# Some geometric properties of nonparametric $\mu$-surfaces in
$\mathbb{R}^{3}$

Michael Bildhauer & Martin Fuchs

###### Abstract

111AMS subject classification: 49Q05, 53A10, 53C42, 58E12

Smooth solutions of the equation

$\operatorname{div}\,\Bigg{\\{}\frac{g^{\prime}\big{(}|\nabla
u|\big{)}}{|\nabla u|}\nabla u\Bigg{\\}}=0$

are considered generating nonparametric $\mu$-surfaces in $\mathbb{R}^{3}$,
whenever $g$ is a function of linear growth satisfying in addition

$\int_{0}^{\infty}sg^{\prime\prime}(s){\rm d}s<\infty\,.$

Particular examples are $\mu$-elliptic energy densities $g$ with exponent
$\mu>2$ (see [1]) and the minimal surfaces belong to the class of
$3$-surfaces.

Generalizing the minimal surface case we prove the closedness of a suitable
differential form $\hat{N}\wedge{\rm d}X$. As a corollary we find an
asymptotic conformal parametrization generated by this differential form.

_Dedicated to S. Hildebrandt_

##  1 Introduction

Starting with the variational point of view we like to mention three scenarios
for variational problems with linear growth conditions.

The most prominent representative is the area minimizing problem

$\displaystyle J[\nabla w]$ $\displaystyle:=$
$\displaystyle\int_{\Omega}F\big{(}\nabla w\big{)}\,{\rm d}x\to\min\,,$
$\displaystyle F(Z)$ $\displaystyle:=$ $\displaystyle\sqrt{1+|Z|^{2}}\,,\quad
Z\in\mathbb{R}^{2}\,,$ (1.1)

within a suitable class of comparison functions $w$:
$\mathbb{R}^{2}\supset\Omega\to\mathbb{R}$.

The are uncountably many contributions to the study of problem (1). We refer
to the monographs [2], [3], [4], [5] and [6] giving a detailed picture of this
classical problem.

There is another well known application involving variational problems with
linear growth conditions: the theory of perfect plasticity. We just mention
[7] as one of a series of papers written by Seregin and the monograph of
Temam, [8] as well as the book [9].

As a third class of variational problems with linear growth conditions we like
to mention the discussion of $\mu$-elliptic integrands with linear growth
introduced in [1]. Depending on the parameter $\mu$, this family includes the
minimal surface case as one example with exponent $\mu=3$ and an approximation
of perfect plasticity is covered for large values of $\mu$.

In [1] and in subsequent papers the question of existence and regularity of
eventually relaxed solutions was studied w.r.t. to different circumstances.
These investigations include also aspects from image analysis (see, e.g.,
[10]), where the model serves as an appropriate TV-approximation. For an
overview on the aspects of existence, relaxation and regularity of solutions
under the assumption of $\mu$-ellipticity we refer to [11] or [12].

While in the case of perfect plasticity and related applications the dual
problem formulated in terms the stress tensor plays the key role, various
geometric features are developed in the minimal surface case. Here the Euler-
Lagrange equation for $C^{2}$-solutions of the variational problem (1), i.e.
the minimal surface equation

$u_{xx}\big{(}1+u_{y}^{2}\big{)}+u_{yy}\big{(}1+u_{x}^{2}\big{)}-2u_{x}u_{y}u_{xy}=0$
(1.2)

serves as a prototype for the study of elliptic PDEs arising in connection
with problems in geometry. In [13] the reader will find an exposition with a
particular focus to the geometric structure of equation (1.2).

Our note on geometrical properties of what we call nonparametric
$\mu$-surfaces is also strongly influenced by the pioneering work on the
minimal surface equation. To be precise, we consider the following theorem
formulated by Dierkes, Hildebrandt and Sauvigny in the notion of differential
forms.

