theta},
\overline{L}_{\theta}^{(m)}\partial_{x}^{m}\overline{v}_{\theta})\\
&=&\!\!\!\int_{\Omega}\left[-\frac{1}{2}(b\overline{K}_{\theta})_{y}(\partial_{x}^{m+1}\overline{v}_{\theta})^{2}
+(b_{x}\overline{K}_{\theta}-mb\partial_{x}\overline{K}_{\theta}-b\overline{C})\partial_{x}^{m+1}
\overline{v}_{\theta}
\partial_{y}\partial_{x}^{m}\overline{v}_{\theta}\right]\nonumber\\
& &\!\!\!+\int_{\Omega}
\left(\frac{1}{2}b_{y}-b(2y+\overline{D})\right)(\partial_{y}\partial_{x}^{m}\overline{v}_{\theta})^{2}\nonumber\\
& &\!\!\!+\int_{\Omega}
\frac{1}{2}\left[b(1+y^{2}+y\overline{D}+m\partial_{x}\overline{C}+\frac{m(m+1)}{2}\partial_{x}^{2}\overline{K}
_{\theta})\right]_{y}(\partial_{x}^{m}\overline{v}_{\theta})^{2}\nonumber\\
&
&\!\!\!+\int_{\partial\Omega}\left[\frac{1}{2}b\overline{K}_{\theta}(\partial_{x}^{m+1}\overline{v}_{\theta})^{2}\nu_{2}
-b\overline{K}_{\theta}\partial_{x}^{m+1}\overline{v}_{\theta}\partial_{y}\partial_{x}^{m}
\overline{v}_{\theta}\nu_{1}\right]\nonumber\\
&
&\!\!\!-\int_{\partial\Omega}\left[\frac{1}{2}b(\partial_{y}\partial_{x}^{m}\overline{v}_{\theta})^{2}\nu_{2}
+\frac{1}{2}b\left(1+y^{2}+y\overline{D}
+m\partial_{x}\overline{C}+\frac{m(m+1)}{2}\partial_{x}^{2}\overline{K}_{\theta}\right)
(\partial_{x}^{m}\overline{v}_{\theta})^{2}\nu_{2}\right].\nonumber\end{aligned}$$ The boundary integral along $\partial\Omega_{2}$ is nonnegative, and according to the choice of $\eta_{\theta}$ it vanishes along $\partial\Omega_{1}$. Moreover, the same calculations as in the proof of Theorem 4.1 apply to the interior integral to yield $$\begin{aligned}
&
&\lambda(\parallel\partial_{x}^{m+1}\overline{v}_{\theta}\parallel+
\parallel\sqrt{|b|}\partial_{y}\partial_{x}^{m}\overline{v}_{\theta}\parallel
+\parallel\sqrt{|b|}\partial_{x}^{m}\overline{v}_{\theta}\parallel)\\
&\leq&
C(\parallel\sqrt{|b|}\partial_{x}^{m}(\overline{f}-\overline{L}_{\theta}\eta_{\theta})
\parallel
+\parallel\sqrt{|b|}\overline{f}^{(m)}(\overline{v}_{\theta})\parallel).\end{aligned}$$

We proceed to estimate each term on the right-hand side separately. First note that since $m\leq N+1$ and $|w|_{C^{N+4}}<1$, we have $$\parallel\sqrt{|b|}\overline{f}^{(m)}(\overline{v}_{\theta})\parallel^{2}
\leq C_{m}\int_{\Omega}\sum_{s=0}^{m-1}e^{-\lambda
y}|\overline{K}_{\theta}|^{-1}[(\partial_{x}^{s}
\overline{v}_{\theta})^{2}+(\partial_{y}\partial_{x}^{s}
\overline{v}_{\theta})^{2}].$$ Next observe that since $\overline{K}$ vanishes (at most) to order $N$ at the origin, there exists a constant $C_{0}>0$ such that $|\overline{K}|\geq C_{0}^{-1}(y-h(x))^{N+1}$ in $\Omega$. Then in light of (4.11), a little calculation shows that $$\begin{aligned}
\parallel\sqrt{|b|}\partial_{x}^{m}(\overline{f}-\overline{L}_{\theta}\eta_{\theta})\parallel^{2}
&\leq&
C_{0}\int_{\Omega}\frac{[\partial_{x}^{m}(\overline{f}-\overline{L}_{\theta}\eta_{\theta})]^{2}}{(y-h(x))^{N+1}}\\
&\leq&
C_{1}\int_{\Omega}[\partial_{y}^{N+1}\partial_{x}^{m}(\overline{f}-\overline{L}_{\theta}\eta_{\theta})]^{2}.\end{aligned}$$ It follows that applying (4.12) and summing from 0 to $m$ produces $$\begin{aligned}
&
&\sum_{s=0}^{m+1}\parallel\partial_{x}^{s}\overline{v}_{\theta}\parallel
+\sum_{s=0}^{m}\parallel\sqrt{|b|}\partial_{y}\partial_{x}^{s}\overline{v}_{\theta}\parallel
+\sum_{s=0}^{m}\parallel\sqrt{|b|}\partial_{x}^{s}\overline{v}_{\theta}\parallel\\
&\leq& C_{m}(\parallel f\parallel_{m+N+1}+\parallel
\phi\parallel_{m+N+2,\partial\Omega_{1}}+\parallel
\psi\parallel_{m+N+1,\partial\Omega_{1}}\nonumber\\
& &+\parallel w\parallel_{m+N+7}(\parallel
f\parallel_{2}+\parallel\phi\parallel_{2,\partial\Omega_{1}}
+\parallel\psi\parallel_{2,\partial\Omega_{1}})),\nonumber\end{aligned}$$ if $\lambda$ is sufficiently large. From this inequality we may obtain an estimate for the $x$-derivatives of $v_{\theta}$ (and hence for $u_{\theta}$), with the help of (4.12). In addition, by solving for $\partial_{y}^{2}u_{\theta}$ in equation (4.2) all remaining derivatives up to and including order $m$ may also be estimated.

We now assume that $m\geq N+2$. In order to isolate terms involving high order derivatives of $w$ we break $\overline{f}^{(m)}$ into two parts. Let $v_{\theta}^{(s)}:=\partial_{x}^{s}v_{\theta}-\eta_{\theta}^{(s)}$ with $\eta_{\theta}^{(s)}$ to be given below, then $\overline{f}^{(m)}=\overline{f}_{1}^{(m)}+\overline{f}_{2}^{(m)}$ where $$\begin{aligned}
\overline{f}_{1}^{(m)}(v_{\theta})&=&
-\sum_{s=N+2}^{m}\left(\begin{array}{c}
m \\
s
\end{array}\right)\partial_{x}^{s}\overline{K}_{\theta}\partial_{x}^{m-s+2}v_{\theta}
-\sum_{s=N+1}^{m}\left(\begin{array}{c}
m \\
s
\end{array}\right)\partial_{x}^{s}(\overline{C}+\partial_{x}\overline{K}_{\theta})
\partial_{x}^{m-s+1}v_{\theta}\\
& &-\sum_{s=N+1}^{m}\left(\begin{array}{c}
m \\
s
\end{array}\right)\partial_{x}^{s}\overline{D}\partial_{y}\partial_{x}^{m-s}v_{\theta}
-y\sum_{s=N}^{m}\left(\begin{array}{c}
m \\
s
\end{array}\right)\partial_{x}^{s}\overline{D}\partial_{x}^{m-s}v_{\theta}\\
& &-\sum_{s=3}^{N+1}\left(\begin{array}{c}
m \\
s
\end{array}\right)\partial_{x}^{s}\overline{K}_{\theta}\eta_{\theta}^{(m-s+2)}
-\sum_{s=2}^{N}\left(\begin{array}{c}
m \\
s
\end{array}\right)\partial_{x}^{s}(\overline{C}+\partial_{x}\overline{K}_{\theta})
\eta_{\theta}^{(m-s+1)}\\
& &-\sum_{s=1}^{N}\left(\begin{array}{c}
m \\
s
\end{array}\right)\partial_{x}^{s}\overline{D}\partial_{y}\eta_{\theta}^{(m-s)}
-y\sum_{s=1}^{N-1}\left(\begin{array}{c}
m \\
s
\end{array}\right)\partial_{x}^{s}\overline{D}\eta_{\theta}^{(m-s)}\end{aligned}$$ and $$\begin{aligned}
\overline{f}_{2}^{(m)}(v_{\theta})&=&
-\sum_{s=3}^{N+1}\left(\begin{array}{c}
m \\
s
\end{array}\right)\partial_{x}^{s}\overline{K}_{\theta}v_{\theta}^{(m-s+2)}
-\sum_{s=2}^{N}\left(\begin{array}{c}
m \\
s
\end{array}\right)\partial_{x}^{s}(\overline{C}+\partial_{x}\overline{K}_{\theta})
v_{\theta}^{(m-s+1)}\\
& &-\sum_{s=1}^{N}\left(\begin{array}{c}
m \\
s
\end{array}\right)\partial_{x}^{s}\overline{D}\partial_{y}v_{\theta}^{(m-s)}
-y\sum_{s=1}^{N-1}\left(\begin{array}{c}
m \\
s
\end{array}\right)\partial_{x}^{s}\overline{D}v_{\theta}^{(m-s)}.\end{aligned}$$ The functions $\eta_{\theta}^{(s)}\in\overline{H}_{0}^{N+3}(\Omega)$, $0\leq
s\leq m$, are defined recursively in the following way. For $0\leq s\leq N+1$ we set $\eta_{\theta}^{(s)}=\partial_{x}^{s}\eta_{\theta}$, and if $N+2\leq s\leq m$ we apply Lemma 4.2 to obtain $\eta_{\theta}^{(s)}$ such that $\eta_{\theta}^{(s)}|_{\partial\Omega_{1}}=\partial_{x}^{s}v_{\theta}|_{\partial\Omega_{1}}$, $\partial_{y}\eta_{\theta}^{(s)}|_{\partial\Omega_{1}}
=\partial_{y}\partial_{x}^{s}v_{\theta}|_{\partial\Omega_{1}}$ with $$\partial_{y}^{t}(\partial_{x}^{s}\overline{f}+\overline{f}^{(s)}_{1}(v_{\theta})
-\overline{L}_{\theta}^{(s)}\eta_{\theta}^{(s)})|_{\partial\Omega_{1}}=0,\text{
}\text{ }\text{ }\text{ }0\leq t\leq N.$$ Note that since $\overline{f}^{(s)}_{1}\in\overline{H}_{0}^{\min(m_{*}-s-6,N+2)}(\Omega)$ and $\partial_{x}^{s}v_{\theta}|_{\partial\Omega_{1}}\in\overline{H}_{0}^{m_{*}-s-7}(\partial\Omega_{1})$, $\partial_{y}\partial_{x}^{s}v_{\theta}|_{\partial\Omega_{1}}\in\overline{H}_{0}^{m_{*}-s-8}(\partial\Omega_{1})$, we must have $N+1\leq m_{*}-s-8$ for the construction of $\eta_{\theta}^{(s)}$ to be valid. In this case the following estimate holds $$\begin{aligned}
\parallel\eta_{\theta}^{(s)}\parallel_{N+3}\!\!\!&\leq&\!\!\!
C(\parallel\partial_{x}^{s}\overline{f}+\overline{f}^{(s)}_{1}(v_{\theta})\parallel_{N+1}
+\parallel\partial_{x}^{s}v_{\theta}\parallel_{N+2,\partial\Omega_{1}}
+\parallel\partial_{y}\partial_{x}^{s}v_{\theta}\parallel_{N+1,\partial\Omega_{1}}\\
& &\!\!\!+\parallel
w\parallel_{N+7}(\parallel\partial_{x}^{s}\overline{f}+\overline{f}^{(s)}_{1}(v_{\theta})\parallel_{2}
+\parallel\partial_{x}^{s}v_{\theta}\parallel_{2,\partial\Omega_{1}}
+\parallel\partial_{y}\partial_{x}^{s}v_{\theta}\parallel_{2,\partial\Omega_{1}}))\nonumber\\
&\leq&\!\!\!
C_{s}(\parallel\overline{f}^{(s)}_{1}(v_{\theta})\parallel_{N+1}+\parallel
f\parallel_{s+N+1}+\parallel\phi\parallel_{s+N+2,\partial\Omega_{1}}
+\parallel\psi\parallel_{s+N+1,\partial\Omega_{1}}\nonumber\\
& &\!\!\!+\parallel w\parallel_{s+N+7}(\parallel
f\parallel_{2}+\parallel\phi\parallel_{2,\partial\Omega_{1}}
+\parallel\psi\parallel_{2,\partial\Omega_{1}})),\nonumber\end{aligned}$$ where we have used $|w|_{C^{N+7}}<1$ and the proof of Lemma 4.2 to estimate $\partial_{y}^{t}\partial_{x}^{s}v_{\theta}|_{\partial\Omega_{1}}$, $t=0,1$.

