Boundary Harmonic Coordinates on Manifolds with Boundary in Low Regularity

Abstract

In this paper, we prove the existence of $H^2$-regular coordinates on Riemannian 3-manifolds with boundary, assuming only $L^2$-bounds on the Ricci curvature, $L^4$-bounds on the second fundamental form of the boundary, and a positive lower bound on the volume radius. The proof follows by extending the theory of Cheeger–Gromov convergence to include manifolds with boundary in the above low regularity setting. The main tools are boundary harmonic coordinates together with elliptic estimates and a geometric trace estimate, and a rigidity argument using manifold doubling. Assuming higher regularity of the Ricci curvature, we also prove corresponding higher regularity estimates for the coordinates.

Appendices

Appendix A. Preliminaries of Geometric Analysis

In this section, we collect basic preliminaries of geometric analysis. The following lemma follows from more general Sobolev embeddings for $W^{s,p}$ spaces, see for example [24] and [1] for details and proofs.

Lemma A.1

(Sobolev embeddings). Let $\Omega ^2 \subset {{\mathbb {R}}}^2$ and $\Omega ^3 \subset {{\mathbb {R}}}^3$ be smooth domains. Then, the following are continuous embeddings,

$$\begin{aligned}&H^{1/2}(\Omega ^2) \hookrightarrow L^4(\Omega ^2), \\&L^2(\Omega ^2) \hookrightarrow H^{-1/2}(\Omega ^2). \end{aligned}$$

Furthermore, if $\Omega ^2$ and $\Omega ^3$ are bounded, then the following embeddings are continuous and compact,

$$\begin{aligned}&H^2(\Omega ^3) \hookrightarrow C^{0,{\alpha }}(\overline{\Omega ^3}) \text { for } {\alpha }\in (0,1/2), \\&H^1(\Omega ^3) \hookrightarrow L^4(\Omega ^3), \\&H^2(\Omega ^3) \hookrightarrow W^{1,4}(\Omega ^3), \\&H^2(\Omega ^3) \hookrightarrow H^1(\Omega ^3), \\&H^{3/2}(\Omega ^2) \hookrightarrow H^{1/2}(\Omega ^2). \end{aligned}$$

The following are standard product estimates, we refer the reader to Section 13.3 in [15, 26] for more details and proofs.

Lemma A.2

(Product estimates). Let $\Omega ^2 \subset {{\mathbb {R}}}^2$ be a smooth domain, and let u and v be functions on $\Omega ^2$. Then,

$$\begin{aligned} \Vert uv \Vert _{H^{1/2}(\Omega ^2)} \lesssim \Vert u \Vert _{H^{5/4}(\Omega ^2)} \Vert v \Vert _{H^{1/2}(\Omega ^2)}. \end{aligned}$$

Moreover, for every integer $m\ge 0$, there is a constant $C_m>0$ such that

$$\begin{aligned} \Vert u v \Vert _{H^{m+3/2}(\Omega ^2)}&\lesssim \Vert u \Vert _{H^{m+3/2}(\Omega ^2)} \Vert v \Vert _{H^{3/2}(\Omega ^2)} + \Vert u \Vert _{H^{3/2}(\Omega ^2)} \Vert v \Vert _{H^{m+3/2}(\Omega ^2)} \\&\quad + C_m ( \Vert u \Vert _{H^{3/2}(\Omega ^2)} + \Vert v \Vert _{H^{3/2}(\Omega ^2)}). \end{aligned}$$

Let $\Omega ^3 \subset {{\mathbb {R}}}^3$ be a smooth domain, and let u and v be functions on $\Omega ^3$. Then for every integer $m\ge 0$ there is a constant $C_m>0$ such that

$$\begin{aligned} \Vert u v \Vert _{H^{m}(\Omega ^3)}&\lesssim \Vert u \Vert _{H^{m}(\Omega ^3)} \Vert v \Vert _{H^{2}(\Omega ^3)} + \Vert u \Vert _{H^{2}(\Omega ^3)} \Vert v \Vert _{H^{m}(\Omega ^3)} \\&\quad + C_m ( \Vert u \Vert _{H^{2}(\Omega ^3)} + \Vert v \Vert _{H^{2}(\Omega ^3)}). \end{aligned}$$

We define now the trace operator for continuous functions.