######  Theorem 1.1.

([6], Theorem Section 2.2) A nonparametric surface
$X(x,y)=\big{(}x,y,z(x,y)\big{)}$, described by a function $z=z(x,y)$ of class
$C^{2}$ on a simply connected domain $\Omega$ of $\mathbb{R}^{2}$, with the
Gauss map $N=(\xi,\eta,\zeta)$ is a minimal surface if and only if the vector-
valued differential form $N\wedge{\rm d}X$ is a total differential, i.e., if
and only there is a mapping $X^{*}\in C^{2}(\Omega,\mathbb{R}^{3})$ such that

$-{\rm d}X^{*}=N\wedge{\rm d}X\,.$ (1.3)

If we write

$X^{*}=(a,b,c)\,,\quad N\wedge{\rm d}X=(\alpha,\beta,\gamma)\,,$

equation (1.3) is equivalent to

$-{\rm d}a=\alpha\,,\quad-{\rm d}b=\beta\,,\quad-{\rm d}c=\gamma\,.$

The particular importance of this theorem is evident by the fact, that it
serves as the main tool to prove that solutions of the minimal surface
equation are analytic functions and that $X^{*}$ induces a diffeomorphism
leading to a conformal representation.

Motivated by the $\mu$-elliptic examples with linear growth mentioned above,
we are faced with the question, whether $C^{2}$-solutions of the corresponding
Euler equations can also be characterized by the closedness of suitable
differential forms.

######  Remark 1.1.

Let us shortly clarify the notion ”nonparametic $\mu$-surface”: $\mu$-elliptic
energy densities $g_{\mu}$ introduced in Example 1.2 provide a typical
motivation for our studies. Going through the details of the proofs it becomes
evident, that we just have to consider $C^{2}$-solutions of (1.9) together
with our assumption (1.8), which roughly speaking corresponds to the case
$\mu>2$.

The limit case $\mu=2$ also plays an important role in studying the regularity
of solutions. While our geometric considerations are based on the finiteness
in condition (1.8), we note that, e.g., in (1.9) of [14] the condition

$\int_{1}^{\infty}sg^{\prime\prime}(s){\rm d}s=\infty$

characterizes the existence of regular solutions taking the boundary data of a
Dirichlet problem. We also like to refer to the introductory remarks of [14]
and to the classical paper [15], where related conditions can be found.

Now let us introduce a more precise notation: given a simply connected domain
$\Omega\subset\mathbb{R}^{2}$ and a $C^{2}$-function $u$:
$\Omega\to\mathbb{R}$ we consider the nonparametric surface $X$:
$\Omega\to\mathbb{R}^{3}$

$X(x,y)=\big{(}x,\,y,\,u(x,y)\big{)}\,,\quad(x,y)\in\Omega\,,$

and denote the asymptotic normal by

$\hat{N}=(\hat{N}_{1}\,\hat{N}_{2}\,\hat{N}_{3})\,,\quad(x,y)\in\Omega\,,$

with components

$\hat{N}_{1}=-\Xi\big{(}|\nabla
u|\big{)}u_{x}\,,\quad\hat{N}_{2}=-\Xi\big{(}|\nabla
u|\big{)}u_{y}\,,\quad\hat{N}_{3}=\Xi\big{(}|\nabla
u|\big{)}+\vartheta\big{(}|\nabla u|\big{)}\,.$ (1.4)

Here we let for $g$: $[0,\infty)\to\mathbb{R}$ and all $t\geq 0$ ($g\in
C^{2}\big{(}[0,\infty)\big{)}$, $g^{\prime}(0)=0$, $g^{\prime\prime}(t)>0$ for
all $t>0$)

$\displaystyle\Xi(t)$ $\displaystyle:=$
$\displaystyle\frac{g^{\prime}(t)}{t}\,,$ (1.5) $\displaystyle\vartheta(t)$
$\displaystyle:=$ $\displaystyle g(t)-tg^{\prime}(t)-\Xi(t)\,.$ (1.6)

The main hypothesis throughout this paper are summarized in the following
Assumption.