Observe that $v_{\theta}^{(s)}$, $N+2\leq s\leq m$, satisfies $$\overline{L}_{\theta}^{(s)}v_{\theta}^{(s)}
=(\partial_{x}^{s}\overline{f}+\overline{f}^{(s)}_{1}(v_{\theta})-\overline{L}_{\theta}^{(s)}\eta_{\theta}^{(s)})
+\overline{f}^{(s)}_{2}(v_{\theta}).$$ Therefore (4.13) applies to yield $$\begin{aligned}
& &\lambda(\parallel\partial_{x}v_{\theta}^{(s)}\parallel+
\parallel\sqrt{|b|}\partial_{y}v_{\theta}^{(s)}\parallel
+\parallel\sqrt{|b|}v_{\theta}^{(s)}\parallel)\\
&\leq&
C(\parallel\sqrt{|b|}(\partial_{x}^{s}\overline{f}+\overline{f}_{1}^{(s)}(v_{\theta})
-\overline{L}^{(s)}_{\theta}\eta_{\theta}^{(s)})\parallel
+\parallel\sqrt{|b|}\overline{f}_{2}^{(s)}(v_{\theta})\parallel).\nonumber\end{aligned}$$ 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 $$\begin{aligned}
\parallel\sqrt{|b|}\overline{f}_{2}^{(s)}(v_{\theta})\parallel^{2}
&\leq& C_{s}\int_{\Omega}\sum_{l=0}^{s-1}e^{-\lambda
y}|\overline{K}_{\theta}|^{-1}[(
v_{\theta}^{(l)})^{2}+(\partial_{y}v_{\theta}^{(l)})^{2}]\\
&=&C_{s}\int_{\Omega}\sum_{l=N+2}^{s-1}e^{-\lambda
y}|\overline{K}_{\theta}|^{-1}[(
v_{\theta}^{(l)})^{2}+(\partial_{y}v_{\theta}^{(l)})^{2}]\nonumber\\
& &+C_{s}\int_{\Omega}\sum_{l=0}^{N+1}e^{-\lambda
y}|\overline{K}_{\theta}|^{-1}[(\partial_{x}^{l}
\overline{v}_{\theta})^{2}+(\partial_{y}\partial_{x}^{l}
\overline{v}_{\theta})^{2}].\nonumber\end{aligned}$$ Next observe again that since $|\overline{K}|\geq
C_{0}^{-1}(y-h(x))^{N+1}$ in $\Omega$, (4.15) implies that $$\begin{aligned}
\parallel\sqrt{|b|}(\partial_{x}^{s}\overline{f}+\overline{f}_{1}^{(s)}(v_{\theta})
-\overline{L}^{(s)}_{\theta}\eta_{\theta}^{(s)})\parallel^{2}
&\leq&
C_{0}\int_{\Omega}\frac{[\partial_{x}^{s}\overline{f}+\overline{f}_{1}^{(s)}(v_{\theta})
-\overline{L}^{(s)}_{\theta}\eta_{\theta}^{(s)}]^{2}}{(y-h(x))^{N+1}}\\
&\leq&
C_{1}\int_{\Omega}[\partial_{y}^{N+1}(\partial_{x}^{s}\overline{f}+\overline{f}_{1}^{(s)}(v_{\theta})
-\overline{L}^{(s)}_{\theta}\eta_{\theta}^{(s)})]^{2}.\nonumber\end{aligned}$$ Furthermore applying the Gagliardo-Nirenberg inequalities (Lemma 5.2), (4.12), (4.14), (4.16), and using $|w|_{C^{2N+4}}<1$ produces $$\begin{aligned}
\parallel\overline{f}_{1}^{(s)}(v_{\theta})\parallel_{N+1}
&\leq& C_{s}((|\overline{K}_{\theta}|_{C^{N+2}}
+|\overline{C}|_{C^{N+1}}+|\overline{D}|_{C^{N+1}})
\parallel v_{\theta}\parallel_{s+1}\\
& &+(\parallel\overline{K}_{\theta}\parallel_{s+N+3}
+\parallel\overline{C}\parallel_{s+N+2}+
\parallel\overline{D}\parallel_{s+N+2})|v_{\theta}|_{C^{0}})\nonumber\\
&
&+C_{s}\left(\sum_{l=s-N}^{N+1}\parallel\eta_{\theta}\parallel_{l+N+2}
+\sum_{l=N+2}^{s-1}\parallel\eta_{\theta}^{(l)}\parallel_{N+2}\right)\nonumber\\
&\leq& C_{s}(\parallel f\parallel_{s+N+1}+\varepsilon\parallel
v_{\theta}\parallel_{s+1}+\parallel\phi\parallel_{s+N+2,\partial\Omega_{1}}
+\parallel\psi\parallel_{s+N+1,\partial\Omega_{1}}\nonumber\\
& &+\parallel w\parallel_{s+N+7}(\parallel
f\parallel_{N+2}+\parallel\phi\parallel_{N+3,\partial\Omega_{1}}
+\parallel\psi\parallel_{N+2,\partial\Omega_{1}})).\nonumber\end{aligned}$$ It follows that we may combine (4.17)-(4.20) and utilize (4.12), (4.14), as well as (4.16) to obtain $$\begin{aligned}
& &\sum_{s=N+2}^{m}\parallel
\partial_{x}v_{\theta}^{(s)}\parallel
+\sum_{s=N+2}^{m}\parallel\sqrt{|b|}\partial_{y}v_{\theta}^{(s)}\parallel
+\sum_{s=N+2}^{m}\parallel\sqrt{|b|}v_{\theta}^{(s)}\parallel\\
&\leq& C_{m}(\parallel f\parallel_{m+N+1}+\varepsilon\parallel
u_{\theta}\parallel_{m+1}+\parallel
\phi\parallel_{m+N+2,\partial\Omega_{1}}+\parallel
\psi\parallel_{m+N+1,\partial\Omega_{1}}\nonumber\\
& &+\parallel w\parallel_{m+N+7}(\parallel
f\parallel_{N+2}+\parallel\phi\parallel_{N+3,\partial\Omega_{1}}
+\parallel\psi\parallel_{N+2,\partial\Omega_{1}})).\nonumber\end{aligned}$$ Since $v_{\theta}^{(s)}=\partial_{x}^{s}v_{\theta}-\eta_{\theta}^{(s)}$, with the help of (4.16) the above inequality yields an estimate for the $x$-derivatives of $u_{\theta}$. The remaining derivatives of $u_{\theta}$ may be estimated in the usual way, by solving for $\partial_{y}^{2}u_{\theta}$ from equation (4.2). Lastly taking $\varepsilon=\varepsilon(m)$ sufficiently small yields the desired result. Q.E.D.