Definition A.3

(Trace operator for continuous functions). Let $n\ge 1$ be an integer and let $\Omega \subset {{\mathbb {R}}}^n$ be a smooth domain. Let f be a continuous scalar function on $\overline{\Omega }$. Denote the restriction of f to ${\partial }\Omega$ by

$$\begin{aligned} \tau (f) := f \vert _{{\partial }\Omega }. \end{aligned}$$

The trace operator $\tau$ extends to Sobolev spaces as follows.

Lemma A.4

(Trace operator for $W^{k,p}$-functions). Let $n\ge 1$ and $k\ge 1$ be integers and $1<p< \infty$ a real. Let $\Omega \subset {{\mathbb {R}}}^n$ be a smooth domain. Then the trace operator $\tau$ extends to a bounded linear operator

$$\begin{aligned} \tau : W^{k,p}(\Omega ) \rightarrow W^{k-1/p,p}({\partial }\Omega ). \end{aligned}$$

Remark A.5

In general, for a smooth domain $\Omega$, the spaces denoted by $W^{k-1/p,p}({\partial }\Omega )$ must be defined as a Besov space, see [1]. However, in this paper, because of the way we apply the Sobolev spaces results to $B^+(x,r)$ and ${\underline{B}}^+(x,r)$, we do not need to introduce these Besov spaces, and can use our definition of $W^{k-1/p,p}(\Omega ^2)$ for a smooth domain $\Omega ^2 \subset {{\mathbb {R}}}^2$, see Definition 2.2.

Remark A.6

We cannot apply Lemmas A.1, A.2 and A.4 directly to the sets $B^+(x,r)$ because they are not smooth domains. Therefore, whenever these lemmas are invoked in Sects. 3, 4 and 5, we tacitly apply the lemmas to the smooth domain $\Omega _{x,r',r}$ between $B^+(x,r')$ and $B^+(x,r)$, see its definition in (2.2), so that the estimates hold on $B^+(x,r')$ for a slightly smaller $r'<r$.

The quadratic non-linearities and $Q_{ij}, Q^{ij} \approx {\partial }g {\partial }g$ satisfy the following properties. The proof follows by standard Sobolev embeddings and product estimates (see Lemmas A.1 and A.2), and is left to the reader.

Lemma A.7

Let $\Omega ^2 \subset {{\mathbb {R}}}^2$ and $\Omega ^3 \subset {{\mathbb {R}}}^3$ be bounded smooth domains. Then the following holds.

• Let and be smooth Riemannian metrics on $\Omega ^2$. If

  

  then

  

• Let $(g_n)_{n \ge 0}$ and g be smooth Riemannian metrics on $\Omega ^3$. If

  $$\begin{aligned} g_n \rightharpoonup g \text { in } H^{2}(\Omega ^3) \text { as } n \rightarrow \infty , \end{aligned}$$

  then

  $$\begin{aligned} \Vert Q_{AB}(g_n) - Q_{AB}(g) \Vert _{L^2(\Omega ^3)} \rightarrow 0 \text { as } n \rightarrow \infty . \end{aligned}$$

• There is $\varepsilon >0$ such that if

  

  then for every integer $m>0$ there is a constant $C_m>0$ such that

  

• There is $\varepsilon >0$ such that if

  $$\begin{aligned} \Vert g-e \Vert _{{{\mathcal {H}}}^{2}(\Omega ^3)} < \varepsilon , \end{aligned}$$

  then for every integer $m>0$ there is a constant $C_m>0$ such that

  $$\begin{aligned} \Vert {\partial }^{m} Q_{AB} \Vert _{L^2(\Omega ^3)}&\lesssim \Vert g-e \Vert _{{{\mathcal {H}}}^{2}(\Omega ^3)} \Vert {\partial }^{m+2} g \Vert _{{{\mathcal {L}}}^2(\Omega ^3)} + \Vert g \Vert _{{{\mathcal {H}}}^{m+1}(\Omega ^3)} \\&\quad + C_m \Vert g-e \Vert _{{{\mathcal {H}}}^{2}(\Omega ^3)}. \end{aligned}$$

The next lemma shows that bounds on the metric components $g^{33},g^{3A},g_{AB}$ imply bounds for all metric components $g_{ij}$.

