In order to prove that the solutions to (
1.5) are the rotationally symmetric solutions to (
1.3) first we compute
\(\partial ^2_{x_N}u\) for functions that are constant along normal directions to
M. More precisely, let
\(\hat{s} \in (0,a)\). Since rotationally symmetric functions satisfy
\(u(\theta , x_N) = u(0, |\theta |, z)\) we simply write
\(u(0,x_{N-1}, x_N) = u(x_{N-1}, x_N)\). Let
\((0, \ldots , 0, G(t), z(t)):= (G(t), z(t))\) denote a local parametrization of
$$\begin{aligned}M \cap \{(x_1, \ldots , x_{N-1},z); x_1= \cdots = x_{N-2}=0\}\end{aligned}$$
around
\((0, \ldots , 0, G(\hat{s}) , z(\hat{s}))\) such that
\((G'(s))^2 + (z'(s))^2 = 1\). By the
\(\epsilon \)-neighborhood theorem, see [
9], there exists
\(\delta >0\) and differentiable functions
\(s:(-\delta , \delta ) \rightarrow \mathbb {R}\) and
\(t:(-\delta , \delta ) \rightarrow \mathbb {R}\) such that
$$\begin{aligned} (G(\hat{s}), z(\hat{s}) + h) = (G(s(h)) - t(h) z'(s(h)), z(s(h)) + t(h) G'(s(h))), \end{aligned}$$
(A.1)
for all
\( h \in (-\delta , \delta )\),
\(s(0) = \hat{s}\) and
\( t(0) = 0\). Differentiating in (
A.1) with respect to
h we have
$$\begin{aligned} \begin{aligned} 0&= G'(s(h))s'(h) - z'(s(h))t'(h) - z''(s(h))s'(h) t(h) \\ 1&= z'(s(h))s'(h) + G'(s(h))t'(h) + t(h)G''(s(h))s'(h). \end{aligned} \end{aligned}$$
(A.2)
Taking
\(h=0\) and using that
\((G'(s))^2 + (z'(s))^2 = 1\) yield
\(t'(0) = G'(\hat{s})\) and
\(s'(0) =z'(\hat{s})\). Differentiating again gives
$$\begin{aligned} \begin{aligned} 0&= G''(\hat{s})(z'(\hat{s}))^2 + G'(\hat{s})s''(0) - z'(\hat{s}) t''(0) - 2z''(\hat{s}) G'(\hat{s}) z'(\hat{s}), \ \hbox {and } \\ 0&= z''(\hat{s}) (z'(\hat{s}))^2 + z'(\hat{s}) s''(0) + G'(\hat{s})t''(0) + 2G''(\hat{s}) G'(\hat{s}) z'(\hat{s}), \end{aligned} \end{aligned}$$
(A.3)
which in turn implies
$$\begin{aligned} \begin{aligned} s''(0)&= -G''(\hat{s})(z'(\hat{s}))^2 G'(\hat{s})+ 2z'(\hat{s})z''(\hat{s})(G'(\hat{s}))^2 - z''(\hat{s})(z'(\hat{s}))^3 \\&\quad - 2G'(\hat{s})G''(\hat{s})(z'(\hat{s}))^2 \\&= -G''(\hat{s}) G'(\hat{s})\left[ (z'(\hat{s}))^2 + 2(G'(\hat{s}))^2 -(z'(\hat{s}))^2 + 2(z'(\hat{s}))^2 \right] \\&= -2G''(\hat{s}) G'(\hat{s}), \ \ \hbox {and} \\ t''(0)&= -z''(\hat{s})((G'(\hat{s}))^2G'(\hat{s}) - G''(\hat{s})(z'(\hat{s}))^2 G'(\hat{s}). \end{aligned} \end{aligned}$$
(A.4)
Assuming now that
u is constant along normal lines in an
\(\epsilon \)-neighborhood of
M and defining
\(v(s) = u((G(s),z(s)) + t\eta (s))\) with
\(\eta (s) = (-z' (s), G'(s))\) we have
$$\begin{aligned} \begin{aligned} \frac{\partial ^2u}{\partial x_N^2}&(G(\hat{s}), z(\hat{s})) = \lim _{h \rightarrow 0}\left( \frac{u((0,G(\hat{s}), z(\hat{s})+h))}{h^2} \right. \\&\quad - \left. \frac{2u(0,G(\hat{s}) , z(\hat{s}))-u((0,G(\hat{s}),z(\hat{s})-h) )}{h^2}\right) \\&= \lim _{h \rightarrow 0}\left( \frac{u(0,G(s(h)) - t(h) z'(s(h)), z(s(h)) + t(h) G'(s(h))) -2u(G(\hat{s}) , z(\hat{s}))}{h^2} \right. \\&\quad + \left. \frac{ u(0,G(s(-h)) - t(-h) z'(r(h)), z(s(-h)) + t(-h) G'(s(-h)))}{h^2}\right) \\&= \lim _{h \rightarrow 0}\left( \frac{v(s(h)) - 2v(s(0)) + v(s(-h))}{h^2} \right) . \end{aligned} \end{aligned}$$
(A.5)
Thus, by L’Hopital rule,
$$\begin{aligned} \begin{aligned} \frac{\partial ^2u}{\partial x_N^2}(G(\hat{s}),y(\hat{s}))&= \lim _{h \rightarrow 0}\left( \frac{v'(s(h)) s'(h) -v'(s(-h))s'(-h)}{2h} \right) \\&= \frac{1}{2} \lim _{h \rightarrow 0}\left( v''(s(h)) (s'(h))^2 + v'(s(h))s''(h) \right. \\&\quad \left. +v''(s(-h))(s'(-h))^2 + v'(s(-h))s''(-h)\right) \\&= (z'(\hat{s}))^2 v''(\hat{s}) -2G''(\hat{s})G'(\hat{s})v'(\hat{s}). \end{aligned} \end{aligned}$$