**Remark 4.1.** *In this section we have focused on the Cauchy problem in the domains $\Omega_{\varrho}^{-}\cap
B_{\sigma}(0)$, $\varrho=1,2$. However analogous existence and regularity results, requiring only slight modifications of the arguments above, hold for the Cauchy problem in the domains $\Omega_{\varrho}^{-}\cap B_{\sigma}(0)$, $\varrho=3,4$, 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 $x=h(y)$).*

**5. 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 $$\Omega_{\kappa}^{+}=\{(\xi^{1},\xi^{2})\mid
0<\xi^{2}<(\tan\delta)\xi^{1},\!\text{ } |\xi|<\sigma\},\text{
}\text{ }\text{ }\text{ }1\leq\kappa\leq\kappa_{0},$$ each hyperbolic region is given by $$\Omega_{\varrho}^{-}=\{(\xi^{1},\xi^{2})\mid
h(\xi^{1})<\xi^{2}<\sigma\},\text{ }\text{ }\text{ }\text{
}\varrho=1,2,$$ or $$\Omega_{\varrho}^{-}= \{(\xi^{1},\xi^{2})\mid
h(\xi^{2})<\xi^{1}<\sigma\},\text{ }\text{ }\text{ }\text{
}\varrho=3,4.$$ If $\overline{\partial}\Omega_{\kappa}^{+}$ denotes the portion of the boundary consisting of the curves $\xi^{2}=0$ and $\xi^{2}=(\tan\delta)\xi^{1}$, $\overline{\partial}\Omega_{\varrho}^{-}$, $\varrho=1,2$, denotes the portion of the boundary given by $\xi^{2}=h(\xi^{1})$, and $\overline{\partial}\Omega_{\varrho}^{-}$, $\varrho=3,4$, denotes either the upper or lower part of the boundary curve $\xi^{1}=h(\xi^{2})$, then we aim to solve $$\Phi(w_{\kappa}^{+})=0\text{ }\text{ }\text{ in }\text{ }\text{
}\Omega_{\kappa}^{+},\text{ }\text{ }\text{ }\text{
}w_{\kappa}^{+}|_{\overline{\partial}\Omega_{\kappa}^{+}}=0,$$ $$\partial^{\alpha}w_{\kappa}^{+}(0,0)=0,\text{ }\text{ }\text{
}\text{ }|\alpha|\leq\alpha_{0},$$ for each $\kappa=1,\ldots,\kappa_{0}$, and $$\Phi(w_{\varrho}^{-})=0\text{ }\text{ }\text{ in }\text{ }\text{
}\Omega_{\varrho}^{-},\text{ }\text{ }\text{ }\text{
}w_{\varrho}^{-}|_{\overline{\partial}\Omega_{\varrho}^{-}}=\phi_{\varrho}^{-},\text{
}\text{ }\text{ }\text{
}\partial_{\nu}w_{\varrho}^{-}|_{\overline{\partial}\Omega_{\varrho}^{-}}=\psi_{\varrho}^{-},$$ $$\partial^{\alpha}w_{\varrho}^{-}(0,0)=0,\text{ }\text{ }\text{
}\text{ }|\alpha|\leq\alpha_{0},$$ for each $\varrho=1,\ldots,\varrho_{0}$, where $\partial_{\nu}$ denotes the outward normal derivative, $\alpha_{0}$ is a large integer, and $\phi_{\varrho}^{-}$, $\psi_{\varrho}^{-}$ 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 $\S 3$ and $\S
4$ 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.

<span style="font-variant:small-caps;">Hyperbolic Regions</span>

In the hyperbolic regions, the linearization $\mathcal{L}(w)$ and the operator $L_{\theta}(w)$ that we invert in section $\S 4$ differ by perturbation terms coming from Lemma 2.1 as well as a regularizing term involving $\theta=|\Phi(w)|_{C^{1}}$. More precisely according to Lemma 2.1 $$\begin{aligned}
\mathcal{L}(w)u&=&\varepsilon
a^{22}(w)L_{\theta}(w)u+\varepsilon\theta
a^{22}(w)\partial_{\xi^{1}}^{2}u\\
&
&+\varepsilon(a^{22}(w))^{-1}\Phi(w)[\partial_{x^{1}}^{2}u-\partial_{x^{1}}\log
(a^{22}(w)\sqrt{|g|})\partial_{x^{1}}u],\nonumber\end{aligned}$$ where $\xi^{i}(x^{1},x^{2})$ 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 $\S
4$ is based on the Sobolev spaces $\overline{H}_{0}^{m}$, the smoothing operators we employ should respect these spaces. For convenience, in the remainder of this section the hyperbolic region in question $\Omega_{\varrho}^{-}$ will be denoted by $\Omega$, and the $\overline{H}_{0}^{m}(\Omega)$ norm will be denoted by $\parallel\cdot\parallel_{m}$.

**Lemma 5.1.** *Given $\mu\geq 1$ there exists a linear smoothing operator $S_{\mu}:L^{2}(\Omega)\rightarrow\overline{H}_{0}^{\infty}(\Omega)$ such that for all $l,m\in\mathbb{Z}_{\geq 0}$ and $u\in
\overline{H}_{0}^{l}(\Omega)$,*

$i)$ $\parallel S_{\mu}u\parallel_{m}\leq C_{l,m}\parallel
u\parallel_{l}$, $m\leq l$,

$ii)$ $\parallel S_{\mu}u\parallel_{m}\leq
C_{l,m}\mu^{m-l}\parallel u\parallel_{l}$, $l\leq m$,

$iii)$ $\parallel u- S_{\mu}u\parallel_{m}\leq
C_{l,m}\mu^{m-l}\parallel u\parallel_{l}$, $m\leq l$.

*Furthermore there exists a linear smoothing operator $S^{'}_{\mu}:L^{2}(\Omega)\rightarrow H^{\infty}(\Omega)$ such that $(i)$, $(ii)$, and $(iii)$ hold whenever $u\in
H^{l}(\Omega)$.*

*Proof.* See appendix B. Q.E.D.

The next lemma contains the so called Gagliardo-Nirenberg inequalities, which will be used frequently throughout this section.

**Lemma 5.2.** *Let $u,v\in
C^{k}(\overline{\Omega})$.*

$i)$ *If $\alpha$ and $\beta$ are multi-indices such that $|\alpha|+|\beta|=m$, then there exists a constant $C_{1}$ depending on $m$ such that* $$\parallel\partial^{\alpha}u\partial^{\beta}v\parallel_{L^{2}(\Omega)}\leq
C_{1}(|u|_{L^{\infty}(\Omega)}\parallel
v\parallel_{H^{m}(\Omega)}+\parallel
u\parallel_{H^{m}(\Omega)}|v|_{L^{\infty}(\Omega)}).$$

$ii)$ *If $\alpha_{1},\ldots,\alpha_{l}$ are multi-indices such that $|\alpha_{1}|+\cdots+|\alpha_{l}|=m$, then there exists a constant $C_{2}$ depending on $l$ and $m$ such that* $$\parallel\partial^{\alpha_{1}}u_{1}\cdots\partial^{\alpha_{l}}u_{l}
\parallel_{L^{2}(\Omega)}\leq
C_{2}\sum_{j=1}^{l}(|u_{1}|_{L^{\infty}(\Omega)}\cdots
\widehat{|u_{j}|}_{L^{\infty}(\Omega)}
\cdots|u_{l}|_{L^{\infty}(\Omega)})\parallel
u_{j}\parallel_{H^{m}(\Omega)},$$ *where $\widehat{|u_{j}|}_{L^{\infty}(\Omega)}$ indicates the absence of $|u_{j}|_{L^{\infty}(\Omega)}$.*

$iii)$ *Let $\mathcal{D}\subset\mathbb{R}^{l}$ be compact and contain the origin, and let $G\in C^{\infty}(\mathcal{D})$. If $u\in H^{m}(\Omega,\mathcal{D})\cap
L^{\infty}(\Omega,\mathcal{D})$, then there exists a constant $C_{3}$ depending on $m$ such that* $$\parallel G\circ u\parallel_{H^{m}(\Omega)}\leq C_{3}|u|_{L^{\infty}(\Omega)}(|G(0)|+\parallel
u\parallel_{H^{m}(\Omega)}).$$

*Proof.* These estimates are standard consequences of the interpolation inequalities, and may be found in, for instance, \[19\]. Q.E.D.

We now set up the underlying iterative procedure. Suppose that $\phi_{\varrho}^{-}\in\overline{H}_{0}^{m_{*}-m_{0}+1}(\overline{\partial}\Omega)$ and $\psi_{\varrho}^{-}\in\overline{H}_{0}^{m_{*}-m_{0}}(\overline{\partial}\Omega)$ for some $m_{0}\geq 0$. Then according to the proof of Lemma 4.2 there exists $w_{0}\in\overline{H}_{0}^{m_{*}-m_{0}+2}(\Omega)$ such that $$w_{0}|_{\overline{\partial}\Omega}=\phi_{\varrho}^{-},\text{
}\text{ }\text{ }\text{
}\partial_{\nu}w_{0}|_{\overline{\partial}\Omega}=\psi_{\varrho}^{-}.$$ Now suppose that in addition to $w_{0}$, functions $w_{1},w_{2},\ldots,w_{n}$ have been defined on $\Omega$, and put $v_{i}=S_{i}w_{i}$, $0\leq i\leq n$, where $S_{i}=S_{\mu^{i}}$. Then define $w_{n+1}=w_{n}+u_{n}$ where $u_{n}$ is the unique solution of $$L_{\theta_{n}}(v_{n})u_{n}=f_{n}\text{ }\text{ }\text{ in }\text{
}\text{ }\Omega,\text{ }\text{ }\text{
}u_{n}|_{\overline{\partial}\Omega}=\partial_{\nu}u_{n}|_{\overline{\partial}\Omega}=0,$$ given by Theorem 4.2, where $\theta_{n}=|\Phi(v_{n})|_{C^{1}}$ and $f_{n}$ will be specified below.

Let $Q_{n}(w_{n},u_{n})$ denote the quadratic error in the Taylor expansion of $\Phi$ at $w_{n}$. Then according to (5.3) we have $$\begin{aligned}
\Phi(w_{n+1})&=&\Phi(w_{n})+\mathcal{L}(w_{n})u_{n}+Q_{n}(w_{n},u_{n})\\
&=&\Phi(w_{n})+\varepsilon
S^{'}_{n}a^{22}(v_{n})L_{\theta_{n}}(v_{n})u_{n}+e_{n},\nonumber\end{aligned}$$ where $$\begin{aligned}
e_{n}&=&(\mathcal{L}(w_{n})-\mathcal{L}(v_{n}))u_{n}+\varepsilon(I-S_{n}^{'})a^{22}(v_{n})L_{\theta_{n}}(v_{n})u_{n}
+\varepsilon\theta_{n}a^{22}(v_{n})\partial_{\xi_{n}^{1}}^{2}u_{n}\\
&
&+Q_{n}(w_{n},u_{n})+\varepsilon(a^{22}(v_{n}))^{-1}\Phi(v_{n})[\partial_{x^{1}}^{2}u_{n}-\partial_{x^{1}}(\log
a^{22}(v_{n})\sqrt{|g|})\partial_{x^{1}}u_{n}],\end{aligned}$$ and $\xi_{n}^{i}$ 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 $\S 4$, we require $f_{n}\in\overline{C}^{\infty}(\Omega)$. 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 $E_{0}=0$, $E_{n}=\sum_{i=0}^{n-1}e_{i}$, and define $$f_{0}=-[\varepsilon S_{0}^{'}a^{22}(v_{0})]^{-1}S_{0}\Phi(w_{0}),$$ $$f_{n}=[\varepsilon
S_{n}^{'}a^{22}(v_{n})]^{-1}(S_{n-1}E_{n-1}-S_{n}E_{n}+(S_{n-1}-S_{n})\Phi(w_{0})).$$ It follows that $$\begin{aligned}
\Phi(w_{n+1})&=&\Phi(w_{0})+\sum_{i=0}^{n}\varepsilon
S_{i}^{'}a^{22}(v_{i})f_{i}+E_{n}+e_{n}\\
&=&(I-S_{n})\Phi(w_{0})+(I-S_{n})E_{n}+e_{n}.\nonumber\end{aligned}$$ 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 $g\in C^{m_{*}}$ and $N$ is as in Theorem 4.2. If $m\leq m_{*}-N-8$, $|v_{n}|_{C^{2N+4}}<1$, and $\varepsilon=\varepsilon(m)$ is sufficiently small then there exists a unique solution $u_{n}\in\overline{H}^{m}_{0}(\Omega)$ of (5.5) which satisfies the estimate* $$\parallel u_{n}\parallel_{m}\leq C_{m}(\parallel
f_{n}\parallel_{m+N}+\parallel v_{n}\parallel_{m+N+6}
\parallel f_{n}\parallel_{N+2}),$$ *for some constant $C_{m}$ independent of $\varepsilon$ and $\theta_{n}$.*