######  Assumption 1.1.

Let $g$: $[0,\infty)\to\mathbb{R}$ be a function of class
$C^{2}\big{(}[0,\infty)\big{)}$ such that $g^{\prime}(0)=0$,
$g^{\prime\prime}(t)>0$ for all $t>0$ and such that with real numbers $a$,
$A>0$, $b$, $B\geq 0$

$at-b\leq g(t)\leq At+B\quad\mbox{for all}\quad t\geq 0\,.$ (1.7)

Moreover, suppose that we have

$\int_{0}^{\infty}sg^{\prime\prime}(s){\rm d}s<\infty\,.$ (1.8)

Before these hypotheses are discussed more detailed in Remark 1.2, we state
our main theorem:

######  Theorem 1.2.

With the notation introduced above we suppose that the general Assumption 1.1
is valid, in particular we have (1.8). Let $u$ denote a function of class
$C^{2}(\Omega)$.

Then $u$ is a solution of

$\operatorname{div}\,\Bigg{\\{}\frac{g^{\prime}\big{(}|\nabla
u|\big{)}}{|\nabla u|}\nabla u\Bigg{\\}}=0$ (1.9)

on a simply connected domain $\Omega\subset\mathbb{R}^{2}$ if and only if the
vector-valued differential form satisfies

$\hat{N}\wedge{\rm d}X=-{\rm d}X^{*}$ (1.10)

for some mapping $X^{*}=:(a,b,c)\in C^{2}\big{(}\Omega;\mathbb{R}^{3}\big{)}$,
i.e. $N\wedge{\rm d}X=:(\alpha,\beta,\gamma)$ is a total differential.

Let us recall the geometrical meaning of the differential form $N\wedge{\rm
d}X$ in the minimal surface case by assuming that the surface is given w.r.t.
isothermal parameters $(v,w)$. In this case we have

$N\wedge{\rm d}X=X_{w}{\rm d}v-X_{v}{\rm d}w$

and the closedness of $N\wedge{\rm d}X$ corresponds to the minimal surface
characterization $\Delta X=0$ whenever $X$ is given in conformal parameters.
In fact it turns out that $X^{*}$ generates a diffeomorphism leading from the
nonparametric representation to conformal parameters and as a consequence to
the analyticity of the solutions and to Bernstein’s theorem.

As it is also emphasized in [6], this approach is of explicit geometrical
nature related to the particular kind of surfaces under consideration, which
is in contrast to the abstract application of Lichtenstein’s mapping theorem
to ensure the existence of a conformal representation. We refer to
Hildebrandt’s beautiful overview [16] and the relation to Plateau’s problem
including quite recent results with von der Mosel [17], [18] and Sauvigny
[19].

In our setting we are not in the minimal surface case having the analyticity
of solutions, i.e. we cannot argue with the help of suitably defined
holomorphic functions. Thus we change the point of view in the sense that we
are not mainly interested in conformal representations. As an application of
our main theorem we are rather interested in the question, which kind of
representation is generated by the mapping $X^{*}$ constructed in Theorem 1.2.
This means that we are looking for some kind of natural parametrisations for
nonparametric $\mu$-surfaces.

It will turn out as a corollary of our main theorem, that both of the
conformality relations are perturbed by the same function $\Theta$ and that
asymptotically the conformality relations are recovered.

Before giving a precise statement of this corollary, we like to include some
additional remarks on our assumptions.

######  Remark 1.2.

  1. i)

Observe that we have an equivalent formulation of the main hypothesis (1.8):
an integration by parts gives

$\int_{0}^{t}sg^{\prime\prime}(s){\rm d}s=s\cdot
g^{\prime}(s)\Big{|}^{t}_{0}-\int_{0}^{t}g^{\prime}(s){\rm
d}s=tg^{\prime}(t)-g(t)+g(0)\,.$

and we may write (1.8) in the form

$\lim_{t\to\infty}\Big{[}g(t)-tg^{\prime}(t)\Big{]}=K:=g(0)-\int_{0}^{\infty}sg^{\prime\prime}(s){\rm
d}s\,.$

Replacing the function $g$ by the function $g-K$ (not changing equation
(1.9)), we may replace w.l.o.g. assumption (1.8) by

$\lim_{t\to\infty}\Big{[}g(t)-tg^{\prime}(t)\Big{]}=0$ (1.11)

and in the following (1.11) can be taken as general assumption.

