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A Nash–Hörmander iteration and boundary elements for the Molodensky problem

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Abstract

We investigate the numerical approximation of the nonlinear Molodensky problem, which reconstructs the surface of the earth from the gravitational potential and the gravity vector. The method, based on a smoothed Nash–Hörmander iteration, solves a sequence of exterior oblique Robin problems and uses a regularization based on a higher-order heat equation to overcome the loss of derivatives in the surface update. In particular, we obtain a quantitative a priori estimate for the error after \(m\) steps, justify the use of smoothing operators based on the heat equation, and comment on the accurate evaluation of the Hessian of the gravitational potential on the surface, using a representation in terms of a hypersingular integral. A boundary element method is used to solve the exterior problem. Numerical results compare the error between the approximation and the exact solution in a model problem.

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Acknowledgments

We thank Lothar Banz for instructive discussions about computational aspects of this work. This work was supported by the cluster of excellence QUEST, the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation and the Danish Science Foundation (FNU) through research grant 10-082866. H. G. thanks the Institut für Angewandte Mathematik for hospitality.

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Correspondence to Heiko Gimperlein.

Appendix

Appendix

We are going to use the following definition of the Hölder-spaces, but will rarely use them for integer exponent \(a\):

Definition 1

(Definition A.3 [5]) Let \(k \in \mathbb{N }_0, k<a \le k+1\) and \(B \subseteq {\mathbb{R }}^n\) compact, convex such that \(\mathring{B} \ne \emptyset \).

Define

$$\begin{aligned}&{\fancyscript{H}}^{a}(B):=\left\{ u \in C^k(B): \Vert u\Vert _0=\sup \limits _{x \in B} |u(x)|<\infty \quad \hbox {and}\right. \\&\quad \left. |u|_a:=\sum _{|\alpha |=k}\sup \limits _{x \ne y \in B} \frac{|\partial ^\alpha u(x)-\partial ^{\alpha }u(y)|}{|x-y|^{a-k}}<\infty \right\} . \end{aligned}$$

We also set \({\fancyscript{H}}^0(B):=C(B)\). Then \({\fancyscript{H}}^a := {\fancyscript{H}}^a(B)\) with the norm \(\Vert \cdot \Vert _a:=\Vert \cdot \Vert _0+|\cdot |_a\) is a Banach space.

For a compact manifold, one defines \({\fancyscript{H}}^a\) by covering it with a finite number of neighborhoods homeomorphic to subsets of \(\mathbb{R }^n\).

Basic interpolation estimates will be used frequently:

$$\begin{aligned} \Vert v\Vert _{\sigma a + (1-\sigma )b} \le C \Vert v\Vert _a^{\sigma } \Vert v\Vert _{b}^{1-\sigma }\quad (\sigma \in (0,1), \ v \in {\fancyscript{H}}^{\max \{a, b\}}). \end{aligned}$$
(27)

Similarly, if \((a,b) \in \mathbb{R }_+^2\) belongs to the convex hull of \((a_1, b_1),\ldots , (a_J,b_J)\in \mathbb{R }_+^2\), then

$$\begin{aligned} \Vert v\Vert _{a}\Vert w\Vert _{b} \le C \sum _{j=1}^{J}\Vert v\Vert _{a_j} \Vert w\Vert _{b_j} \quad (v \in {\fancyscript{H}}^{\max \{a_j\}}, \ w \in {\fancyscript{H}}^{\max \{b_j\}}). \end{aligned}$$
(28)

1.1 Proof of Theorem 2

For ease of presentation, we set \(u = (u^{(1)},u^{(2)}):=(W, \varphi ) : S^2 \rightarrow \mathbb{R } \times \mathbb{R }^3, f := (G,W) : S^2 \rightarrow \mathbb{R }^3 \times \mathbb{R }\). The map from \((W,\varphi )\) to the corresponding \(G\) is denoted by \(\Gamma \). The Molodensky problem assumes the form \(\Phi (u) := (\Gamma (u), u^{(1)})=f\).

In this notation, the Nash–Hörmander iteration reads as

$$\begin{aligned} u_{m+1}=u_m + \Delta _m {{\dot{u}}}_m, \quad {{\dot{u}}}_m = \Psi (v_m) g_m, \quad v_m=S_{\theta _m} u_m, \end{aligned}$$

\(\Psi \) being the inverse operator to the linearization of \(\Phi \) around \(v_m\). To show that the Algorithm 1 is a reformulation of the one in [5], we show that the Eqs. (15) and (16) are equivalent to usual definition

$$\begin{aligned} \Delta _0 g_0 \!=\! S_{\theta _0} f, \quad g_m \!=\! \Delta _m^{-1} ((S_{\theta _m}\!-\!S_{\theta _{m-1}})(f-E_{m-1})- \Delta _{m-1}S_{\theta _m}e_{m-1})\quad (m>0).\nonumber \\ \end{aligned}$$
(29)

Here, the errors \(e_m= e_m' + e_m''\) are defined as the sum of a smoothing error

$$\begin{aligned} e_m'=(\Phi '(u_m)-\Phi '(v_m)){\dot{u}}_m \end{aligned}$$
(30)

and a linearization error

$$\begin{aligned} e_m'' = \Delta _m^{-1}(\Phi (u_{m}+\Delta _m {{\dot{u}}}_m)-\Phi (u_m)-\Delta _m\Phi '(u_m){\dot{u}}_m). \end{aligned}$$
(31)

The total error is \(E_m=e_0+\cdots +e_{m-1}\) (\(e_0 = E_0 = 0\)).