*Proof.* This will follow from Theorem 4.2 with $\phi=\psi\equiv 0$. The only difference is that the Sobolev norms appearing in Theorem 4.2 are with respect to the coordinates $\xi_{n}^{i}(x^{1},x^{2})$ of Lemma 2.1. In order to obtain the current estimate from that of Theorem 4.2 we may utilize (2.4). Note that $\delta$ (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 $\{w_{n}\}_{n=0}^{\infty}$ to a solution of (5.2). Let $\rho$ be a positive number that will be chosen as large as possible, and set $\mu=\varepsilon^{-\frac{1}{2\rho}}$, $\mu_{n}=\mu^{n}$. Furthermore, note that $\Phi(w_{0})\in\overline{H}_{0}^{m_{*}-m_{0}}(\Omega)$. The convergence of $\{w_{n}\}_{n=0}^{\infty}$ will follow from the following eight statements, valid for $0\leq m\leq m_{*}-N-8$ unless specified otherwise. These statements shall be proven by induction on $n$, for some constants $C_{1},\ldots,C_{5}$ independent of $n$ and $\varepsilon$, but dependent on $m$.

I$_{n}$: $\parallel u_{n-1}\parallel_{m}\leq\varepsilon\mu_{n-1}^{m+N+2-\rho}$,

II$_{n}$: $\parallel w_{n}\parallel_{m}\leq\begin{cases}
C_{1}\varepsilon & \text{if $m+N+2-\rho\leq-1/2$},\\
C_{1}\varepsilon\mu_{n}^{m+N+2-\rho} & \text{if $m+N+2-\rho\geq
1/2$,}
\end{cases}$

III$_{n}$: $\parallel w_{n}\parallel_{2N+6}\leq C_{1}\varepsilon,\text{ }
   \text{ }\parallel v_{n}\parallel_{2N+6}\leq
   C_{3}\varepsilon$,

IV$_{n}$: $\parallel w_{n}-v_{n}\parallel_{m}\leq
   C_{2}\varepsilon\mu_{n}^{m+N+2-\rho}$,

V$_{n}$: $\parallel v_{n}\parallel_{m}\leq\begin{cases}
C_{3}\varepsilon & \text{if $m+N+2-\rho\leq-1/2$},\\
C_{3}\varepsilon\mu_{n}^{m+N+2-\rho} & \text{if $m+N+2-\rho\geq
1/2$,}
\end{cases}$ $m<\infty$,

VI$_{n}$: $\parallel e_{n-1}\parallel_{m}\leq
   C_{4}\varepsilon^{3}\mu_{n-1}^{m-\rho}$, $m\leq\min(m_{*}-N-10,m_{*}-m_{0})$,

VII$_{n}$: $\parallel f_{n}\parallel_{m}
   \leq C_{5}\varepsilon^{2}(1+\mu^{\rho-m})\mu_{n}^{m-\rho}$, $m<\infty$,

VIII$_{n}$: $\parallel\Phi(w_{n})\parallel_{m}+\parallel\Phi(v_{n})\parallel_{m}
   \leq\varepsilon\mu_{n}^{m+N+4-\rho}$, $m\leq\min(m_{*}-N-10,m_{*}-m_{0})$.

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.

**Proposition 5.1.** *If $3N+8< \rho< m_{*}-6$, $0\leq m\leq m_{*}-N-8$, and $\varepsilon$ is sufficiently small, then* I$_{n+1}$, II$_{n+1}$, III$_{n+1}$, IV$_{n+1}$, *and* V$_{n+1}$ *hold.*

*Proof.* I$_{n+1}$: First note that by III$_{n}$, $$|v_{n}|_{C^{2N+4}}\leq C\parallel v_{n}
\parallel_{2N+6}\leq CC_{3}\varepsilon<1$$ for small $\varepsilon$. Therefore if $m\leq m_{*}-N-8$ we may apply Theorem 5.1 to obtain a solution $u_{n}\in\overline{H}_{0}^{m}(\Omega)$ of (5.5) which satisfies the estimate (5.9). When $m+2N+8-\rho\geq 1/2$ this may be combined with V$_{n}$, VII$_{n}$, and $\rho\geq 2N+8$ to obtain $$\begin{aligned}
\parallel u_{n}\parallel_{m}&\leq& C_{m}(\parallel
f_{n}\parallel_{m+N}+\parallel v_{n}\parallel_{m+N+6}\parallel
f_{n}\parallel_{N+2})\\
&\leq&
C_{m}(C_{5}\varepsilon^{2}(1+\mu^{\rho-m})\mu_{n}^{m+N-\rho}+C_{3}C_{5}\varepsilon^{3}
(1+\mu^{\rho-N-2})\mu_{n}^{m+2N+8-\rho}\mu_{n}^{N+2-\rho})\\
&\leq&\varepsilon\mu_{n}^{m+N+2-\rho}\end{aligned}$$ for small $\varepsilon$. When $m+2N+8-\rho\leq-1/2$, the estimate $\parallel v_{n}\parallel_{m+N+6}\leq C_{3}\varepsilon$ placed in the above calculation gives the desired result.

II$_{n+1}$: Since $w_{n+1}=\sum_{i=0}^{n}u_{i}$, we have $$\parallel w_{n+1}\parallel_{m}\leq\sum_{i=0}^{n}\parallel u_{i}
\parallel_{m}\leq\varepsilon\sum_{i=0}^{n}\mu_{i}^{m+N+2-\rho}.$$ Hence, if $m+N+2-\rho\leq-1/2$ then $$\parallel
w_{n+1}\parallel_{m}\leq\varepsilon\sum_{i=0}^{\infty}(\mu^{i})^{-1/2}
\leq\varepsilon\sum_{i=0}^{\infty}(2^{i})^{-1/2}:=C_{1}\varepsilon,$$ and if $m+N+2-\rho\geq 1/2$ then $$\parallel w_{n+1}\parallel_{m}\leq\varepsilon\mu_{n+1}^{m+N+2-\rho}
\sum_{i=0}^{n}(\frac{\mu_{i}}{\mu_{n+1}})^{m+N+2-\rho}
\leq\varepsilon\mu_{n+1}^{m+N+2-\rho}\sum_{i=0}^{\infty}(\mu^{-i})^{1/2}
\leq C_{1}\varepsilon\mu_{n+1}^{m+N+2-\rho}.$$

III$_{n+1}$: By the largeness assumption on $\rho$ we have $3N+8-\rho\leq-1/2$. Therefore II$_{n+1}$ and V$_{n+1}$ (proven below) imply that $$\parallel w_{n+1}\parallel_{2N+6}\leq C_{1}\varepsilon
\text{ }\text{ }\text{ and }\text{ }
\parallel v_{n+1}\parallel_{2N+6}\leq
C_{3}\varepsilon.$$

IV$_{n+1}$: Since $\rho< m_{*}-6$ we have $(m_{*}-N-8)+N+2-\rho\geq 1/2$. Therefore Lemma 5.1 and II$_{n+1}$ yield, $$\begin{aligned}
\parallel w_{n+1}-v_{n+1}\parallel_{m}&=&\parallel
(I-S_{n+1})w_{n+1}\parallel_{m}\\
&\leq& C_{m}\mu^{m-(m_{*}-N-8)}_{n+1}\parallel w_{n+1}\parallel_{m_{*}-N-8}\\
&\leq& C_{m}\mu^{m-(m_{*}-N-8)}_{n+1}C_{1}\varepsilon\mu^{(m_{*}-N-8)+N+2-\rho}_{n+1}\\
&:=& C_{2}\varepsilon\mu^{m+N+2-\rho}_{n+1}.\end{aligned}$$

V$_{n+1}$: From Lemma 5.1 and $\rho< m_{*}-6$ we have for all $m\geq 0$, $$\parallel \!v_{n+1}\!\parallel_{m}=\parallel\!
S_{n+1}w_{n+1}\!\parallel_{m}\leq C_{m}
\begin{cases}
\parallel \!w_{n+1}\!\parallel_{\rho-N-3}
& \text{if }m+N+2-\rho\leq-1/2,\\
\mu_{n+1}^{m+N+1-\rho}\!\parallel\! w_{n+1}\!\parallel_{\rho-N-1}
& \text{if }m+N+2-\rho\geq 1/2.
\end{cases}$$ V$_{n+1}$ now follows from II$_{n+1}$. Q.E.D.