  2. ii)

The convexity of $g$ immediately gives for all $t\geq 0$

$g(0)\geq g(t)-tg^{\prime}(t)\,.$ (1.12)

Moreover, the function $g(t)-tg^{\prime}(t)$ is a decreasing function in
$[0,\infty)$ since we have for all $t\geq 0$

$\frac{{\rm d}}{{\rm
d}t}\Big{[}g(t)-tg^{\prime}(t)\Big{]}=-tg^{\prime\prime}(t)\leq 0\,.$ (1.13)

Thus, by (1.12), (1.13) and (1.11)

$g(0)\geq g(t)-tg^{\prime}(t)\geq 0\quad\mbox{for all}\quad t\geq 0\,,$ (1.14)

  3. iii)

From the convexity of $g$ and the linear growth in the sense of (1.7) we
obtain the existence of

$\lim_{t\to\infty}g^{\prime}(t)=:g_{\infty}^{\prime}=\lim_{t\to\infty}\frac{g(t)}{t}\,.$

Writing

$\frac{g(t)g^{\prime}(t)}{t}-\big{(}g^{\prime}(t)\big{)}^{2}=R(t)\,,\quad\mbox{i.e}\quad
g^{\prime}(t)\Big{[}g(t)-tg^{\prime}(t)\Big{]}=tR(t)\,,$

assumption (1.11) in addition gives

$R(t)=o\big{(}t^{-1}\big{)}\,.$ (1.15)

  4. iv)

W.l.o.g. let us suppose $g^{\prime}_{\infty}=1$ and define for $t\geq 0$ the
functions

$g^{\prime}(t)=:1-h(t)\,,\quad\Xi(t)=\frac{1}{t}\big{[}1-h(t)\big{]}=\frac{g^{\prime}(t)}{t}\geq
0\,.$ (1.16)

Then $0\leq h(t)<1$ for all $t\in\mathbb{R}$ and $\lim_{t\to\infty}h(t)=0$.

Now we formulate

######  Corollary 1.1.

Suppose that $g$: $[0,\infty)\to\mathbb{R}$ satisfies Assumption 1.1 and
suppose that $u$: $\mathbb{R}^{2}\supset\Omega\to\mathbb{R}$ is a
$C^{2}$-solution of (1.9), where $\Omega$ now in addition is assumed to be
convex.

Then the function $X^{*}$ described in Theorem 1.2 generates an asymptotically
conformal parametrization $\chi$: $\hat{\Omega}\to\mathbb{R}^{3}$ of the
surface $\operatorname{graph}u$ in the following sense:

  1. i)

There is a function $\Theta$: $[0,\infty)\to\mathbb{R}$ such that

$\displaystyle\partial_{\hat{x}}\chi\cdot\partial_{\hat{y}}\chi$
$\displaystyle=$
$\displaystyle\frac{1}{\big{(}\operatorname{det}D\Lambda\big{)}^{2}}\Theta\big{(}|\nabla
u|\big{)}u_{x}u_{y}\,,$
$\displaystyle|\partial_{\hat{x}}\chi|^{2}-|\partial_{\hat{y}}\chi|^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{\big{(}\operatorname{det}D\Lambda\big{)}^{2}}\Theta\big{(}|\nabla
u|\big{)}\big{[}u^{2}_{x}-u^{2}_{y}\big{]}\,.$

Here the diffeomorphism $\Lambda$:
$\Omega\to\hat{\Omega}\subset\mathbb{R}^{2}$ is given by