Lemma A.8

(Control of all metric components). Let $\Omega \subset {{\mathbb {R}}}^3$ be a smooth domain and let g be a smooth Riemannian metric on $\Omega$. There is $\varepsilon >0$ such that if

$$\begin{aligned} \Vert g-e \Vert _{{{\mathcal {H}}}^2(\Omega )} < \varepsilon , \end{aligned}$$

then for every integer $m\ge 0$, there is a constant $C_{m}>0$ such that

$$\begin{aligned} \Vert g \Vert _{{{\mathcal {H}}}^m(\Omega )} \lesssim \sum \limits _{A,B=1,2} ( \Vert g_{AB} \Vert _{{{\mathcal {H}}}^m(\Omega )} +\Vert g^{33} \Vert _{{{\mathcal {H}}}^m(\Omega )} + \Vert g^{3A} \Vert _{{{\mathcal {H}}}^m(\Omega )} )+ C_{m} \Vert g-e \Vert _{{{\mathcal {H}}}^2(\Omega )}. \end{aligned}$$

(A.1)

Proof

It suffices to control the components $g_{3A}$ and $g_{33}$. By (2.5), (2.7), and the general property $g_{3k} g^{k3} = 1$,

$$\begin{aligned} g_{3A} = g_{AC} \frac{g^{3C}}{g^{33}}, \, \, g_{33} = \frac{1}{g^{33}}\left( 1- g_{3A}g^{3A} \right) . \end{aligned}$$

From this, the product estimates of Lemma A.2 imply (A.1). Details are left to the reader.

$\square$

Appendix B. Global Elliptic Estimates

In this section, we collect global elliptic estimates for Dirichlet and Neumann problems, see Sections B.1 and B.2, respectively. These estimates are applied in Sect. 3 to construct boundary harmonic coordinates, and in Sect. 5 to derive higher regularity estimates for given boundary harmonic coordinates.

The estimates are standard, see for example [9] or [20], but we give some proofs for completeness. In the following, the constant in $\lesssim$ is allowed to depend on the domain $\Omega$.

1.1 B.1. Global estimates for Dirichlet data

We have the following standard elliptic estimates, see for example Theorem 9.13 from [9] and Lemma A.1.

Theorem B.1

(Standard global elliptic estimate). Let $n \in \{ 2,3 \}$ and let $\Omega \subset {{\mathbb {R}}}^n$ be a bounded smooth domain. For a real $n<p<\infty$, let $g \in {{\mathcal {W}}}^{1,p}(\Omega )$ be a Riemannian metric such that $\triangle _g = g^{ij}{\partial }_i {\partial }_j$. Then, for all $1<p'<\infty$ and $u \in W^{2,p'}(\Omega )$, we have

$$\begin{aligned} \Vert u \Vert _{W^{2,p'}(\Omega )} \lesssim \Vert \triangle _g u \Vert _{L^{p'}(\Omega )} + \Vert u \Vert _{W^{2-1/p',p'}({\partial }\Omega )}. \end{aligned}$$

From Theorem B.1 we get the following standard higher regularity estimate, see for example Theorem 8.13 in [9].

Corollary B.2

(Standard global higher regularity elliptic estimate). Let $\Omega \subset {{\mathbb {R}}}^3$ be a bounded smooth domain. Let the Riemannian metric $g \in {{\mathcal {H}}}^2(\Omega )$ be such that $\triangle _g = g^{ij} {\partial }_i {\partial }_j$. Then for all $u \in H^{3}(\Omega )$,

$$\begin{aligned} \Vert u \Vert _{H^3(\Omega )} \lesssim \Vert \triangle _g u \Vert _{H^1(\Omega )} + \Vert u \Vert _{H^{5/2}({\partial }\Omega )}. \end{aligned}$$

We now turn to the Laplace–Beltrami operator of a general Riemannian metric in three dimensions.