Write $e_{n}=e^{'}_{n}+e^{''}_{n}+e^{'''}_{n}$, where $$\begin{aligned}
e^{'}_{n}&=&(\mathcal{L}(w_{n})-\mathcal{L}(v_{n}))u_{n},\\
e^{''}_{n}&=&\varepsilon(I-S_{n}^{'})a^{22}(v_{n})L_{\theta_{n}}(v_{n})u_{n}
+\varepsilon\theta_{n}a^{22}(v_{n})\partial_{\xi_{n}^{1}}^{2}u_{n}\\
&
&+\varepsilon(a^{22}(v_{n}))^{-1}\Phi(v_{n})[\partial_{x^{1}}^{2}u_{n}-\partial_{x^{1}}(
\log
a^{22}(v_{n})\sqrt{|g|})\partial_{x^{1}}u_{n}],\\
e^{'''}_{n}&=&Q_{n}(w_{n},u_{n}).\end{aligned}$$

**Proposition 5.2.** *If the hypotheses of Proposition 5.1 hold in addition to $n>0$, $\rho\geq 2N+12$, and $0\leq m\leq \min(m_{*}-N-10,m_{*}-m_{0})$ then* VI$_{n+1}$ *holds.*

*Proof.* We will estimate $e^{'}_{n}$, $e^{''}_{n}$, and $e^{'''}_{n}$ separately. According to (2.3) we may write $$(\mathcal{L}(w_{n})-\mathcal{L}(v_{n}))u_{n}
=\varepsilon\sum_{i,j}A_{ij}\partial_{x^{i}x^{j}}u_{n}
+\varepsilon\sum_{i}A_{i}\partial_{x^{i}}u_{n}.$$ Then Lemma 5.2 $(i)$ and $(iii)$, I$_{n+1}$, and IV$_{n}$ show that $$\begin{aligned}
\parallel e_{n}^{'}\parallel_{m}&\leq& \varepsilon C_{m,1}
[(\sum_{i,j}\parallel A_{ij}\parallel_{m}+\sum_{i}\parallel
A_{i}\parallel_{m})\parallel
u_{n}\parallel_{4}\\
& &+(\sum_{i,j}\parallel A_{ij}\parallel_{2}+\sum_{i}\parallel
A_{i}\parallel_{2})\parallel u_{n}\parallel_{m+2}]\\
&\leq& \varepsilon C_{m,2}(\parallel
w_{n}-v_{n}\parallel_{m+2}\parallel u_{n}\parallel_{4}+\parallel
w_{n}-v_{n}\parallel_{4}\parallel u_{n}\parallel_{m+2})\\
&\leq&
C_{m,3}\varepsilon^{3}\mu_{n}^{m+N+4-\rho}\mu_{n}^{N+6-\rho}\\
&\leq&C_{m,3}\varepsilon^{3}\mu_{n}^{m-\rho}.\end{aligned}$$ Note that we have used $\rho\geq 2N+10$, as well as $m\leq
m_{*}-N-10$ 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$_{n}$, $$\begin{aligned}
\parallel\varepsilon\theta_{n}a^{22}(v_{n})\partial_{\xi_{n}^{1}}^{2}u_{n}
\parallel_{m}\!\!\!\!&\leq&\!\!\!\!\varepsilon\theta_{n}C_{m,4}(\parallel
a^{22}(v_{n})\parallel_{m}\parallel\partial_{\xi_{n}^{1}}^{2}u_{n}\parallel_{2}
+\parallel
a^{22}(v_{n})\parallel_{2}\parallel\partial_{\xi_{n}^{1}}^{2}u_{n}\parallel_{m})\\
&\leq&\!\!\!\!\varepsilon\theta_{n}C_{m,5}[(1+\parallel
v_{n}\parallel_{m+2})\parallel u_{n}\parallel_{4}+(1+\parallel
v_{n}\parallel_{4})\parallel u_{n}\parallel_{m+2}]\\
&\leq&\!\!\!\!\varepsilon^{2}\mu_{n}^{3-\rho}C_{m,6}[(1\!+C_{3}\varepsilon\mu_{n}^{m+N+4-\rho})\varepsilon
\mu_{n}^{N+4-\rho}\!+(1\!+C_{3}\varepsilon)\varepsilon\mu_{n}^{m+N+4-\rho}]\\
&\leq&\!\!\!\!C_{m,7}\varepsilon^{3}\mu_{n}^{m-\rho}\end{aligned}$$ if $\mu$ is large and $m+N+4-\rho\geq 1/2$. If $m+N+4-\rho\leq
-1/2$ then we may use the estimate $\parallel
v_{n}\parallel_{m+2}\leq C_{3}\varepsilon$ to obtain the same outcome. Another application of Lemma 5.2 gives $$\begin{aligned}
\parallel\varepsilon(a^{22}(v_{n}))^{-1}\Phi(v_{n})\partial_{x^{1}}^{2}u_{n}\parallel_{m}&\leq&
\varepsilon C_{m,8}[\parallel\Phi(v_{n})\parallel_{m}(1+\parallel
v_{n}\parallel_{4})\parallel u_{n}\parallel_{4}\\
& &+\parallel\Phi(v_{n})\parallel_{2}(1+\parallel
v_{n}\parallel_{m+2})\parallel u_{n}\parallel_{4}\\
& &+\parallel\Phi(v_{n})\parallel_{2}(1+\parallel
v_{n}\parallel_{4})\parallel u_{n}\parallel_{m+2}]\\
&\leq&C_{m,9}\varepsilon^{3}\mu_{n}^{m-\rho}\end{aligned}$$ after noting that $m\leq\min(m_{*}-N-10,m_{*}-m_{0})$ is required for VIII$_{n}$ to be valid, and similar methods yield $$\begin{aligned}
\parallel\varepsilon(a^{22}(v_{n}))^{-1}\Phi(v_{n})\partial_{x^{1}}(
\log a^{22}(v_{n})\sqrt{|g|})\partial_{x^{1}}u_{n}\parallel_{m}
\leq C_{m,10}\varepsilon^{3}\mu_{n}^{m-\rho}.\end{aligned}$$ Moreover, if $l=\rho+2\leq m_{*}-2$ and $n>0$ then we may apply Lemma 5.1 and recall that $\mu=\varepsilon^{-\frac{1}{2\rho}}$ to obtain $$\begin{aligned}
&
&\parallel\varepsilon(I-S_{n}^{'})a^{22}(v_{n})L_{\theta_{n}}(v_{n})u_{n}\parallel_{m}\\
&\leq&\!\!\!\!\varepsilon
C_{m,11}[\parallel(I-S_{n}^{'})a^{22}(v_{n})\parallel_{m}\parallel
f_{n}\parallel_{2}+\parallel(I-S_{n}^{'})a^{22}(v_{n})\parallel_{2}\parallel f_{n}\parallel_{2}]\\
&\leq&\!\!\!\!\varepsilon C_{m,12}[\mu_{n}^{m-l}(1+\parallel
v_{n}\parallel_{l+2})\varepsilon^{2}(1\!+\mu^{\rho-2})\mu_{n}^{2-\rho}+\mu_{n}^{2-l}(1+\parallel
v_{n}\parallel_{l+2})\varepsilon^{2}(1\!+\mu^{\rho-m})\mu_{n}^{m-\rho}]\\
&\leq&\!\!\!\! C_{m,13}\varepsilon^{3}\mu_{n}^{m-\rho}.\end{aligned}$$ Therefore $$\parallel e^{''}_{n}\parallel\leq
C_{m,14}\varepsilon^{3}\mu^{m-\rho}_{n}.$$

We now estimate $e^{'''}_{n}$. We have $$e^{'''}_{n}=Q_{n}(w_{n},u_{n})=\int_{0}^{1}(1-t)\frac{\partial^{2}}
{\partial t^{2}}\Phi(w_{n}+tu_{n})dt.$$ Apply Lemma 5.2 $(i)$ and $(ii)$, as well as the Sobolev Lemma to obtain $$\begin{aligned}
\parallel e^{'''}_{n}\parallel_{m}&\leq&\int_{0}^{1}
\sum_{|\alpha|,|\beta|\leq
2}\parallel\nabla_{\overline{\alpha}\overline{\beta}}
\Phi(w_{n}+tu_{n})\partial^{\alpha}u_{n}\partial^{\beta}u_{n}\parallel
_{m}dt\\
&\leq&\int_{0}^{1}C_{m,15}(\parallel\nabla^{2}\Phi(w_{n}+tu_{n})\parallel_{2}
\parallel u_{n}\parallel_{4}\parallel u_{n}\parallel_{m+2}\\
& &+\parallel\nabla^{2}\Phi(w_{n}+tu_{n})\parallel_{m}
\parallel u_{n}\parallel_{4}^{2})dt,\end{aligned}$$ where $\overline{\alpha}=\partial^{\alpha}(w_{n}+tu_{n})$ and $\overline{\beta}=\partial^{\beta}(w_{n}+tu_{n})$. The notation $\nabla^{2}\Phi$ represents the collection of second partial derivatives with respect to the variables $\overline{\alpha}$ and $\overline{\beta}$. Furthermore it is easy to see that $|\nabla^{2}\Phi(0)|=O(\varepsilon)$. Therefore using Lemma 5.2 $(iii)$, I$_{n+1}$, and II$_{n}$, we have $$\begin{aligned}
\parallel e^{'''}_{n}\parallel_{m}&\leq& C_{m,16}[(\varepsilon+\parallel
w_{n}\parallel_{4}+\parallel u_{n}\parallel_{4})\parallel u_{n}
\parallel_{4}\parallel u_{n}\parallel_{m+2}\\
& &+(\varepsilon+\parallel w_{n}
\parallel_{m+2}+\parallel u_{n}\parallel_{m+2})\parallel u_{n}
\parallel^{2}_{4}]\\
&\leq&C_{m,17}\varepsilon^{3}\mu^{m-\rho}_{n}\end{aligned}$$ if $\rho\geq 2N+12$. Combining the estimates of $e^{'}_{n}$, $e^{''}_{n}$, and $e^{'''}_{n}$ yields the desired result. Q.E.D. According to the above proposition if $\rho+1\leq m_{*}-m_{0}$ (in addition to the other required restrictions) then $E_{n}\in H^{\rho+1}(\Omega)$ and the following estimate holds, which will be utilized in the next proposition: $$\begin{aligned}
\parallel E_{n}\parallel_{\rho+1}\leq\sum_{i=0}^{n-1}\parallel
e_{i}\parallel_{\rho+1}\!\!\!&\leq&\!\!\!
C_{4}\varepsilon^{3}\sum_{i=0}^{n-1}
\mu_{i}\\
&\leq&\!\!\!
C_{4}\varepsilon^{3}(\sum_{i=0}^{\infty}\mu^{-1}_{i})\mu^{n} \leq
C_{4}\varepsilon^{3}(\sum_{i=0}^{\infty}2^{-i})\mu_{n}.\nonumber\end{aligned}$$

**Proposition 5.3.** *If the hypotheses of Proposition 5.2 hold and $\rho+1\leq m_{*}-m_{0}$, then* VII$_{n+1}$ *holds for all $0\leq m<\infty$.*