In our case, \(\Phi '(u){\dot{u}} = (\Gamma '(u){\dot{u}}, {\dot{u}}^{(1)})\), so that with \({\dot{G}}_m = \Gamma '(W_m, \varphi _m)({\dot{W}}_m, {\dot{\varphi }}_m)\)

$$\begin{aligned} e_m'&= \Phi '(W_m, \varphi _m)({\dot{W}}_m, {\dot{\varphi }}_m)-\Phi '(\widetilde{W}_m, \widetilde{\varphi }_m)({\dot{W}}_m, {\dot{\varphi }}_m)\\&= (\Gamma '(W_m, \varphi _m)({\dot{W}}_m, {\dot{\varphi }}_m), {\dot{W}}_m) - (\Gamma '(\widetilde{W}_m, \widetilde{\varphi }_m)({\dot{W}}_m, {\dot{\varphi }}_m), {\dot{W}}_m)\\&= ({\dot{G}}_m - g_m^{(1)},0). \end{aligned}$$

Similarly, for the linearization error we have

$$\begin{aligned} \Delta _m e_m'' = \Phi (W_m\!+\!\Delta _m{\dot{W}}_m, \varphi _m\!+\! \Delta _m{\dot{\varphi }}_m)\!-\!\Phi (W_m, \varphi _m)\!-\!\Delta _m\Phi '(W_m, \varphi _m)({\dot{W}}_m, {\dot{\varphi }}_m), \end{aligned}$$

with second component \(W_m+\Delta _m{\dot{W}}_m - W_m - \Delta _m{\dot{W}}_m = 0\). The first component, by definition, is \(G_{m+1}-G_m-\Delta _m {\dot{G}}_m\).

As a consequence, the second components of the errors \(e_m\) and \(E_m\) vanish. Equation (29) yields

$$\begin{aligned} \Delta _0 g_0^{(2)}\!=\!S_{\theta _0}(W\!-\!W_0),\quad \Delta _m g_m^{(2)}\!=\! S_{\theta _m}(W\!-\!W_0)\!-\!S_{\theta _{m-1}} (W\!-\!W_0) \,\, (m\!>\!0).\qquad \quad \end{aligned}$$
(32)

Concerning the first component of \(g_m\), we use \(\Delta _0 g_0^{(1)} = S_{\theta _0}(G-G_0)\) and our above computation:

$$\begin{aligned} \Delta _1 g_1^{(1)}&= (S_{\theta _1}-S_{\theta _{0}})(G-G_0)-\Delta _0 S_{\theta _1}({\dot{G}}_0 - g_0^{(1)})-S_{\theta _1}(G_1-G_0-\Delta _0{\dot{G}}_0) \\&= S_{\theta _1}(G-G_1+\Delta _0g_0^{(1)})-S_{\theta _0}(G-G_0). \end{aligned}$$

Similarly we obtain a recursion formula for \(m>1\):

$$\begin{aligned} \Delta _m g_m^{(1)}=S_{\theta _m} \left( G-G_m+\sum _{j=0}^{m-1}\Delta _jg_j^{(1)} \right) -S_{\theta _{m-1}}\left( G-G_{m-1}+ \sum _{j=0}^{m-2}\Delta _jg_j^{(1)}\right) .\nonumber \\ \end{aligned}$$
(33)

Equations (32) and (33) show the equivalence of Algorithm 1 to the formulation in [5].

Proof of Theorem 2

By the above reduction, it is sufficient to analyze Hörmander’s iterative method and derive an a priori estimate for the error after \(m\) steps. Our proof relies on certain estimates in Hörmander’s qualitative analysis. Recall that the smoothing operator is generated by \((-1)^k \Delta ^k\).

The proof is given in several steps. We are going to rely on a number of auxiliary results.

Given a sufficiently large, fixed \(a_\Phi \), we recall the following continuity estimates for \(\Phi ''\) and the inverse of the linearization \(\Psi \) from ([5], equations (2.1.5/6)):

  1. 1.

    For all \(\epsilon >0, 0\le a \le a_\Phi \) and \(u, v,w\in C^\infty \) with \(\Vert u\Vert _{2+\epsilon }<C\) we have:

    $$\begin{aligned} \Vert \Phi ''(u;v,w)\Vert _{a+2\epsilon }&\lesssim \Vert v\Vert _{a+2+3\epsilon }\Vert w\Vert _{0}\!+\!\Vert v\Vert _{0}\Vert w\Vert _{a+2+3\epsilon }\nonumber \\&+ \Vert v\Vert _0\Vert w\Vert _0\Vert u\Vert _{a+3+2\epsilon } \end{aligned}$$
    (34)
  2. 2.

    For all \(\epsilon >0, 0\le a \le a_\Phi \) and \(v,g\in C^\infty \) with \(\Vert v\Vert _{2+\epsilon }<C\)

    $$\begin{aligned} \Vert \Psi (v)g\Vert _{a+\epsilon } \lesssim \Vert g\Vert _{a+\epsilon } + \Vert g\Vert _{\epsilon } \Vert v\Vert _{a+2+\epsilon } \end{aligned}$$
    (35)

\(\square \)

The first lemma translates a bound for the first \(m\) increments \({\dot{u}}_j\) into properties of \(U_m= \sum _{j=0}^m \triangle _j{\dot{u}}_j\).

Lemma 1

Let \(\epsilon >0,\alpha +\epsilon \notin \mathbb{N }, {-}\epsilon \le \alpha _{-} \le \alpha \le \alpha _{+}\). Assume that \(2k > \alpha +\epsilon - a\) and that for some \(\delta >0, m \ge 0\)

$$\begin{aligned} \Vert \dot{u}_j \Vert _{a+\epsilon } \le \delta \theta _j^{a-\alpha -1} \quad \quad \forall 0\le j \le m \ \forall a \in [\alpha _-,\alpha _{+}]. \end{aligned}$$
(36)

Then \(U_m= \sum _{j=0}^m \triangle _j {\dot{u}}_j \in {\fancyscript{H}}^{\alpha +\epsilon }\) satisfies

$$\begin{aligned} \Vert U_m\Vert _a&\le C_1 \delta \quad \quad \forall a \le \alpha +\epsilon ,\end{aligned}$$
(37)
$$\begin{aligned} \Vert U_m-S_{\theta _{m+1}}U_m\Vert _a&\le C_2 \delta \theta _{m+1}^{a-\alpha -\epsilon }\quad \quad \forall \, 0\le a\le \alpha _{+} +\epsilon ,\end{aligned}$$
(38)
$$\begin{aligned} \Vert S_{\theta _{m+1}}U_m\Vert _a&\le {C}_3 \delta \theta _{m+1}^{(a-\alpha -\epsilon )_{+}}\quad \quad \forall \, 0 \le a \le a_0 \end{aligned}$$
(39)

for fixed \(a_0\).