(A.6)
Now we proceed to the computation of
\(\frac{\partial ^2u}{\partial x_{N-1}^2}\). Using again the
\(\epsilon \)-neighborhood theorem, for
h small enough there exist differentiable functions
s(
h) and
t(
h) such that
$$\begin{aligned} (G(\hat{s}) + h, z(\hat{s})) = (G(s(h)), z(s(h)) + t(h) (-z'(s(h)), G'(s(h))). \end{aligned}$$
(A.7)
Differentiating with respect to
h,
$$\begin{aligned} \begin{aligned} 1&= G'(s(h))s'(h) - z'(s(h))t'(h) - z''(s(h))s'(h) t(h) \\ 0&= z'(s(h))s'(h) + G'(s(h))t'(h) + t(h)G''(s(h))s'(h). \end{aligned} \end{aligned}$$
(A.8)
Also,
\(t(0) = 0\) and
\(s(0) = \hat{s}\). Hence
\(1 = G'(\hat{s}) s'(0) - t'(0) z'(\hat{s})\) and
\( 0= z'(\hat{s}) s'(0) + t'(0) G'(\hat{s})\). Thus
$$\begin{aligned} s'(0)=G'(\hat{s}), \ \ \hbox { and } \ \ t'(0) = -z'(\hat{s}). \end{aligned}$$
(A.9)
Differentiating in (
A.8) with respect to
h, replacing
h by 0 and using (
A.9) we have
$$\begin{aligned} \begin{aligned} 0&= G''(\hat{s})(G'(\hat{s}))^2 + G'(\hat{s})s''(0) - z'(\hat{s}) t''(0) + 2z''(\hat{s}) G'(\hat{s}) z'(\hat{s}), \ \hbox {and } \\ 0&= z''(\hat{s}) (G'(\hat{s}))^2 + z'(\hat{s}) s''(0) + G'(\hat{s})t''(0) - 2G''(\hat{s}) G'(\hat{s}) z'(\hat{s}). \end{aligned} \end{aligned}$$
(A.10)
Using that
\(z'(s)z''(s) = - G'(s) G''(s)\) for all
s we have
$$\begin{aligned} \begin{aligned} s''(0)&= 2G'(\hat{s}) G''(\hat{s}) \ \ \hbox {and} \\ t''(0)&= \frac{(G'(\hat{s}))^2G''(\hat{s})}{ z'(\hat{s})}. \end{aligned} \end{aligned}$$
(A.11)
Since
u is constant along normal lines in an
\(\epsilon \)-neighborhood of
M, defining
\(v(s) = u(0,G(s),z(s)) + t\eta (s))\) with
\(\eta (s) = ( 0,-z' (s), G'(s))\) we have
$$\begin{aligned} \begin{aligned} \frac{\partial ^2u}{\partial x_{N-1}^2}&(0, G(\hat{s}), y(\hat{s})) = \lim _{h \rightarrow 0}\left( \frac{u(0,G(\hat{s})+h, z(\hat{s}) )}{h^2} \right. \\&\quad - \left. \frac{2u(0,G(\hat{s}) , z(\hat{s}))-u(0, G(\hat{s}) - h, z(\hat{s}) )}{h^2}\right) \\&= \lim _{h \rightarrow 0}\left( \frac{u(0, G(s(h)) - t(h) z'(s(h)), z(s(h)) + t(h) G'(s(h)))- 2u(0,G(\hat{s}) , z(\hat{s})) }{h^2} \right. \\&\quad + \left. \frac{ u(0,G(s(-h)) - t(-h) z'(r(h)), z(s(-h)) + t(-h) G'(s(-h)))}{h^2}\right) \\&= \lim _{h \rightarrow 0}\left( \frac{v(s(h)) - 2v(s(0)) + v(s(-h))}{h^2} \right) . \end{aligned} \end{aligned}$$
(A.12)
Thus, by L’Hopital rule,
$$\begin{aligned} \begin{aligned} \frac{\partial ^2u}{\partial x_{N-1}^2}(G(\hat{s}),y(\hat{s}))&= \lim _{h \rightarrow 0}\left( \frac{v'(s(h)) s'(h) -v'(s(-h))s'(-h)}{2h} \right) \\&= \frac{1}{2} \lim _{h \rightarrow 0}\left( v''(s(h)) (s'(h))^2 + v'(s(h))s''(h) \right. \\&\quad \left. +v''(s(-h))(s'(-h))^2 + v'(s(-h))s''(-h)\right) \\&= (G'(\hat{s}))^2 v''(\hat{s}) +2G''(\hat{s})G'(\hat{s})v'(\hat{s}). \end{aligned} \end{aligned}$$
(A.13)
From (
A.9), (
A.6), and (
A.12),
$$\begin{aligned} \begin{aligned} \Delta _M(u)&= \frac{\partial ^2u}{\partial x_N^2}( G(s), z(s)) + \frac{\partial ^2u}{\partial x_{N-1}^2}( G(s), z(s)) + \frac{(N-2)}{G(s)} \frac{\partial u}{\partial x_{N-1}}(G(s), z(s)) \\&= v''(s) + \frac{(N-2)G'(s)}{G(s)} v'(s), \end{aligned} \end{aligned}$$
(A.14)
where we have used that the Laplacian operator
\(\Delta \) in
\(\mathbb {R}^{N-1}\) for radial functions is given by
\(\partial ^2_r+ ((N-2)/r)\partial _r\). This proves that rotational symmetric solutions to (
1.3) are given by solutions to (
1.5) subject to (
1.6).