$\Lambda(x,y)=\left(\begin{array}[]{c}x\\\
y\end{array}\right)+\left(\begin{array}[]{c}b(x,y)\\\\[4.30554pt]
-a(x,y)\end{array}\right)\,,\quad(x,y)\in\Omega\,,$ (1.17)

and we define $\chi$: $\hat{\Omega}\to\mathbb{R}^{3}$ by

$\chi:\,(\hat{x},\hat{y})\mapsto\Big{(}\Lambda^{-1}(\hat{x},\hat{y}),u\circ\Lambda^{-1}(\hat{x},\hat{y})\Big{)}\,.$
(1.18)

  2. ii)

There is a constant $c>0$ such that for all $(x,y)\in\Omega$

${\rm det}\,D\Lambda\geq c\big{(}1+|\nabla u|\big{)}\,.$ (1.19)

  3. iii)

If we suppose for some $\mu>2$ that for all $t\geq 0$

$\big{|}g(t)-tg^{\prime}(t)\big{|}\leq c_{1}t^{2-\mu}\,,\quad 0\leq
1-g^{\prime}(t)\leq c_{2}t^{1-\mu}\,,$ (1.20)

with constants $c_{1}$, $c_{2}>0$, then we have for all $t\gg 1$

$|\Theta(t)|\leq d_{1}t^{2-\mu}+d_{2}t^{-1}$ (1.21)

with some real numbers $d_{1}$, $d_{2}>0$.

Let us close this introduction with three intuitive examples we have in mind.

######  Example 1.1.

The most prominent one is the minimal surface example given by

$g_{\min}(t)=\sqrt{1+t^{2}}\,,$

for which

$\displaystyle\Xi_{\min}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1+t^{2}}}\,,$ $\displaystyle
g_{\min}(t)-tg_{\min}^{\prime}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1+t^{2}}}=\Xi_{\min}(t)\,,$ $\displaystyle
h_{\min}(t)$ $\displaystyle=$ $\displaystyle
1-\frac{t}{\sqrt{1+t^{2}}}=\frac{1}{t\sqrt{1+t^{2}}+(1+t^{2})}\,.$

With the help of this example we can always check our results by comparing to
the classical ones.

######  Example 1.2.

We fix $\mu>1$, $\mu\not=2$, and consider

$g_{\mu}(t)=t+\frac{1}{\mu-2}(1+t)^{2-\mu}\,,\quad t\geq 0\,.$ (1.22)

The particular choice $g_{\mu}$ is suitable for image analysis problems
discussed in [10], since for large $\mu$ we have convergence to the total
variation energy.

For $g_{\mu}$ we have

$\Xi_{\mu}(t)=\frac{1}{t}\Big{[}1-(1+t)^{1-\mu}\Big{]}\,,\quad
h_{\mu}(t)=(1+t)^{1-\mu}\,.$

We note that the growth of $h_{\mu}$ corresponds to the minimal surface case
if $\mu=3$. Moreover, we may use direct calculations to obtain

$\displaystyle g_{\mu}(t)-tg_{\mu}^{\prime}(t)$ $\displaystyle=$
$\displaystyle t+\frac{1}{\mu-2}(1+t)^{2-\mu}-t\Big{[}1-(1+t)^{1-\mu}\Big{]}$
$\displaystyle=$
$\displaystyle(1+t)^{1-\mu}\Bigg{[}\frac{1}{\mu-2}(1+t)+t\Bigg{]}$
$\displaystyle=$
$\displaystyle(1+t)^{1-\mu}\Bigg{[}\frac{\mu-1}{\mu-2}t+\frac{1}{\mu-2}\Bigg{]}\,.$

Hence, condition (1.11) is satisfied whenever $\mu>2$.

######  Example 1.3.

In [1] a variant of the above family is introduced by letting (again $\mu>1$
is fixed)

$\hat{g}_{\mu}(t):=\int_{0}^{t}\int_{0}^{s}\big{(}1+\tau^{2}\big{)}^{-\frac{\mu}{2}}{\rm
d}\tau{\rm d}s\,,\quad t\geq 0\,.$

This variant is of particular interest in our context since the case $\mu=3$
exactly corresponds to the minimal surface case.