Proposition B.3

Let $\Omega \subset {{\mathbb {R}}}^3$ be bounded smooth domain and $g \in {{\mathcal {W}}}^{1,6}(\Omega )$ a Riemannian metric. Let u be the solution to

$$\begin{aligned} \triangle _g u&= f \text { in } \Omega ,\\ u&= 0\text { on } {\partial }\Omega . \end{aligned}$$

for $f \in L^6$. Then $u \in W^{2,6}(\Omega )$ and

$$\begin{aligned} \Vert u \Vert _{W^{2,6}(\Omega )} \lesssim \Vert f \Vert _{L^6(\Omega )}. \end{aligned}$$

(B.1)

Proof

First, we note that by the Lax-Milgram theorem and Lemma A.1, there exists a solution $u \in H^1(\Omega )$ with

$$\begin{aligned} \Vert u \Vert _{H^1(\Omega )} \lesssim \Vert f \Vert _{H^{-1}(\Omega )} \lesssim \Vert f \Vert _{L^6(\Omega )}, \end{aligned}$$

and consequently also

$$\begin{aligned} \Vert u \Vert _{L^6(\Omega )} \lesssim \Vert f \Vert _{L^6(\Omega )}. \end{aligned}$$

(B.2)

By Theorem B.1,

$$\begin{aligned} \begin{aligned} \Vert u \Vert _{W^{2,6}(\Omega )}&\lesssim \Vert g^{ij} {\partial }_i {\partial }_j u \Vert _{L^{6}(\Omega )} \\&\lesssim \Vert f \Vert _{L^{6}(\Omega )} + \Vert \triangle _g u-g^{ij} {\partial }_i {\partial }_j u \Vert _{L^{6}(\Omega )} \\&\lesssim \Vert f \Vert _{L^{6}(\Omega )} + \Vert {\partial }g {\partial }u \Vert _{L^{6}(\Omega )} \\&\lesssim \Vert f \Vert _{L^{6}(\Omega )} + \Vert g \Vert _{{{\mathcal {W}}}^{1,6}(\Omega )} \Vert {\partial }u \Vert _{C^0(\Omega )} . \end{aligned} \end{aligned}$$

(B.3)

In the following, we control the term $\Vert {\partial }u \Vert _{C^0(\Omega )}$ on the right-hand side of (B.3) by a bootstrap argument. We remark that this argument is taken from [20]. Using that $g \in {{\mathcal {W}}}^{1,6}(\Omega ) \subset {{\mathcal {C}}}^0(\Omega )$ in $n=3$, we have by partial integration on $\Omega$,

$$\begin{aligned} \Vert {\partial }u \Vert _{L^2(\Omega )} \lesssim \Vert \nabla u \Vert _{L^2(\Omega )} \lesssim \Vert u \Vert ^{1/2}_{L^2(\Omega )} \Vert \triangle _g u \Vert ^{1/2}_{L^2(\Omega )} \lesssim \Vert f \Vert _{L^6(\Omega )}. \end{aligned}$$

(B.4)

By using Theorem B.1 with $p'=3/2$, this implies further

$$\begin{aligned} \begin{aligned} \Vert u \Vert _{W^{2,3/2}(\Omega )}&\lesssim \Vert g^{ij} {\partial }_i {\partial }_j u \Vert _{L^{3/2}(\Omega )} \\&\lesssim \Vert \triangle _g u\Vert _{L^{3/2}(\Omega )} + \Vert {\partial }g {\partial }u \Vert _{L^{3/2}(\Omega )} \\&\lesssim \Vert f \Vert _{L^{3/2}(\Omega )} + \Vert g \Vert _{{{\mathcal {W}}}^{1,6}(\Omega )} \Vert {\partial }u \Vert _{L^2(\Omega )} \\&\lesssim \Vert f \Vert _{L^{3/2}(\Omega )} + \Vert g \Vert _{{{\mathcal {W}}}^{1,6}(\Omega )} \Vert f \Vert _{L^{6}(\Omega )}. \\ \end{aligned} \end{aligned}$$

(B.5)

By Lemma A.1 and (B.5), we have

$$\begin{aligned} \Vert {\partial }u \Vert _{L^3(\Omega )}&\lesssim \Vert {\partial }u \Vert _{W^{1,3/2}(\Omega )} \\&\lesssim \Vert f \Vert _{3/2(\Omega )} + \Vert g \Vert _{{{\mathcal {W}}}^{1,6}(\Omega )} \Vert f \Vert _{L^6(\Omega )} \\&\lesssim \Vert f \Vert _{L^{6}(\Omega )}. \end{aligned}$$

Compared to (B.4), we bootstrapped the regularity of ${\partial }u$.