*Proof.* By (5.7) as well as Lemma 5.2 $(i)$ and $(iii)$, $$\begin{aligned}
\parallel f_{n+1}\parallel_{m}&\leq&\varepsilon^{-1}
C_{m}(\parallel
S_{n}E_{n}-S_{n+1}E_{n+1}+(S_{n}-S_{n+1})\Phi(w_{0})\parallel_{m}\\
& &+\parallel v_{n+1}\parallel_{m+2}\parallel
S_{n}E_{n}-S_{n+1}E_{n+1}+(S_{n}-S_{n+1})\Phi(w_{0})\parallel_{2}).\nonumber\end{aligned}$$ Next observe that (2.2) together with (5.14) below yields $$\parallel\Phi(w_{0})\parallel_{\rho+1}\leq C(\varepsilon^{3}+
\parallel w_{0}\parallel_{\rho+3})\leq C\varepsilon^{3}.$$ Then (5.10) implies that for all $m\geq \rho+1$, $$\begin{aligned}
& &\!\!\!\parallel
S_{n}E_{n}-S_{n+1}E_{n+1}+(S_{n}-S_{n+1})\Phi(w_{0})\parallel_{m}\\
&\leq& \!\!\!C_{m}(\mu_{n}^{m-\rho-1}\parallel
E_{n}\parallel_{\rho+1}+\mu_{n+1}^{m-\rho-1}\parallel
E_{n+1}\parallel_{\rho+1}
+(\mu_{n}^{m-\rho-1}+\mu_{n+1}^{m-\rho-1})\parallel\Phi(w_{0})\parallel_{\rho+1})\nonumber\\
&\leq&\!\!\!
C_{m}\varepsilon^{3}(1+\mu^{\rho-m})\mu_{n+1}^{m-\rho}.\nonumber\end{aligned}$$ If $m<\rho+1$, then applying similar methods along with VI$_{n+1}$ to $$\begin{aligned}
& &\parallel
S_{n}E_{n}-S_{n+1}E_{n+1}+(S_{n}-S_{n+1})\Phi(w_{0})\parallel_{m}\\
&\leq& \parallel (I-S_{n})E_{n}\parallel_{m}+\parallel
(I-S_{n+1})E_{n}\parallel_{m}+\parallel
S_{n+1}e_{n}\parallel_{m}\\
& &+\parallel (I-S_{n})\Phi(w_{0})\parallel_{m}+\parallel
(I-S_{n+1})\Phi(w_{0})\parallel_{m},\end{aligned}$$ produces the same estimate found in (5.13). Therefore plugging into (5.11) produces $$\begin{aligned}
\parallel f_{n+1}\parallel_{m}&\leq&
C_{m}[\varepsilon^{2}(1+\mu^{\rho-m})\mu^{m-\rho}_{n+1}
+\varepsilon^{3}(1+\mu^{\rho-2})\mu^{m+N+6-2\rho}_{n+1}]\\
&\leq& C_{m}\varepsilon^{2}(1+\mu^{\rho-m})\mu^{m-\rho}_{n+1},\end{aligned}$$ if $m+N+4-\rho\geq 1/2$. If $m+N+4-\rho\leq-1/2$ and $m\geq 2$, then using $\parallel v_{n+1}\parallel_{m+2}\leq C_{3}\varepsilon$ in the estimate above gives the desired result. Moreover if $0\leq
m<2$, then in place of (5.11) we use the estimate $$\parallel f_{n+1}\parallel_{m}\leq\varepsilon^{-1}C_{m}\parallel
S_{n}E_{n}-S_{n+1}E_{n+1}+(S_{n}-S_{n+1})\Phi(w_{0})\parallel_{m}$$ combined with the above method to obtain the desired result. Lastly if $m+N+4-\rho=0$, then replace $\parallel
v_{n+1}\parallel_{m+2}$ in (5.11) by $\parallel
v_{n+1}\parallel_{m+3}$ and follow the above method. Q.E.D.

**Proposition 5.4.** *If the hypotheses of Proposition 5.3 hold and $\rho+1=\min(m_{*}-N-10,m_{*}-m_{0})$, then* VIII$_{n+1}$ *holds for $0\leq m\leq\min(m_{*}-N-10,
m_{*}-m_{0})$.*

*Proof.* By (5.8) and VI$_{n+1}$ and $m\leq\rho+1$, we have $$\begin{aligned}
\parallel\Phi(w_{n+1})\parallel_{m}
&\leq&\parallel(I-S_{n})\Phi(w_{0})\parallel_{m}+\parallel(I-S_{n})E_{n}\parallel
_{m}+\parallel e_{n}\parallel_{m}\\
&\leq&
C_{m}(\mu^{m-\rho-1}_{n}\parallel\Phi(w_{0})\parallel_{\rho+1}+
\mu^{m-\rho-1}_{n}\parallel
E_{n}\parallel_{\rho+1}+C_{4}\varepsilon^{3}\mu^{m-\rho}_{n}).\end{aligned}$$ Applying the estimate (5.10) along with (5.12) and $\varepsilon\mu^{\rho-m}\leq\varepsilon^{1/2}$ produces $$\parallel\Phi(w_{n+1})\parallel_{m}\leq
C_{m}\varepsilon^{3}
\mu^{\rho-m}\mu_{n+1}^{m-\rho}\leq\frac{1}{3}\varepsilon\mu_{n+1}^{m-\rho},$$ if $\varepsilon$ is sufficiently small. Lastly a similar estimate may be obtained for $\Phi(v_{n+1})$ by writing $$\begin{aligned}
\parallel\Phi(v_{n+1})\parallel_{m}&\leq&\parallel\Phi(w_{n+1})\parallel_{m}
+\parallel\Phi(v_{n+1})-\Phi(w_{n+1})\parallel_{m}\\
&\leq&\frac{1}{3}\varepsilon\mu_{n+1}^{m-\rho}+\varepsilon\parallel
v_{n+1}-w_{n+1}\parallel_{m+2}\\
&\leq&(\frac{1}{3}+C_{2}\varepsilon^{2})\mu_{n+1}^{m+N+4-\rho}.\end{aligned}$$ Q.E.D.

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: $$\parallel\phi_{\varrho}^{-}\parallel_{m_{*}-m_{0}+1,\overline{\partial}\Omega}+
\parallel\psi_{\varrho}^{-}\parallel_{m_{*}-m_{0},\overline{\partial}\Omega}\leq
C\varepsilon^{l},\text{ }\text{ }\text{ }\text{ }l\geq 3.$$ Then according to (5.4), II$_{0}$, III$_{0}$, IV$_{0}$, and V$_{0}$ are trivial as long as $\varepsilon$ is small enough. Furthermore applying (5.12) and again taking $\varepsilon$ 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$_{1}$, $$\parallel u_{0}\parallel_{m}\leq C_{0}\varepsilon^{2},\text{ }\text{ }\text{ }\text{ }
m\leq m_{*}-N-8.$$ Now the proof of Proposition 5.2 may be appropriately modified to show that VI$_{1}$ is valid. This completes the proof by induction.

In view of the hypotheses of Propositions 5.1-5.4, we will choose $$\rho=\min(m_{*}-N-10,m_{*}-m_{0})-1.$$ Since $\rho\geq 3N+9$ we must then have $$m_{*}\geq\max(3N+16, 3N+m_{0}+10).$$ The following corollary yields a solution of (5.2) with $\alpha_{0}=m_{*}-m_{0}-N-6$.

**Corollary 5.1.** *If $m_{0}\geq N+10$ then under the above assumptions $w_{n}\rightarrow w$ *in* $\overline{H}_{0}^{m_{*}-m_{0}-N-4}(\Omega)$. Furthermore $\Phi(w_{n})\rightarrow 0$ *in* $C^{0}(\Omega)$.*

*Proof.* When $m_{0}\geq N+10$ we have $\rho-1=m_{*}-m_{0}-2$. Then for $m+N+2\leq \rho-1$ and $i>j$, I$_{n}$ implies that $$\parallel w_{i}-w_{j}\parallel_{m}\leq
\sum_{n=j}^{i-1}\parallel u_{n}\parallel_{m}\leq
\varepsilon\sum_{n=j}^{i-1}\mu_{n}^{m+N+2-\rho}\leq\varepsilon
\sum_{n=j}^{i-1}\mu^{-n}.$$ Hence, $\{w_{n}\}_{n=0}^{\infty}$ is Cauchy in $\overline{H}_{0}^{m}(\Omega)$ for all $m\leq
m_{*}-m_{0}-N-4$.

Lastly by the Sobolev Lemma and VIII$_{n}$, $$|\Phi(w_{n})|_{C^{0}(\Omega)}\leq
C\parallel\Phi(w_{n})\parallel_{2}\leq
\varepsilon\mu^{N+6-\rho}_{n}.$$ The desired conclusion follows since $\rho>N+6$. Q.E.D.

<span style="font-variant:small-caps;">Elliptic Regions</span>

Here we shall set up the iteration procedure for problem (5.1). For convenience we will denote the domain $\Omega_{\kappa}^{+}$ by $\Omega$. Set $w_{0}=0$ and suppose that functions $w_{1},\ldots,w_{n}$ have been defined on $\Omega$. If $S_{i}=S_{\mu^{i}}$ are smoothing operators given by Lemma 5.1, then we put $v_{i}=S_{i}w_{i}$, $0\leq i\leq n$, and define $w_{n+1}=w_{n}+u_{n}$ where $u_{n}$ is the solution of $$L(v_{n})u_{n}=f_{n}\text{ }\text{ }\text{ in }\text{ }\text{
}\Omega,\text{ }\text{ }\text{ }\text{
}u_{n}|_{\overline{\partial}\Omega}=0,$$ 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 $\mathcal{L}(w_{n})$ the linearization of (5.1), then according to (5.3) (with $\theta_{n}=0$) we have $$\begin{aligned}
\Phi(w_{n+1})&=&\Phi(w_{n})+\mathcal{L}(w_{n})u_{n}+Q_{n}(w_{n},u_{n})\\
&=&\Phi(w_{n})+\varepsilon
S_{n}^{'}a^{22}(v_{n})L(v_{n})u_{n}+e_{n},\end{aligned}$$ with $$\begin{aligned}
e_{n}&=&(\mathcal{L}(w_{n})-\mathcal{L}(v_{n}))u_{n}+\varepsilon(I-S_{n}^{'})
a^{22}(v_{n})L(v_{n})u_{n}
+Q_{n}(w_{n},u_{n})\\
&
&+\varepsilon(a^{22}(v_{n}))^{-1}\Phi(v_{n})[\partial_{x^{1}}^{2}u_{n}-\partial_{x^{1}}(
\log a^{22}(v_{n})\sqrt{|g|})\partial_{x^{1}}u_{n}].\end{aligned}$$ Lastly we set $E_{0}=0$, $E_{n}=\sum_{i=0}^{n-1}e_{i}$, and define $f_{n}$ according to (5.7).

It is clear that similar arguments as those used for the hyperbolic regions will show that $\{w_{n}\}_{n=0}^{\infty}$ 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 $\S 3$, we need to extend the coefficients of $L(v_{n})$ outside of $\Omega$ 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 $\mathbb{R}^{2}$, with Lipschitz smooth boundary. Then there exists a linear operator $E_{X}:L^{2}(X)\rightarrow
L^{2}(\mathbb{R}^{2})$ such that:*

$i)$ $E_{X}(u)|_{X}=u$,

$ii)$ $E_{X}:H^{m}(X)\rightarrow H^{m}(\mathbb{R}^{2})$ *continuously for each $m\in\mathbb{Z}_{\geq 0}$*.