We refer to [5, Lemma 2.2.1]) for the proof.

The lemma implies that under the given assumptions all iterates \(u_m\) will remain in a neighborhood of \(u_0\). We may therefore localize and appeal to estimates valid near \(u_0\) and justify the linearization of the problem. A quantitative formulation of the localization is as follows:

Corollary 1

Let \({\widetilde{\epsilon }}>0, \mu \le \alpha + \epsilon \), and \(a, \delta \) as above. Define

$$\begin{aligned} V_s&:= \lbrace u \in C^{\infty }: \Vert u-u_0\Vert _\mu \le s \rbrace ,\\ {\widetilde{V}}_s&:= \lbrace u \in C^{\infty }: \Vert u\Vert _\mu \le s \rbrace . \end{aligned}$$

Then there exist \(C, C'>0\) (which depend only on \(C_1\) and the constants in Theorem 3(i), (iii’)) such that for all \(\theta \ge C(\frac{{\widetilde{\epsilon }}}{\Vert u_0\Vert _a})^{\frac{1}{a-\mu }}\) and all \(\delta <C' \tilde{\epsilon }\):

  1. (a)

    \(S_\theta u_0 \in V_{{\widetilde{\epsilon }}/2}\)

  2. (b)

    If \(a=\mu \le \alpha +\epsilon \), then \(U_k,S_\theta U_k \in {\widetilde{V}}_{{\widetilde{\epsilon }}/2}, u_{k+1}=u_0 +U_k \in V_{{\widetilde{\epsilon }}}\) and \(S_{\theta _{k+1}} u_{k+1} \in V_{{\widetilde{\epsilon }}}\).

We note some related estimates for the smoothed iterates \(v_m=S_{\theta _m}u_m\), which in particular hold for \(b=\alpha +\epsilon \):

Corollary 2

Under the above assumptions there holds:

$$\begin{aligned} \Vert u_j-v_j\Vert _c&\le C (\Vert u_0\Vert _b \theta _j^{c-b} + \delta \theta _j^{c-\alpha -\epsilon })\quad \forall \, c\le \alpha +\epsilon ,\ 0\le b - c<2k \qquad \end{aligned}$$
(40)
$$\begin{aligned} \Vert v_j\Vert _c&\le C(\Vert u_0\Vert _b \theta _j^{c-b} + \delta \theta _j^{(c-\alpha -\epsilon )_+})\quad \forall \, c\le c_0, \ b\le c \end{aligned}$$
(41)

Proof

Indeed, using Property (iii’) of \(S_\theta \) and Lemma 1,

$$\begin{aligned} \Vert u_j-v_j\Vert _c&\le \Vert u_0-S_{\theta _j}u_0\Vert _{c}+ \Vert U_{j-1}-S_{\theta _j}U_{j-1}\Vert _c\nonumber \\&\le C \Vert u_0\Vert _b \theta _j^{c-b} +C \delta \theta _j^{c-\alpha -\epsilon }. \end{aligned}$$
(42)

Similarly, \(\Vert v_j\Vert _c \le \Vert S_{\theta _j}u_0\Vert _c+ \Vert S_{\theta _j}U_{j-1}\Vert _c \le C \theta _j^{c-b}\Vert u_0\Vert _b +C \delta \theta _j^{(c-\alpha -\epsilon )_{+}}\). \(\square \)

The a priori estimate of Theorem 2 will be shown by induction in \(m\).

As hypothesis we assume (36), that \(\Vert f\Vert _{\alpha +\epsilon }\) is small and that the assumptions of Lemma 1 are verified for a suitable \(\delta \). We are going to deduce the corresponding assertions with \(m\) replaced by \(m+1\).

Together with the induction hypothesis, Lemma 1 and the above corollaries provide bounds of the Hölder norms of \(u_j, v_j\) and \(u_j-v_j\) for \(0\le j\le m\). To estimate \({\dot{u}}_{m+1}\) in terms of these data, note that by definition of \({\dot{u}}_{m+1}\) and \(g_{m+1}\),

$$\begin{aligned} \Vert {\dot{u}}_{m+1}\Vert _{a+\epsilon }&\le C(\Vert g_{m+1}\Vert _{a+\epsilon }+\Vert g_{m+1}\Vert _{\epsilon } \theta _{m+1}^{(2+a-\alpha )_{+}})\\ g_{m+1}&= \widetilde{S}_m(f-E_m)-\frac{\triangle _m}{\triangle _{m+1}} S_{\theta _{m+1}}e_m \end{aligned}$$

where \(E_m=\sum _{j=0}^{m-1} \triangle _j e_j\) and \(\widetilde{S}_m = \frac{S_{\theta _{m+1}}-S_{\theta _{m}}}{\theta _{m+1}-\theta _{m}}\). Writing \({\widetilde{S}}_m\) as an average of \(\frac{d}{d\theta } S_\theta ,{\widetilde{S}}_m f\) can be bounded using Property (iv) of the smoothing operator,

$$\begin{aligned} \Vert {\tilde{S}}_m f\Vert _b \le C \theta _m^{b-c-1} \Vert f\Vert _c, \end{aligned}$$

for any \(b,c\). Properties (ii) and (iv) of the smoothing operator give similar estimates for the other terms:

$$\begin{aligned} \Vert S_{\theta _{m+1}}e_m\Vert _{b}&\le C \theta _m^{(b-c')_{+}} \Vert e_m\Vert _{c'}\end{aligned}$$
(43)
$$\begin{aligned} \Vert {\widetilde{S}}_mE_m\Vert _{b}&\le \theta _m^{b-c''-1} \sum _{j=0}^{m-1} \triangle _j \Vert e_j\Vert _{c''}. \end{aligned}$$
(44)