By continuing to bootstrap the regularity of ${\partial }u$ with Theorem B.1 and Lemma A.1 as above (with $p'=2$, $p'=3$ and finally $p'=24/5$), we get

$$\begin{aligned} \Vert {\partial }u \Vert _{W^{1,24/5}(\Omega ) } \lesssim \Vert f \Vert _{L^{6}(\Omega )}, \end{aligned}$$

and consequently, by Lemma A.1 and (B.2),

$$\begin{aligned} \Vert {\partial }u \Vert _{C^0(\Omega )}&\lesssim \Vert {\partial }u \Vert _{W^{1,24/5}(\Omega ) }, \\&\lesssim \Vert f \Vert _{L^{6}(\Omega )}. \end{aligned}$$

Plugging this bound into (B.3) proves (B.1). This finishes the proof of Proposition B.3.

$\square$

Corollary B.4

(Global estimates for the Dirichlet problem in. $H^3$, $n=3$). Let $\Omega \subset {{\mathbb {R}}}^3$ be bounded smooth domain and $g \in {{\mathcal {H}}}^2(\Omega )$ a Riemannian metric. Let u be the solution to

$$\begin{aligned} \begin{aligned} \triangle _g u&= f \text { in } \Omega ,\\ u&= h\text { on } {\partial }\Omega \end{aligned} \end{aligned}$$

(B.6)

for $f \in H^1(\Omega )$ and $h \in H^{5/2}({\partial }\Omega )$. Then we have $u \in H^3(\Omega )$ and

$$\begin{aligned} \Vert u \Vert _{H^3(\Omega )} \lesssim \Vert f \Vert _{H^1(\Omega )} + \Vert h \Vert _{H^{5/2}({\partial }\Omega )}. \end{aligned}$$

Proof

First, we reduce to homogeneous Dirichlet data. By Paragraph 7.41 in [1], there exists ${\tilde{h}} \in H^3(\Omega )$, such that

$$\begin{aligned} \tau ({{\tilde{h}}}) = h \text { on } {\partial }\Omega \end{aligned}$$

and

$$\begin{aligned} \Vert {{\tilde{h}}} \Vert _{H^3(\Omega )} \lesssim \Vert h \Vert _{H^{5/2}({\partial }\Omega )}. \end{aligned}$$

Therefore we can reduce (B.6) to the study of

$$\begin{aligned} \begin{aligned} \triangle _g u&= {\tilde{f}} \text { in } \Omega ,\\ u&= 0 \text { on } {\partial }\Omega \end{aligned} \end{aligned}$$

(B.7)

where ${\tilde{f}} := f + \triangle _g {\tilde{h}}$ is bounded by

$$\begin{aligned} \begin{aligned} \Vert {\tilde{f}} \Vert _{H^1(\Omega )}&\lesssim \Vert f \Vert _{H^1(\Omega )} + \Vert \triangle _g {\tilde{h}} \Vert _{H^1(\Omega )} \\&\lesssim \Vert f \Vert _{H^1(\Omega )} + \Vert g {\partial }^2 {\tilde{h}} \Vert _{H^1(\Omega )} + \Vert {\partial }g {\partial }{\tilde{h}} \Vert _{H^1(\Omega )}\\&\lesssim \Vert f \Vert _{H^1(\Omega )} + \Vert g \Vert _{{{\mathcal {H}}}^2(\Omega )} \Vert {\tilde{h}} \Vert _{H^3(\Omega )} . \end{aligned} \end{aligned}$$

(B.8)

By the Lax-Milgram theorem there exists a unique solution u to (B.7) satisfying

$$\begin{aligned} \Vert u \Vert _{H^1(\Omega )}&\lesssim \Vert {\tilde{f}} \Vert _{L^2(\Omega )}. \end{aligned}$$