Here we immediately verify (1.8) if we again have $\mu>2$.

##  2 The $\mu$-surface equation

We suppose that a $F$: $\mathbb{R}^{2}\to\mathbb{R}$ is given by

$F(Z)=g\big{(}|Z|\big{)}\,,$

where the function $g$: $[0,\infty)\to\mathbb{R}$ satisfies the main
Assumption 1.1. We observe that

$\nabla
F(Z)=\frac{g^{\prime}\big{(}|Z|\big{)}}{|Z|}Z=\Xi\big{(}|Z|\big{)}Z\,.$

and note that the variational problem

$J[w]:=\int_{\Omega}F\big{(}\nabla w\big{)}\,{\rm d}x\to\min$

w.r.t. a suitable class of comparison functions leads to the Euler equation

${\rm div}\,\Big{\\{}\nabla F(\nabla u)\Big{\\}}={\rm
div}\,\Big{\\{}\Xi\big{(}|\nabla u|\big{)}\nabla u\Big{\\}}=0\,.$ (2.1)

Here and in the following we suppose that we have a solution $u$:
$\mathbb{R}^{2}\supset\Omega\to\mathbb{R}$ to (2.1) which is at least of class
$C^{2}(\Omega)$, $\Omega$ denoting an open set in $\mathbb{R}^{2}$.

We write (2.1) in an explicit way:

$\displaystyle\Delta u\,\Xi\big{(}|\nabla
u|\big{)}+u_{x}\partial_{x}\Xi\big{(}|\nabla
u|\big{)}+u_{y}\partial_{y}\Xi\big{(}|\nabla u|\big{)}$ $\displaystyle=$
$\displaystyle\Delta u\,\Xi\big{(}|\nabla u|\big{)}$
$\displaystyle+u_{x}\Bigg{[}\frac{\Xi^{\prime}\big{(}|\nabla
u|\big{)}}{|\nabla
u|}(u_{x}u_{xx}+u_{y}u_{xy})\Bigg{]}+u_{y}\Bigg{[}\frac{\Xi^{\prime}\big{(}|\nabla
u|\big{)}}{|\nabla u|}(u_{x}u_{xy}+u_{y}u_{yy})\Bigg{]}\,,$

which shows

$\displaystyle\operatorname{div}\Big{\\{}\Xi\big{(}|\nabla u|\big{)}\nabla
u\Big{\\}}$ $\displaystyle=$ $\displaystyle u_{xx}\Bigg{[}\Xi\big{(}|\nabla
u|\big{)}+\frac{\Xi^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla
u|}u_{x}^{2}\Bigg{]}$ (2.2) $\displaystyle+u_{yy}\Bigg{[}\Xi\big{(}|\nabla
u|\big{)}+\frac{\Xi^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla
u|}u_{y}^{2}\Bigg{]}$
$\displaystyle+u_{xy}u_{x}u_{y}2\frac{\Xi^{\prime}\big{(}|\nabla
u|\big{)}}{|\nabla u|}\,.$

######  Example 2.1.

In the minimal surface case we have the expressions

$\displaystyle\Xi_{\min}\big{(}|\nabla
u|\big{)}+\frac{\Xi^{\prime}_{\min}\big{(}|\nabla u|\big{)}}{|\nabla
u|}u_{x}^{2}$ $\displaystyle=$
$\displaystyle\frac{1+u_{y}^{2}}{\big{(}1+|\nabla u|^{2}\big{)}^{3/2}}\,,$
$\displaystyle\Xi_{\min}\big{(}|\nabla
u|\big{)}+\frac{\Xi^{\prime}_{\min}\big{(}|\nabla u|\big{)}}{|\nabla
u|}u_{y}^{2}$ $\displaystyle=$
$\displaystyle\frac{1+u_{x}^{2}}{\big{(}1+|\nabla u|^{2}\big{)}^{3/2}}\,,$
$\displaystyle 2\frac{\Xi^{\prime}_{\min}\big{(}|\nabla u|\big{)}}{|\nabla
u|}$ $\displaystyle=$ $\displaystyle-2\frac{1}{\big{(}1+|\nabla
u|^{2}\big{)}^{3/2}}\,,$

thus (2.2) is equivalent to equation (1.2).