**Theorem 5.2.** *Suppose that $g\in C^{m_{*}}$. If $m\leq \frac{1}{3}(m_{*}-8)$, $|v_{n}|_{C^{6}}<1$, and $\delta=\delta(m)$, $\varepsilon=\varepsilon(m,\delta)$ are sufficiently small, then there exists a solution $u_{n}\in\overline{H}^{m}_{0}(\Omega)$ of (5.15) which satisfies the estimate* $$\parallel u_{n}\parallel_{m}\leq\delta^{-1}C_{m}(\parallel
f_{n}\parallel_{m+2+\gamma}+\sum_{i+j+l\leq
m+23+\gamma}(1+\parallel v_{n}\parallel_{i})\parallel
v_{n}\parallel_{j}\parallel f_{n}\parallel_{l})$$ *for some constant $C_{m}$ independent of $\delta$ and $\varepsilon$ and where $2m<\gamma<m_{*}-m-6$.*

*Proof.* This will follow from Theorem 3.2. However we must first change to the coordinates $\xi_{n}^{i}(x,y)$ of Lemma 2.1, and then change to polar coordinates so that $$\Omega=\{(r,\theta)\mid 0<r<\sigma,0<\theta<\delta\}.$$ Set $$\Omega^{1}=\{(r,\theta)\mid 0<r<\sigma+1,0<\theta<\delta\}$$ and let $\varphi=\varphi(r)$ be a smooth nonnegative cut-off function with $\varphi(r)\equiv 1$ for $0<r<\sigma$, and $\varphi(r)\equiv 0$ for $\sigma+1<r$. If we cut-off the coefficients of $L(E_{\Omega}v_{n})$ as in (3.1), we may use Theorem 3.2 to solve $$L(E_{\Omega}v_{n})u_{n}=E_{\Omega}f_{n}\text{ }\text{ }\text{ in
}\text{ }\text{ }\Omega^{1},\text{ }\text{ }\text{ }\text{
}u_{n}|_{\overline{\partial}\Omega^{1}}=0,$$ with $$\parallel u_{n}\parallel_{(m,\gamma),\Omega^{1}}^{'}\leq C_{m}(\parallel
E_{\Omega}f_{n}\parallel^{'}_{m+2+\gamma,\Omega^{1}}+\parallel
E_{\Omega}v_{n}\parallel^{'}_{m+6,\Omega^{1}}
\parallel E_{\Omega}f_{n}\parallel^{'}_{5+\gamma,\Omega^{1}})$$ for $m\leq m_{*}-6$ where $\gamma> 2m$ and $\parallel\cdot\parallel^{'}$ indicates that the norm is with respect to these polar coordinates. By Lemma 5.3 $$\parallel E_{\Omega}f_{n}\parallel^{'}_{m+2+\gamma,\Omega^{1}}
\leq C_{m}\parallel f_{n}\parallel^{'}_{m+2+\gamma,\Omega},\text{
}\text{ }\text{ }\text{ }\parallel
E_{\Omega}v_{n}\parallel^{'}_{m+6,\Omega^{1}} \leq C_{m}\parallel
v_{n}\parallel^{'}_{m+6,\Omega}.$$ Therefore with the help of (2.4) and Lemma 5.2 it follows that $$\parallel u_{n}\parallel_{m,\Omega}\leq\delta^{-1}C_{m}(\parallel
f_{n}\parallel_{m+2+\gamma,\Omega}+\sum_{i+j+l\leq
m+23+\gamma}(1+\parallel v_{n}\parallel_{i,\Omega})\parallel
v_{n}\parallel_{j,\Omega}
\parallel f_{n}\parallel_{l,\Omega}).$$ The result is now obtained by noting that $\max(m+2+\gamma,5+\gamma)\leq m_{*}-2$ is required to apply (2.4). Q.E.D.

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 $\rho$, $\gamma$, and $m_{*}$ in the elliptic regions: $$\rho\geq 2\gamma+54,\text{ }\text{ }\text{ }\text{
}\rho+1=\frac{1}{3}(m_{*}-14).$$ Choosing the largest possible value for $\gamma$ and noting that the hypothesis of Theorem 5.2 requires $2m<\gamma$, implies that we must have $m\leq\frac{1}{12}(m_{*}-185)$. The following corollary produces a solution of (5.1) for $\alpha_{0}=\frac{1}{12}m_{*}-18$.

**Corollary 5.2.** *If $m_{*}\geq 192$ then $w_{n}\rightarrow w$ in $\overline{H}_{0}^{\frac{1}{12}m_{*}-16}(\Omega)$ with* $$\parallel w\parallel_{\frac{1}{12}m_{*}-16}\leq
C\varepsilon^{3}.$$ *Furthermore $\Phi(w_{n})\rightarrow 0$ in $C^{0}(\Omega)$.*

*Proof.* The same arguments used for Corollary 5.1 apply. Moreover, we use the analogue of II$_{n}$ to obtain the estimate for $w$. Q.E.D.

<span style="font-variant:small-caps;">Proof of Theorem 1.1</span>

Here we shall construct a solution of (2.1) in a full neighborhood of the origin. First consider the case in which there are exactly two elliptic regions, each bordering two hyperbolic regions. Then on each elliptic region $\Omega_{\kappa}^{+}$ let $w_{\kappa}^{+}\in\overline{H}_{0}^{\frac{1}{12}m_{*}-16}(\Omega_{\kappa}^{+})$ be the solution of (5.1) given by Corollary 5.2. On each boundary of the hyperbolic regions $\overline{\partial}\Omega_{\varrho}^{-}$ set $\phi_{\varrho}^{-}=0$, $\psi_{\varrho}^{-}=-\partial_{\nu}w_{\kappa(\varrho)}^{+}|_{\overline{\partial}
\Omega_{\kappa(\varrho)}^{+}}\in\overline{H}_{0}^{\frac{1}{12}m_{*}-18}
(\overline{\partial}\Omega_{\kappa(\varrho)})$ where $\Omega_{\kappa(\varrho)}^{+}$ is the bordering elliptic region, and note that (5.14) is valid with $m_{0}=\frac{11}{12}m_{*}+18$. Then Corollary 5.1 yields a solution $w_{\varrho}^{-}\in\overline{H}^{m_{*}-m_{0}-N-4}_{0}(\Omega_{\varrho}^{-})$ of (5.2). Under the hypotheses of Corollaries 5.1 and 5.2 we require $m_{*}\geq \max(192,3N+m_{0}+10)$ or rather $m_{*}\geq
36(N+10)$.

Suppose that $\Omega_{\kappa}^{+}$ borders on $\Omega_{\varrho}^{-}$. Then since the common boundary curve $\Upsilon$ is noncharacteristic for (2.1) (according to our original choice of approximate solution $z_{0}$), the functions $w_{\kappa}^{+}$ and $w_{\varrho}^{-}$ agree along with their derivatives up to and including order $\frac{1}{12}m_{*}-N-24$ along $\Upsilon$. It follows that the individual solutions $\{w_{\kappa}^{+}\}_{\kappa=1}^{\kappa_{0}}$ and $\{w_{\varrho}^{+}\}_{\varrho=1}^{\varrho_{0}}$ combine to form a $C^{\frac{1}{12}m_{*}-N-24}$ solution of (2.1) on some neighborhood of the origin.

Now consider the general case in which elliptic and hyperbolic regions are allowed to border regions of the same type. If an elliptic region borders another elliptic region, they may be combined to form a single elliptic region which contains a single curve of degeneracy on the interior. By appropriately regularizing the linearized equation in this combined region to eliminate the interior degeneracy, we may apply the theory of section $\S 3$ to obtain Theorem 5.2, and hence a solution of (5.1) in this combined region. Therefore we may assume that each elliptic region is bordered by hyperbolic regions (unless no hyperbolic regions are present). On the other hand, if two hyperbolic regions share a common boundary, for instance $\Omega_{1}^{-}$ and $\Omega_{3}^{-}$, then Cauchy data may be prescribed appropriately on the portion of $\partial\Omega_{1}^{-}$ which is shared with $\partial\Omega_{3}^{-}$, so that the solution on both regions may be glued together. Moreover, Cauchy data may be arbitrarily prescribed on the portion of $\partial\Omega_{3}^{-}$ emanating from the origin and which is not shared with $\partial\Omega_{1}^{-}$. It follows that in the general case, the solutions of the elliptic and hyperbolic regions may be patched together in the usual way.

**6. Appendix A**

The purpose of this appendix is to show existence for the ODE occurring in the proof of Theorem 3.1: $$\begin{aligned}
&
&\sum_{s=0}^{m}\lambda^{-s}(-1)^{s}\partial_{r}^{s}(a_{\lambda,\gamma-2(s-1)}
\partial_{r}^{s}\zeta)=v,\\
& &\zeta(r,0)=\zeta(r,\delta)=0,\text{ }\text{ }\text{
}\partial_{r}^{s}\zeta(\sigma,\theta)=0,\text{ }\text{ }\text{
}0\leq s\leq m-1,\nonumber\\
&
&\int_{r=r_{0}}(\partial_{r}^{s}\partial_{\theta}^{l}\zeta)^{2}\leq
r_{0}^{\gamma-2s}C,\text{ }\text{ }\text{ }0\leq s\leq 2m-1,\text{
}\text{ }\text{ }0\leq l<\infty,\nonumber\end{aligned}$$ where $v\in\widehat{C}^{\infty}(\Omega)$, $\zeta\in
H^{(m,\infty,\gamma+2)}(\Omega)\cap C^{\infty}(\Omega)$, $r_{0}$ is sufficiently small, and all other definitions/notation may be found in section $\S 3$.

First note that $\eta\mapsto(\eta,(\lambda\theta^{2}-1)^{-1}v)$ is a bounded linear functional on $H^{(m,0,\gamma+2)}(\Omega)$, and thus by the Riesz representation theorem there exists a unique $\zeta\in H^{(m,0,\gamma+2)}(\Omega)$ such that $$(\eta,\zeta)_{(m,0,\gamma+2)}=(\eta,(\lambda\theta^{2}-1)^{-1}v)\text{
}\text{ }\text{ all }\text{ }\text{ }\eta\in
H^{(m,0,\gamma+2)}(\Omega),$$ where $(\cdot,\cdot)_{(m,0,\gamma+2)}$ denotes the inner product on $H^{(m,0,\gamma+2)}(\Omega)$. It follows that $\zeta$ is a weak solution of (6.1), and according to the basic regularity theory for ODEs we have $\zeta\in C^{\infty}(\Omega)$. Furthermore the desired boundary behavior of the solution at $\theta=0,\delta$ arises from the requirement that $v(r,0)=v(r,\delta)=0$, and the vanishing at $r=\sigma$ is a result of the trace theorem for Sobolev spaces.