Hence

$$\begin{aligned} \Vert g_{m+1}\Vert _{a+\varepsilon }&\lesssim \theta _m^{a+\varepsilon -c-1} \Vert f\Vert _c + \theta _m^{a+\varepsilon -c''-1} \sum _{j=0}^{m-1} \triangle _j \Vert e_j\Vert _{c''}\nonumber \\&+\, \theta _m^{(a+\varepsilon -c')_{+}} \Vert e_m\Vert _{c'}, \end{aligned}$$
(45)
$$\begin{aligned} \theta _{m+1}^{(2+a-\alpha )_{+}} \Vert g_{m+1}\Vert _{\varepsilon }&\lesssim \theta _m^{\varepsilon -C-1+(2+a-\alpha )_{+}} \Vert f\Vert _C + \theta _m^{\varepsilon -C''-1+(2+a-\alpha )_{+}} \sum _{j=0}^{m-1} \triangle _j \Vert e_j\Vert _{C''} \nonumber \\&+\,\theta _m^{(\varepsilon -C')_{+}+(2+a-\alpha )_{+}} \Vert e_m\Vert _{C'}, \end{aligned}$$
(46)

giving a bound on \({\dot{u}}_{m+1}\). If we choose \(c=\alpha +\varepsilon , C=\alpha -a+\varepsilon +(2+a-\alpha )_+\le \alpha +\varepsilon \), the terms involving \(f\) are dominated by \(\theta _{m+1}^{E}\Vert f\Vert _{\alpha + \epsilon }\)

To estimate \(e_j\), we will consider the smoothing error \(e'_j\) and the linearization error \(e_j''\) separately. At the end of this section we use (34) and (35) to bound \(e'_j\) resp. \(e_j''\) in terms of the Hölder norms of \(u_j, v_j\) and \(u_j-v_j\) for \(0\le j \le m\), which are controlled by the induction hypothesis. A computation eventually results in

$$\begin{aligned} \Vert {\dot{u}}_{m+1}\Vert _{a+\epsilon } \le C \theta _{m+1}^{E}(A \Vert f\Vert _{\alpha + \epsilon } + \delta \sigma (1+\delta +\delta ^2)) \end{aligned}$$

for small \(\sigma \in (0,1)\) whenever \(\theta _0 =\theta _0(\sigma , u_0, \alpha , S_\theta )\) is sufficiently large. \(A\) only depends on the constants in (35) and Property (iv) of the smoothing operator. Both \(\theta _0\) and \(A\) are, in principle, explicit. We choose \(\sigma = \frac{1}{2C}\frac{1}{1+\delta +\delta ^2}\).

Then for all \(f\) in the ball \(\lbrace u: \Vert u\Vert _{\alpha + \epsilon } \le \frac{\delta }{2AC} \rbrace \) we have

$$\begin{aligned} \Vert {\dot{u}}_{m+1}\Vert _{a+\epsilon } \le \delta \theta _{m+1}^E. \end{aligned}$$
(47)

On the other hand, in the first step the solution to the linearized problem \({\dot{u}}_0=\triangle _0^{-1}\Psi (S_{\theta _0}u_0) S_{\theta _0} f\) is easily estimated using (35) and the smoothing properties

$$\begin{aligned} \triangle _0 \Vert {\dot{u}}_0\Vert _{a+\epsilon }&\le C' (\Vert S_{\theta _0}f\Vert _{a+\epsilon } + \Vert S_{\theta _0}f\Vert _{\epsilon } \Vert S_{\theta _0}u_0\Vert _{a+2+\epsilon })\\&\le C'' \theta _0(1+\theta _0^{-a} \Vert u_0\Vert _{2+\epsilon +a}) \Vert f\Vert _{\alpha + \epsilon } \theta _0^{E}. \end{aligned}$$

We now denote by \(\mathfrak{C }\) the maximum of \(C''\triangle _0^{-1} \theta _0(1+\theta _0^{-a} \Vert u_0\Vert _{2+\epsilon +a})\) and the previous constant \(2AC\) and choose \(\delta = \mathfrak{C }\Vert f\Vert _{\alpha +\epsilon }\). Since \(\Vert f\Vert _{\alpha +\epsilon }\) was sufficiently small by hypothesis, so is \(\delta \), and (47) is fulfilled. By induction, we deduce

$$\begin{aligned} \Vert {\dot{u}}_{m+1}\Vert _{a+\epsilon } \le \mathfrak{C } \Vert f\Vert _{\alpha +\epsilon } \theta _{m+1}^E \quad \forall \,k\ge 0. \end{aligned}$$
(48)

If \(u\) denotes the exact solution, we obtain

$$\begin{aligned} \Vert u-u_m\Vert _{a+\epsilon }&\le \sum _{j=m+1}^\infty \triangle _j \Vert {\dot{u}}_j \Vert _{a+\epsilon } \le \mathfrak{C } \Vert f\Vert _{\alpha +\epsilon } \sum _{j=m+1}^\infty \triangle _j \theta _j^{E}\\&\le C_{\tau }\mathfrak{C }\Vert f\Vert _{\alpha +\epsilon } \theta _m^{E+1+\tau } \end{aligned}$$

for any \(\tau >0\) small such that \(E+1+\tau <0\). As \(u\) was the exact solution, this yields the assertion of Theorem 2.

Note that

$$\begin{aligned} \Phi (u_{m+1}) \!-\! \Phi (u) \!=\! \Phi (u_{m+1}) \!-\! \Phi (u_0) \!-\! f = (S_{\theta _m} f -f) \!+\! \triangle _m e_m + (E_m-S_{\theta _m} E_m). \end{aligned}$$

We have shown that the right hand side converges to \(0\) in \({\mathcal{H }}^{a+\epsilon }\), so that also \(\Phi (u_{m})\) converges to \(\Phi (u)\).