In particular, by Lemma A.1,

$$\begin{aligned} \Vert u \Vert _{L^6(\Omega )} \lesssim \Vert u \Vert _{H^1(\Omega )} \lesssim \Vert {\tilde{f}} \Vert _{L^2(\Omega )}. \end{aligned}$$

Applying Corollary B.2 and Proposition B.3 to (B.7), we get

$$\begin{aligned} \Vert u \Vert _{H^3(\Omega )}&\lesssim \Vert g^{ij} {\partial }_i {\partial }_j u \Vert _{H^1(\Omega )} + \Vert u \Vert _{H^{5/2}({\partial }\Omega )} \\&\lesssim \Vert \triangle _g u \Vert _{H^1(\Omega )} + \Vert {\partial }g {\partial }u \Vert _{H^1(\Omega )} \\&\lesssim \Vert {\tilde{f}} \Vert _{H^1(\Omega )} + \Vert g \Vert _{{{\mathcal {H}}}^2(\Omega )} \Vert {\partial }u \Vert _{C^0({\overline{\Omega }})} \\&\lesssim \Vert {\tilde{f}} \Vert _{H^1(\Omega )} + \Vert g \Vert _{{{\mathcal {H}}}^2(\Omega )} \Vert u \Vert _{W^{1,6}(\Omega )} \\&\lesssim \Vert {\tilde{f}} \Vert _{H^1(\Omega )} + \Vert g \Vert _{{{\mathcal {H}}}^2(\Omega )} \Vert {\tilde{f}} \Vert _{L^6(\Omega )} \\&\lesssim \Vert {\tilde{f}} \Vert _{H^1(\Omega )}, \\&\lesssim \Vert f \Vert _{H^1(\Omega )} + \Vert h \Vert _{H^{5/2}({\partial }\Omega )}, \end{aligned}$$

where we used Lemma A.1 and (B.8). This finishes the proof of Corollary B.4. $\quad \square$

1.2 B.2. Global estimates for Neumann data

In this section we collect standard global elliptic estimates for given Neumann boundary data. These estimates are applied in Sect.  5 to the metric components $g^{33}$ and $g^{3A}$ in boundary harmonic coordinates.

We recall the standard elliptic estimates in a smooth domain, see for example (2.3.3.1) in [10].

Theorem B.5

Let $1<p<\infty$ be a real and $\Omega$ be a bounded smooth domain. Let $g \in C^{0,1}(\overline{\Omega })$ be a Riemannian metric on $\overline{\Omega }$. Let further $B(u):= b^i {\partial }_i u$ be a boundary operator with $b^i$ Lipschitz on ${\partial }\Omega$ and $g(b,N)\ge c$ for some $c>0$. Then for any $u \in W^{2,p}(\Omega )$, we have

$$\begin{aligned} \Vert u \Vert _{W^{2,p}(\Omega )} \lesssim \Vert \triangle _g u \Vert _{L^{p}(\Omega )} + \Vert B(u) \Vert _{W^{1-1/p,p}({\partial }\Omega )}. \end{aligned}$$

In the special case of boundary harmonic coordinates and Neumann boundary data, the proof of Theorem B.5 generalises to the following result.

Theorem B.6

(Global elliptic estimates for the Neumann problem, $n=3$). Let $3<p< \infty$ be a real and let $\Omega \subset {{\mathbb {R}}}^3$ be a bounded smooth domain. Let $g \in {{\mathcal {W}}}^{1,p}(\Omega )$ be a Riemannian metric on $\Omega$ such that $\triangle _g = g^{ij} {\partial }_i {\partial }_j$. Then for every $1<p'<\infty$ and $u \in W^{2,p'}(\Omega )$,

$$\begin{aligned} \Vert u \Vert _{W^{2,p'}(\Omega )} \lesssim \Vert \triangle _g u \Vert _{L^{p'}(\Omega )} +\Vert N(u) \Vert _{W^{1-1/p',p'}({\partial }\Omega )}. \end{aligned}$$

(B.9)

Proof

The proof of Theorem B.5 in [10] uses first an analysis of the Neumann problem for a constant coefficient operator and second a continuity argument. We claim that both the same steps go through in the low regularity setting. Indeed, on the one hand, by Lemma A.1 and $p>3$,

$$\begin{aligned} g^{ij} \in W^{1,p} \subset C^{0,{\alpha }} \end{aligned}$$

for some ${\alpha }>0$. Therefore, at each point, $g^{ij} {\partial }_i {\partial }_j$ is a well-defined pointwise elliptic operator.