Lastly we observe that since $v$ vanishes in a neighborhood of $r=0$, the solution $\zeta$ 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 $\zeta$ must be a linear combination of $2m$ functions of the form: $r^{\alpha}(\log r)^{\beta}$ where $\alpha\in\mathbb{C}$ and $\beta\in\mathbb{Z}_{\geq 0}$. However according to Lemma 3.1 $$\int_{r=r_{0}}(\partial_{r}^{s}\partial_{\theta}^{l}\zeta)^{2}\leq
r_{0}^{\gamma-2s}C\parallel\zeta\parallel_{(m,l+1,\gamma+2)}^{2},\text{
}\text{ }\text{ }s\leq m-1,\text{ }\text{ }\text{ }0\leq l<\infty.$$ Therefore each term in the linear combination must satisfy $$r^{\alpha}(\log r)^{\beta}=O(r^{\gamma/2})\text{ }\text{ }\text{
as }\text{ }\text{ }r\rightarrow 0.$$ The desired boundary behavior at $r=0$ now follows from (6.2) and (6.3).

**7. Appendix B**

The purpose of this section is to construct the smoothing operators $S_{\mu}$ of Lemma 5.1. The construction will differ from the standard one for $S_{\mu}^{'}$ (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 $\Omega$. Fix $\widehat{\chi}\in C^{\infty}_{c}(\mathbb{R}^{2})$ such that $\widehat{\chi}\equiv 1$ on some neighborhood of the origin, and let $$\chi(x)=\int_{\mathbb{R}^{2}}e^{2\pi i\xi\cdot
x}\widehat{\chi}(\xi)d\xi$$ be its inverse Fourier Transform. Then $\chi$ is a Schwartz function and satisfies $$\int_{\mathbb{R}^{2}}\chi(x)dx=1,\text{ }\text{ }\text{ }\text{
}\int_{\mathbb{R}^{2}}x^{\alpha}\chi(x)dx=0,\text{ }\text{ }\text{
}\text{ }|\alpha|>0.$$ Furthermore let $\eta\in C^{\infty}(\mathbb{R}^{2})$ be a radial function vanishing to all orders at the origin, and satisfying $$\eta(x)=\begin{cases}
1 & \text{if $|x|> 1$},\\
0 & \text{if $|x|<\frac{1}{2}$}.
\end{cases}$$ For $\mu\geq 1$ we will write $\eta_{\mu}(x)=\eta(\mu x)$, $\chi_{\mu}(x)=\mu^{2}\chi(\mu x)$, and define smoothing operators $\overline{S}_{\mu}:L^{2}(\mathbb{R}^{2})\rightarrow\overline{H}_{0}^{\infty}(\mathbb{R}^{2})$ by $$(\overline{S}_{\mu}u)(x)=\eta_{\mu}(x)(\chi_{\mu}*u)(x)=
\mu^{2}\eta(\mu x)\int_{\mathbb{R}^{2}}\chi(\mu(x-y))u(y)dy.$$ Here the space $\overline{H}_{0}^{l}(\mathbb{R}^{2})$ is the completion of $\overline{C}_{c}^{\infty}(\mathbb{R}^{2})$ in the Sobolev norm $\parallel\cdot\parallel_{l}$, where $\overline{C}_{c}^{\infty}(\mathbb{R}^{2})$ denotes all $C_{c}^{\infty}(\mathbb{R}^{2})$ functions vanishing in a neighborhood of the origin.

We now proceed to show statements $(i)$, $(ii)$, and $(iii)$ of Lemma 5.1 with respect to $\overline{S}_{\mu}$. Note that it is sufficient to prove these for $u\in\overline{C}_{c}^{\infty}(\mathbb{R}^{2})$ as this space of functions is dense in $\overline{H}^{l}_{0}(\mathbb{R}^{2})$. We begin with $(ii)$. Let $$u(y)=\sum_{|\alpha|<|\rho|}\frac{1}{|\alpha|!}\partial^{\alpha}u(x)
(y-x)^{\alpha}+\frac{1}{(|\rho|-1)!}\sum_{|\alpha|=|\rho|}
(y-x)^{\alpha}\int_{0}^{1}(1-t)^{|\rho|-1}\partial^{\alpha}u(x+t(y-x))dt$$ be a Taylor expansion of $u$ with integral remainder. Then according to the properties of $\chi$, $$\begin{aligned}
& &\!\!\!(\chi_{\mu}*u)(x)\\
&=&\!\!\!u(x)+
\frac{\mu^{2}}{(|\rho|-1)!}\sum_{|\alpha|=|\rho|}\int_{\mathbb{R}^{2}}\int_{0}^{1}\!
\chi(\mu(y\!-\!x))(1\!-\!t)^{|\rho|-1}(y\!-\!x)^{\alpha}\partial^{\alpha}u(x\!+t(y\!-\!x))dtdy.\nonumber\end{aligned}$$ Suppose that $l\leq |\sigma|\leq m$, and notice that $$\parallel\partial^{\sigma}\overline{S}_{\mu}u\parallel\leq\sum_{\beta+\gamma=\sigma}
\parallel\partial^{\beta}\eta_{\mu}\partial^{\gamma}(\chi_{\mu}*u)\parallel.$$ If $|\rho|=l-|\gamma|> 0$ we may apply (7.1) to find $$\begin{aligned}
\parallel\partial^{\beta}\eta_{\mu}\partial^{\gamma}(\chi_{\mu}*u)\parallel^{2}&=&
\mu^{2|\beta|}\int_{\mathbb{R}^{2}}(\partial^{\beta}\eta)^{2}(\mu
x)
(\chi_{\mu}*\partial^{\gamma}u)^{2}(x)dx\\
&\leq&
C_{1}\mu^{2|\beta|}\int_{\mathbb{R}^{2}}(\partial^{\beta}\eta)^{2}(\mu
x)
(\partial^{\gamma}u)^{2}(x)dx+C_{2}\mu^{2(|\beta|+|\gamma|-l)}\parallel
u\parallel_{l}^{2}.\end{aligned}$$ Under the current assumptions $|\beta|\neq 0$ which implies that $\mathrm{supp}\text{
}\partial^{\beta}\eta_{\mu}\subset\{|x|<\mu^{-1}\}$, so applying the Taylor expansion of $\partial^{\gamma}u$ at $x=0$ with $|\rho|=l-|\gamma|$ and recalling that $u$ vanishes to all orders at the origin, yields $$\int_{\mathbb{R}^{2}}(\partial^{\beta}\eta)^{2}(\mu x)
(\partial^{\gamma}u)^{2}(x)dx\leq C_{3}
\int_{B_{\mu^{-1}}(0)}(\partial^{\gamma}u)^{2}(x)dx\leq
C_{4}\mu^{2(|\gamma|-l)}\parallel u\parallel_{l}^{2}.$$ Moreover the case $|\gamma|\geq l$ may be treated by Young’s inequality: $$\begin{aligned}
\parallel\partial^{\beta}\eta_{\mu}\partial^{\gamma}(\chi_{\mu}*u)\parallel^{2}&\leq&
C_{5}\mu^{2(|\beta|+|\gamma|-|\tau|)}\int_{\mathbb{R}^{2}}(\partial^{\gamma-\tau}
\chi_{\mu}*\partial^{\tau}u)^{2}(x)dx\\
&\leq&C_{6}\mu^{2(m-l)}\parallel u\parallel^{2}_{l},\end{aligned}$$ where $|\tau|=l$. Therefore $(ii)$ follows once $(i)$ is established, and $(i)$ is established by similar arguments which will be omitted here.

We now show $(iii)$. Let $|\alpha|=m\leq l$ and observe that $$\begin{aligned}
\parallel\partial^{\alpha}(u-\overline{S}_{\mu}u)\parallel&\leq&
\parallel\partial^{\alpha}[(1-\eta_{\mu})u]\parallel+
\parallel\partial^{\alpha}[\eta_{\mu}(u-\chi_{\mu}*u)]\parallel\\
&\leq&\sum_{\beta+\gamma=\alpha}(\parallel\partial^{\beta}(1-\eta_{\mu})\partial^{\gamma}u
\parallel+\parallel\partial^{\beta}\eta_{\mu}\partial^{\gamma}(u-\chi_{\mu}*u)\parallel).\end{aligned}$$ According to the standard construction \[17\], $$\begin{aligned}
\parallel\partial^{\beta}\eta_{\mu}\partial^{\gamma}(u-
\chi_{\mu}*u)\parallel&\leq& C_{7}\mu^{|\beta|}\parallel
u-\chi_{\mu}*u\parallel_{|\gamma|}\\
&\leq& C_{8}\mu^{|\beta|}\mu^{|\gamma|-l}\parallel
u\parallel_{l}\\
&=& C_{8}\mu^{m-l}\parallel u\parallel_{l}.\end{aligned}$$ Furthermore since $\mathrm{supp }\text{
}\partial^{\beta}(1-\eta_{\mu})\subset\{|x|<\mu^{-1}\}$ we may apply the same methods used to establish (7.2) to obtain $$\parallel\partial^{\beta}(1-\eta_{\mu})\partial^{\gamma}u
\parallel\leq C_{9}\mu^{m-l}\parallel u\parallel_{l}.$$ It follows that $(iii)$ holds.

The desired smoothing operators on $\Omega$ may be obtained from $\overline{S}_{\mu}$ in the following way. If $\Omega$ is a bounded convex Lipschitz domain, then Lemma 5.3 yields an extension operator $E:H^{m}(\Omega)\rightarrow H^{m}(\mathbb{R}^{2})$. We then define smoothing operators $S_{\mu}:L^{2}(\Omega)\rightarrow\overline{H}_{0}^{\infty}(\Omega)$ by $S_{\mu}u=(\overline{S}_{\mu}Eu)|_{\Omega}$. As $E$ is bounded, it is clear that Lemma 5.1 will also hold for $S_{\mu}$.

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<span style="font-variant:small-caps;">Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556</span>

*E-mail address*: `<PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>`

<span style="font-variant:small-caps;">Department of Mathematics, Stony Brook University, Stony Brook, NY 11794</span>

*E-mail address*: `<PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>`