To complete the proof, it remains to estimate \(e_m\) and to translate the result into a bound on \({\dot{u}}_{m+1}\). For the \(j\)-th smoothing error

$$\begin{aligned} e'_j = (\Phi '(u_j)-\Phi '(v_j)){\dot{u}}_j = \int _0^1 \Phi ''(v_j + t(u_j-v_j); u_j-v_j, {\dot{u}}_j)~dt, \end{aligned}$$

Equation (34) implies

$$\begin{aligned} \Vert e_j'\Vert _{2\epsilon +a}&\lesssim \Vert u_j-v_j\Vert _{2+3\epsilon +a} \Vert {\dot{u}}_j\Vert _0 + \Vert u_j-v_j\Vert _{0}\Vert {\dot{u}}_j\Vert _{2+3\epsilon +a}\\&+\,2\Vert u_j-v_j\Vert _{0}\Vert {\dot{u}}_j\Vert _0(\Vert u_j\Vert _{3+2\epsilon +a}+ \Vert v_j\Vert _{3+2\epsilon +a}). \end{aligned}$$

Using Lemma 1, \(\Vert u_j\Vert _b\le \Vert u_0\Vert _b + C \delta \), and the estimates for \(v_j\) and \(u_j-v_j\) from (40), (41), we obtain

$$\begin{aligned} \Vert e_j'\Vert _{2\epsilon +a}&\lesssim (||u_0||_b \theta _j^{2+3\epsilon +a-b}+\delta \theta _j^{2+2\epsilon +a-\alpha }) \delta \theta _j^{-\alpha -1}\nonumber \\&+\,(\Vert u_0\Vert _{c_1}\theta _j^{-c_1}+\delta \theta _j^{-\alpha -\epsilon }) \delta \theta _j^{1+2\epsilon +a-\alpha } \nonumber \\&+\,(\Vert u_0\Vert _{c_2}\theta _j^{-c_2}+\delta \theta _j^{-\alpha -\epsilon })\delta \theta _j^{-\alpha -1} (\Vert u_0\Vert _{3+2\epsilon +a} +\delta +\delta \theta _j^{(a-\alpha -\epsilon )_{+}}).\nonumber \\ \end{aligned}$$
(49)

Similarly, as the remainder of the first Taylor approximation, \(e''_j\) is also controlled by \(\Phi ''\), and analogous estimates result in

$$\begin{aligned} \Vert e''_j\Vert _{2\epsilon +a}&\lesssim \triangle _j \left\{ \Vert {\dot{u}}_j\Vert _{2+3\epsilon +a}\Vert {\dot{u}}_j\Vert _0+ \Vert {\dot{u}}_j\Vert ^2_0(\Vert u_j\Vert _{3+2\epsilon +a}+ \Vert {\dot{u}}_j\Vert _{3+2\epsilon +a})\right\} \nonumber \\&\lesssim \triangle _j \left\{ \delta \theta _j^{1+3\epsilon +a-\alpha } \delta \theta _j^{-\alpha -1}\!+\!(\delta \theta _j^{-\alpha -1})^2 ( \Vert u_0\Vert _{3+2\epsilon +a} \!+\!\delta \!+\!\delta \theta _j^{(3+\epsilon +a-\alpha )_+}\!\right\} .\nonumber \\ \end{aligned}$$
(50)

To estimate \(g_{m+1}\) in (45) and (46), it remains bound sums \(\sum _{j=0}^{m-1} \triangle _j \{\Vert e'_j\Vert _{b}+\Vert e''_j\Vert _{b}\}\). We consider a generic term of the form \(\sum _{j=0}^{m-1} \triangle _j \theta _j^{-d} F(\delta ,u_0)\) for suitable \(F\) obtained from (49) resp. \(\sum _{j=0}^{m-1} \triangle _j^2 \theta _j^{-d} F(\delta ,u_0)\) from (50). Concerning the former, if \(d>1\), we have for any small \(\tau >0\)

$$\begin{aligned} \sum _{j=0}^{m-1} \triangle _j \theta _j^{-d}\le \theta _0^{-d+1+\tau } \sum _{j=0}^{m-1} \triangle _j \theta _j^{-1-\tau } \le C_\tau \theta _0^{-d+1+\tau }, \end{aligned}$$

with \(C_\tau \) independent of \(\theta _0\ge \theta _0^{\min } >0\) and \(\kappa > \kappa _{\min }>0\). Here we have used that

$$\begin{aligned} \triangle _j\theta _j^{-1-\tau } \lesssim \kappa ^{-1} \theta _j^{1-\kappa -1-\tau }=\kappa ^{-1} (\theta _0^{\kappa }+j)^{-1-\frac{\tau }{\kappa }}. \end{aligned}$$

For \(d<1\),

$$\begin{aligned} \sum _{j=0}^{m-1} \triangle _j \theta _j^{-d}\le \theta _k^{-d+1+\tau } \sum _{j=0}^{m-1} \triangle _j \theta _j^{-1-\tau } \le C_\tau \theta _m^{-d+1+\tau }. \end{aligned}$$

Finally, for \(d=1\)

$$\begin{aligned} \sum _{j=0}^{m-1} \triangle _j \theta _j^{-1} \le C_\tau \theta _m^{\tau }. \end{aligned}$$

As for the term coming from (50)

$$\begin{aligned} \sum _{j=0}^{m-1}\triangle _j^2 \theta _j^{-d} \le C_\tau \kappa ^{-1} \theta _0^{2- d-\kappa +\tau }. \end{aligned}$$

Estimating the sums in (45) resp. (46) thus increases the exponent on \(\theta _m\) resp. \(\theta _0\) in the estimates of \(e'_m\) by at most \(1+\tau \). From the estimates of \(e''_m\), one obtains \(\theta _0\) raised to a power which is arbitrarily negative for large \(\kappa \).