On the other hand, inspecting the continuity argument in the proof of Theorem B.5 in [10], it follows that in the case $\triangle _g = g^{ij} {\partial }_i {\partial }_j$ and $B=N$, the regularity $g^{ij} \in C^{0,{\alpha }}$ suffices to derive (B.9), see in particular the estimates (2.3.3.5) and (2.3.3.6) in [10] where the regularity of $g^{ij}$ is used. This finishes the proof of Theorem B.6. $\quad \square$

Appendix C. Interior Elliptic Estimates

In this section we derive interior elliptic estimates in fractional regularity and control the Dirichlet problem for the Laplace–Beltrami operator.

We recall the following interior estimates, see Proposition 3.4 in [15].

Theorem C.1

(Interior estimates in $H^{3/2}$). Let $\Omega ' \subset \subset \Omega \subset {{\mathbb {R}}}^2$ be two bounded smooth domains, and let $g \in {{\mathcal {H}}}^{3/2}(\Omega )$ be a Riemannian metric. Then for all $u \in H^{3/2}(\Omega )$,

$$\begin{aligned} \Vert u \Vert _{H^{3/2}(\Omega ')} \lesssim \Vert \triangle _g u \Vert _{H^{-1/2}(\Omega )} + \Vert u \Vert _{H^{1/2}(\Omega )}. \end{aligned}$$

Moreover, we have the following interior estimates. They are derived analogously to Proposition B.3, see also Theorem A.3 in [20].

Proposition C.2

(Interior estimates in $W^{2,4}$). Let $\Omega ' \subset \subset \Omega \subset {{\mathbb {R}}}^2$ be two bounded smooth domains. Let $g \in {{\mathcal {W}}}^{1,4}(\Omega )$ be a Riemannian metric. Then for all $u \in W^{2,4}(\Omega )$,

$$\begin{aligned} \Vert u \Vert _{W^{2,4}(\Omega ')} \lesssim \Vert \triangle _g u \Vert _{L^{4}(\Omega )} + \Vert u \Vert _{L^{4}(\Omega )} . \end{aligned}$$

By combining Theorem C.1 and Proposition C.2, we have the following stronger interior elliptic estimates for the Laplace–Beltrami operator.

Corollary C.3

(Interior estimates in $H^{5/2}$, $n=2$). Let $\Omega ' \subset \subset \Omega \subset {{\mathbb {R}}}^2$ be two bounded smooth domains. Let $g \in {{\mathcal {H}}}^{3/2}(\Omega )$ be a Riemannian metric. Then for all $u \in H^{5/2}(\Omega )$, we have

$$\begin{aligned} \Vert u \Vert _{H^{5/2}(\Omega ')} \lesssim \Vert \triangle _g u \Vert _{H^{1/2}(\Omega )} + \Vert u \Vert _{H^{3/2}(\Omega )}. \end{aligned}$$

(C.1)

Proof

Let $g \in {{\mathcal {H}}}^{3/2}(\Omega )$ and let

$$\begin{aligned} \Omega ' \subset \subset \Omega _2 \subset \subset \Omega _1 \subset \subset \Omega \end{aligned}$$

be bounded open subsets with smooth boundary. By Theorem C.1,

$$\begin{aligned} \Vert u \Vert _{H^{3/2}(\Omega _1)} \lesssim \Vert \triangle u \Vert _{{H^{-1/2}(\Omega )}} + \Vert u \Vert _{H^{1/2}(\Omega )}. \end{aligned}$$

(C.2)

By Proposition C.2 and (C.2),

$$\begin{aligned} \begin{aligned} \Vert u \Vert _{W^{2,4}(\Omega _2)}&\lesssim \Vert \triangle _g u \Vert _{L^4(\Omega _1)} + \Vert u \Vert _{L^4(\Omega _1)} \\&\lesssim \Vert \triangle _g u \Vert _{H^{1/2}(\Omega )} + \Vert u \Vert _{H^{1/2}(\Omega )}. \end{aligned} \end{aligned}$$