As a result

$$\begin{aligned}&\theta _m^{a+\varepsilon -c''-1} \sum _{j=0}^{m-1} \triangle _j \Vert e'_j\Vert _{c''}\nonumber \\&\quad \lesssim \delta \Vert u_0\Vert _b \theta _m^{a+\varepsilon -c''-1} \theta _{m/0}^{3+\varepsilon +c''-b+\tau }+\delta ^2 \theta _m^{a+\varepsilon -c''-1} \theta _{m/0}^{2+c''-2\alpha +\tau }\nonumber \\&\qquad +\,\delta \Vert u_0\Vert _{c_1} \theta _m^{a+\varepsilon -c''-1}\theta _{m/0}^{2-c_1+c''-\alpha +\tau } + \delta ^2\theta _m^{a+\varepsilon -c''-1}\theta _{m/0}^{2- \varepsilon +c''-2\alpha +\tau }\nonumber \\&\qquad +\,\delta \Vert u_0\Vert _{c_2}\Vert u_0\Vert _{3+c''} \theta _m^{a+\varepsilon -c''-1}\theta _{m/0}^{-c_2-\alpha +\tau }\nonumber \\&\qquad +\,\delta ^2\Vert u_0\Vert _{3+c''}\theta _m^{a+\varepsilon -c''-1} \theta _{m/0}^{-\varepsilon -2\alpha +\tau }+\delta ^2 \Vert u_0\Vert _{c_2} \theta _m^{a+\varepsilon -c''-1} \theta _{m/0}^{-c_2-\alpha +\tau }\nonumber \\&\qquad +\,\delta ^3\theta _m^{a+\varepsilon -c''-1} \theta _{m/0}^{-\varepsilon -2\alpha +\tau }+ \delta ^2 \Vert u_0\Vert _{c_2} \theta _m^{a+\varepsilon -c''-1}\theta _{m/0}^{-c_2-\alpha +(c''-\alpha -3\varepsilon )_++\tau }\nonumber \\&\qquad +\,\delta ^3\theta _m^{a+\varepsilon -c''-1} \theta _{m/0}^{-\varepsilon -2\alpha +(c''- \alpha -3\varepsilon )_++\tau }. \end{aligned}$$
(51)

Here \(\theta _{m/0}\) is \(\theta _{m}\) or \(\theta _{0}\), depending on whether its exponent is greater or smaller \(\tau \). Choosing e.g. \(c''=\alpha +2\varepsilon ,\, b=3+\varepsilon +c''+2\tau ,\, c_1=2+c''-\alpha +2\tau \) and \(c_2=0\), the exponents of \(\theta _{m/0}\) are negative and the exponent of \(\theta _m\) is strictly smaller than \(E=a-\alpha -1\). Similarly, we obtain

$$\begin{aligned} \theta _m^{a+\varepsilon -c''-1} \sum _{j=0}^{m-1} \triangle _j \Vert e''_j\Vert _{c''}&\lesssim \delta ^2 \theta _m^{a+\varepsilon -c''-1} \theta _{0}^{\varepsilon + c''-2\alpha +2-\kappa +\tau }\nonumber \\&+\,\delta ^2 (\Vert u_0\Vert _{3+c''} + \delta ) \theta _m^{a+\varepsilon -c''-1} \theta _{0}^{-2\alpha -\kappa +\tau } \nonumber \\&+\,\delta ^3 \theta _m^{a+\varepsilon -c''-1} \theta _{0}^{-2\alpha -\kappa +\tau }, \end{aligned}$$
(52)

where the exponents of the \(\theta _0\) and \(\theta _m\) have the same properties as in (51).

It remains to estimate the term \(\theta _m^{(a+\varepsilon -c')_{+}} \Vert e_m\Vert _{c'}\) in (45). We choose \(c'=a+\varepsilon \) and, in (49), set \(c_1=c_2\) equal to the above \(c''\) obtain

$$\begin{aligned} \theta _m^{(a+\varepsilon -c')_{+}} \Vert e_m\Vert _{c'}&\lesssim (||u_0||_b \theta _m^{2+2\epsilon +a-b}+\delta \theta _m^{2+\epsilon +a-\alpha }) \delta \theta _m^{-\alpha -1}\\&+\, (\Vert u_0\Vert _{c''}\theta _m^{-c''}+\delta \theta _m^{-\alpha -\epsilon }) \delta \theta _m^{1+\epsilon +a-\alpha } \\&+\, (\Vert u_0\Vert _{c''}\theta _m^{-c''}+\delta \theta _m^{-\alpha -\epsilon })\delta \theta _m^{-\alpha -1} (\Vert u_0\Vert _{3+\epsilon +a} +\delta +\delta \theta _m^{(a-\alpha )_{+}})\\&+\, \delta ^2 \triangle _m \theta _m^{2\epsilon +a-2\alpha } + \delta ^2 \triangle _m \theta _m^{-2\alpha -2}(\Vert u_0\Vert _{3+\varepsilon +a} +\delta )\\&+\,\delta ^3 \triangle _m \theta _m^{-2\alpha -2+(3+a-\alpha )_+}. \end{aligned}$$

The exponent of \(\theta _m\) is again strictly smaller than \(E\).

The analysis of (46) is analogous. This completes the proof of Theorem 2. \(\square \)

1.2 Proof of Theorem 3

We consider the operator \(A\) as an unbounded operator on the Hölder space \({\fancyscript{H}}^a\) with domain \(D(A)={\fancyscript{H}}^{a+2}\) (if \(a \notin \mathbb{N }_0\)). Using the nonpositivity of \(A\) and [29, Theorem 9.3], we see that \(A-\lambda \) is invertible for \(\lambda \in {\mathcal{S }}_{\theta }= \lbrace \lambda \in \mathbb{C }{\setminus }\{0\}: |\hbox {arg} \lambda | <\theta \rbrace , \, \theta \in (\pi /2,\pi )\), and that \((A-\lambda )^{-1}\) is a pseudodifferential operator, depending on the parameter \(\lambda \), whose symbol decays as \(\frac{C}{|\lambda |}\). The mapping properties [30, Proposition 8.6] of such operators in Hölder spaces, which are analogous to those for Sobolev spaces, therefore imply

$$\begin{aligned} \Vert (A-\lambda )^{-1} u\Vert _{a} \le \frac{C}{|\lambda |} \Vert u\Vert _{a}, \quad \forall \, \lambda \in S_{\theta ,0}. \end{aligned}$$
(53)

Equation (53) allows to define the analytic semigroup generated by \(A\),

$$\begin{aligned} e^{tA}u:= \frac{1}{2\pi i} \int _{\gamma _{r,\eta }} e^{t\lambda } (A -\lambda )^{-1} u \,d\lambda , \quad t>0, \end{aligned}$$
(54)

where \(r>0, \eta \in ]\pi /2,\pi [ \), and \(\gamma _{r,\eta }\) is the curve \(\lbrace \lambda \in \mathbb{C }: |\hbox {arg} \lambda |=\eta , |\lambda | \ge r\rbrace \cup \lbrace \lambda \in \mathbb{C }: |\hbox {arg} \lambda |\le \eta , |\lambda | = r\rbrace \), oriented counterclockwise. \(e^{tA}u\) does not depend on the choice of \(r\) and \(\eta \). We recall some basic properties of analytic semigroups (Proposition 2.1.1, [31]):

Proposition 2

  1. (i)

    \(\Vert e^{tA} u\Vert _a \le C_0 \Vert u\Vert _a, \ \forall t \ge 0\).

  2. (ii)

    \(e^{tA}e^{sA}=e^{(t+s)A},\ \forall \,t, s \ge 0\).

  3. (iii)

    \(\lim \limits _{t \rightarrow 0^{+}} \Vert e^{tA}u - u\Vert _a=0,\ \forall \, u \in \overline{D(A)}\).

  4. (iv)

    There are constants \(C_l\), such that

    $$\begin{aligned} \Vert t^l A^l e^{tA} u\Vert _{a} \le C_l \Vert u\Vert _a, \quad 0<t\le 1. \end{aligned}$$
    (55)
  5. (v)

    \(t \mapsto e^{tA}\) is a real-analytic function from \((0,\infty )\) to the Banach space of bounded linear operators on \({\mathcal{H }}^a\) (with norm given by the operator norm) and

    $$\begin{aligned} \frac{d^l}{dt^l}e^{tA}=A^l e^{tA},\quad t>0. \end{aligned}$$
    (56)

Concerning Theorem 3, we first consider Property (0). Using Proposition 2(iii) and setting \(S_\theta =e^{tA}\) and \(t=\theta ^{-2k}\) we have

$$\begin{aligned} \lim \limits _{\theta \rightarrow \infty } S_\theta u=u, \quad \forall u \in \overline{{\fancyscript{H}}^{2+a}} \end{aligned}$$
(57)

and thus, Property (0) holds.

Using Proposition 2(i) and the fact that \(S_\theta =e^{tA}\) is a continuous operator on \({\fancyscript{H}}^{b}\) we have

$$\begin{aligned} \Vert e^{tA} u\Vert _b \lesssim \Vert u\Vert _b \lesssim \Vert u\Vert _a, \quad \forall \,b \le a \end{aligned}$$

and thus also Property (i).

In order to prove Property (ii), note that it suffices to show the assertion for \(0 < t \le 1\), or equivalently \(\theta \ge 1\). We use that \((A -1)^{-1} : {\fancyscript{H}}^a \rightarrow {\fancyscript{H}}^{a+2k}\) is continuous, \(\Vert (A-1)^{-1} u\Vert _{{\fancyscript{H}}^{a+2k}}\lesssim \Vert u\Vert _{{\fancyscript{H}}^a}\). We then have

$$\begin{aligned} \Vert v\Vert _{a+2k} \lesssim \Vert (A-1)v\Vert _a \lesssim \Vert Av\Vert _a + \Vert v\Vert _a. \end{aligned}$$

We first set \(l=1\) and \(v=t e^{tA}u\) and deduce

$$\begin{aligned} \Vert t e^{tA}u\Vert _{a+2k} \lesssim \Vert (A - 1)t e^{tA}u\Vert _a \lesssim \Vert t A e^{tA}u\Vert _a + \Vert te^{tA}u\Vert _a \end{aligned}$$
(58)

and by using Proposition 2(i) and Proposition 2(iv) we have

$$\begin{aligned} \Vert te^{tA}u\Vert _a \le \Vert e^{tA}u\Vert _a \lesssim \Vert u\Vert _a, \quad 0 < t \le 1, \end{aligned}$$

and finally by (58) we obtain

$$\begin{aligned} \Vert t e^{tA}u\Vert _{a+2k} \lesssim \Vert u\Vert _a. \end{aligned}$$

By iterating this argument \(l\)-times using

$$\begin{aligned} \Vert t^l e^{tA}u\Vert _{a+2kl} =l^{l} \left\| \frac{t}{l} e^{t/l}A \cdot _{\cdots } \cdot \frac{t}{l} e^{t/l}A u\right\| _{a+2kl} \end{aligned}$$

we have

$$\begin{aligned} \Vert t^l e^{tA}u\Vert _{a+2kl} \lesssim \Vert u\Vert _a. \end{aligned}$$

Setting \(b= a+2kl,\, t= \theta ^{-2k}\), Property (ii) holds for this specific \(b\).

For an arbitrary \(b,{\tilde{b}}:= a+2kl \ge b\), write \(b=\sigma a +(1-\sigma ){\tilde{b}}\). The interpolation estimate (27) gives

$$\begin{aligned} \Vert e^{tA} u\Vert _b \le \Vert e^{tA} u\Vert _a^{\lambda } \Vert e^{tA} u\Vert _{\tilde{b}}^{1-\lambda } \lesssim t^{-l(1-\lambda )} \Vert u\Vert _a^\lambda \ \Vert u\Vert _a^{1-\lambda }, \end{aligned}$$

and we deduce

$$\begin{aligned} \Vert e^{tA} u\Vert _b \lesssim t^{-(1-\lambda )l} \Vert u\Vert _a= t^{-(b-a)/2k} \Vert u\Vert _a. \end{aligned}$$

Setting now \(S_\theta :=e^{tA}\) with \( t =\theta ^{-2k}\) we have proved

$$\begin{aligned} \Vert S_\theta u\Vert _b \lesssim \theta ^{b-a} \Vert u\Vert _a \end{aligned}$$

and thus, Property (ii) holds.