(C.3)

Applying now Theorem C.1 to ${\partial }u$ and using (C.2) and (C.3), we get

$$\begin{aligned} \begin{aligned} \Vert {\partial }u \Vert _{H^{3/2}(\Omega ')}&\lesssim \Vert \triangle _g {\partial }u \Vert _{H^{-1/2}(\Omega _2)} + \Vert {\partial }u \Vert _{H^{1/2}(\Omega _2)} \\&\lesssim \Vert \triangle _g u \Vert _{H^{1/2}(\Omega _2)} +\Vert [\triangle _g, {\partial }] u \Vert _{H^{-1/2}(\Omega _2)} + \Vert u \Vert _{H^{3/2}(\Omega _2)}. \end{aligned} \end{aligned}$$

(C.4)

The commutator term equals schematically

$$\begin{aligned}{}[\triangle _g, {\partial }] u = g^2 {\partial }^2 g {\partial }u + g^2 {\partial }g {\partial }^2 u. \end{aligned}$$

By Lemma A.1, the first term of the commutator is bounded by (C.3) as

$$\begin{aligned} \Vert g^2 {\partial }^2 g {\partial }u \Vert _{H^{-1/2}(\Omega _2)}&\lesssim \Vert {\partial }^2 g {\partial }u \Vert _{H^{-1/2}(\Omega _2)} \\&\lesssim \Vert {\partial }^2 g \Vert _{{{\mathcal {H}}}^{-1/2}(\Omega _2)} \Vert {\partial }u \Vert _{W^{1,4}(\Omega _2)} \\&\lesssim \Vert g \Vert _{{{\mathcal {H}}}^{3/2}(\Omega _2)} \Vert u \Vert _{W^{2,4}(\Omega _2)} \\&\lesssim \Vert g \Vert _{{{\mathcal {H}}}^{3/2}(\Omega )} \Big ( \Vert \triangle _g u \Vert _{H^{1/2}(\Omega )} + \Vert u \Vert _{H^{1/2}(\Omega )} \Big ). \end{aligned}$$

Similarly, the second term of the commutator is bounded by

$$\begin{aligned} \Vert g^2 {\partial }g {\partial }^2 u \Vert _{H^{-1/2}(\Omega _2)}&\lesssim \Vert {\partial }g {\partial }^2 u \Vert _{H^{-1/2}(\Omega _2)} \\&\lesssim \Vert {\partial }g \Vert _{{{\mathcal {H}}}^{1/2}(\Omega _2)} \Vert {\partial }^2 u \Vert _{W^{1,4}(\Omega _2)} \\&\lesssim \Vert g \Vert _{{{\mathcal {H}}}^{3/2}(\Omega )} \Big (\Vert \triangle _g u \Vert _{H^{1/2}(\Omega )} + \Vert u \Vert _{H^{1/2}(\Omega )} \Big ). \end{aligned}$$

Plugging this into (C.4) and summing over all coordinate derivatives proves (C.1). This finishes the proof of Proposition C.3. $\quad \square$

The proof of the next corollary of Proposition C.3 follows by the Lax-Milgram theorem and is left to the reader.

Corollary C.4

(Interior estimates for the Dirichlet problem in. $H^{5/2}$, $n=2$). Let $\Omega \subset {{\mathbb {R}}}^2$ be a bounded smooth domain and let $g \in H^{3/2}(\Omega )$ be a Riemannian metric on $\Omega$. Then for every $f \in H^{1/2}(\Omega )$, there exists a unique solution u to

$$\begin{aligned} \triangle _g u&= f \text { in } \Omega ,\\ u&= 0\text { on } {\partial }\Omega . \end{aligned}$$

Moreover, for every smooth domain $\Omega ' \subset \subset \Omega$, we have $u \in H^{5/2}(\Omega ')$ and

$$\begin{aligned} \Vert u \Vert _{H^{5/2}(\Omega ')} \lesssim \Vert f \Vert _{H^{1/2}(\Omega )}. \end{aligned}$$