For Property (iv) we first use \( t =\theta ^{-2k}\) and observe

$$\begin{aligned} \frac{d}{d\theta } e^{tA} u= \frac{dt}{d\theta } \frac{d}{dt} e^{tA}u=-2k t^{1/2k} (tAe^{tA}u)=-\frac{2k}{\theta } tAe^{tA}u. \end{aligned}$$

The same proof as for Property (ii) yields, setting \(S_\theta =e^{tA}\):

$$\begin{aligned} \left\| \frac{d}{d\theta }S_\theta u\right\| _b=\frac{2k}{\theta } \Vert tAe^{tA} u\Vert _b \lesssim \frac{2k}{\theta } \theta ^{b-a} \Vert u\Vert _a = 2k \theta ^{b-a-1} \Vert u\Vert _a. \end{aligned}$$

Finally, given the continuity of \(S_\theta \) on \({\fancyscript{H}}^a\), it suffices to show Property \((iii')\) for \(b \ne a\). Note that \(1-A: {\fancyscript{H}}^{a+2k} \rightarrow {\fancyscript{H}}^{a}\) is an isomorphism. With \(1-e^{t\lambda } = -\lambda \int _0^t e^{\lambda s}\,ds\), we have for \(u \in C^\infty ,\, v=(1-A)^{\frac{a-b}{2k}}u\) and \(t\in (0,1]\)

$$\begin{aligned} u- e^{tA} u&= \frac{1}{2\pi i} \int _{\gamma _{r,\eta }} (1-e^{t\lambda }) (A -\lambda )^{-1} (1-A)^{-\frac{a-b}{2k}} v \,d\lambda \\&= -\frac{1}{2\pi i} \int _{\gamma _{r,\eta }} \lambda \int _0^t e^{\lambda s}\,ds\ (A -\lambda )^{-1} (1-\lambda )^{-\frac{a-b}{2k}}v \,d\lambda . \end{aligned}$$

The double integral is absolutely convergent for \(\frac{a-b}{2k} \in (0,1)\). After interchanging the order of integration and using the triangle inequality as well as \(\Vert (A -\lambda )^{-1}v\Vert _b \lesssim \frac{\Vert v\Vert _b}{\lambda }\), the right hand side is smaller than a constant times

$$\begin{aligned} \int _0^t \int _{\gamma _{r,\eta }} e^{s\ \mathrm{Re}\ \lambda } |1-\lambda |^{-\frac{a-b}{2k}} \Vert v\Vert _b \,|d\lambda |\,ds. \end{aligned}$$

If \(r<1\), we may bound \(|1-\lambda |^{-1} \le C_r (1+|\lambda |)^{-1}\) for all \(\lambda \in \gamma _{r,\eta }\). It therefore remains to estimate

$$\begin{aligned} \int _0^t \int _{\gamma _{r,\eta }} e^{s\ \mathrm{Re}\ \lambda } (1+|\lambda |)^{-\frac{a-b}{2k}} \Vert v\Vert _b \,|d\lambda |\,ds. \end{aligned}$$

We split the integral \(\int _{\gamma _{r,\eta }} = I_r + I_+ + I_-\) into integrals over \(\gamma _r = \lbrace \lambda = r e^{i \sigma }\in \mathbb{C }: |\sigma |\le \eta \rbrace \), \(\gamma _+ = \lbrace \lambda =\rho e^{i \eta } \in \mathbb{C }: \rho \ge r\rbrace \) resp. \(\gamma _-=\lbrace \lambda =\rho e^{-i \eta } \in \mathbb{C }: \rho \ge r\rbrace \) and consider the three terms separately. The first integral,

$$\begin{aligned} I_r=\int _0^t \int _{-\eta }^{\eta } e^{s\cos (\sigma )} \,d\sigma \,ds\ (1+r)^{-\frac{a-b}{2k}} \Vert v\Vert _b, \end{aligned}$$

is bounded by \(t (2 \eta ) e (1+r)^{-\frac{a-b}{2k}} \Vert v\Vert _b\) and hence of order \(t\). For the second and third integrals,

$$\begin{aligned} I_{\pm }=\int _0^t \int _{r}^{\infty } e^{-s\rho |\cos (\eta )|} (1+\rho )^{-\frac{a-b}{2k}} \Vert v\Vert _b\ \,d\rho \,ds \end{aligned}$$

the change of variables \(\rho \mapsto \frac{\rho }{s |\cos (\eta )|}\) leads to

$$\begin{aligned} I_{\pm }=\int _0^t \int _{sr|\cos (\eta )|}^{\infty } e^{-\rho } \left( \frac{s |\cos (\eta )|}{s |\cos (\eta )|+\rho }\right) ^{\frac{a-b}{2k}} \Vert v\Vert _b\ \frac{1}{s |\cos (\eta )|}\,d\rho \,ds, \end{aligned}$$

or

$$\begin{aligned} I_{\pm }\le \int _0^t (s |\cos (\eta )|)^{\frac{a-b}{2k}-1}\int _{0}^{\infty } e^{-\rho } {\rho }^{-\frac{a-b}{2k}} \Vert v\Vert _b\, \,d\rho \,ds \ \lesssim t^{\frac{a-b}{2k}} \Vert v\Vert _b. \end{aligned}$$

Using \(t=\frac{1}{\theta ^{2k}}\) and \(\Vert v\Vert _b=\Vert (1-A)^{\frac{a-b}{2k}}u\Vert _b\lesssim \Vert u\Vert _a\), (iii’) follows.

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Costea, A., Gimperlein, H. & Stephan, E.P. A Nash–Hörmander iteration and boundary elements for the Molodensky problem. Numer. Math. 127, 1–34 (2014). https://doi.org/10.1007/s00211-013-0579-8

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