##  3 Proof of the main theorem

With the definition of the asymptotic normal $\hat{N}$ one computes
$\hat{N}\wedge{\rm d}X=(\alpha,\beta,\gamma)$ with components

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle-\Xi\big{(}|\nabla
u|\big{)}u_{y}\,{\rm d}u-\Big{[}\Xi\big{(}|\nabla
u|\big{)}+\vartheta\big{(}|\nabla u|\big{)}\Big{]}{\rm d}y$
$\displaystyle\beta$ $\displaystyle=$ $\displaystyle\Big{[}\Xi\big{(}|\nabla
u|\big{)}+\vartheta\big{(}|\nabla u|\big{)}\Big{]}{\rm d}x+\Xi\big{(}|\nabla
u|\big{)}u_{x}{\rm d}u$ $\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\Xi\big{(}|\nabla u|\big{)}u_{y}{\rm d}x-\Xi\big{(}|\nabla
u|\big{)}u_{x}{\rm d}y\,.$ (3.1)

We observe

${\rm d}u=u_{x}{\rm d}x+u_{y}{\rm d}y$

and obtain

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle-\Xi\big{(}|\nabla
u|\big{)}u_{y}u_{x}\,{\rm d}x-\Xi\big{(}|\nabla u|\big{)}u_{y}u_{y}{\rm
d}y-\Big{[}\Xi\big{(}|\nabla u|\big{)}+\vartheta\big{(}|\nabla
u|\big{)}\Big{]}{\rm d}y$ (3.2) $\displaystyle=$
$\displaystyle-\Xi\big{(}|\nabla u|\big{)}u_{x}u_{y}{\rm
d}x-\Big{[}\Xi\big{(}|\nabla
u|\big{)}\big{(}1+u_{y}^{2}\big{)}+\vartheta\big{(}|\nabla
u|\big{)}\Big{]}{\rm d}y\,.$

In the same way we get

$\beta=\Big{[}\Xi\big{(}|\nabla
u|\big{)}\big{(}1+u_{x}^{2}\big{)}+\vartheta\big{(}|\nabla
u|\big{)}\Big{]}{\rm d}x+\Xi\big{(}|\nabla u|\big{)}u_{x}u_{y}{\rm d}y\,.$
(3.3)

From (3) – (3.3) we obtain the equation

$\displaystyle{\rm d}\alpha$ $\displaystyle=$
$\displaystyle\partial_{y}\Big{[}\Xi\big{(}|\nabla
u|\big{)}u_{x}u_{y}\Big{]}{\rm d}x\wedge{\rm d}y$ (3.4)
$\displaystyle-\partial_{x}\Big{[}\Xi\big{(}|\nabla
u|\big{)}\big{(}1+u_{y}^{2}\big{)}+\vartheta\big{(}|\nabla
u|\big{)}\Big{]}{\rm d}x\wedge{\rm d}y$ $\displaystyle=:$
$\displaystyle\big{[}\partial_{y}\psi_{1}-\partial_{x}\psi_{2}\big{]}{\rm
d}x\wedge{\rm d}y\,.$

The exterior derivative of the form $\beta$ is given by

$\displaystyle{\rm d}\beta$ $\displaystyle=$
$\displaystyle-\partial_{y}\Big{[}\Xi\big{(}|\nabla
u|\big{)}\big{(}1+u_{x}^{2}\big{)}+\vartheta\big{(}|\nabla
u|\big{)}\Big{]}{\rm d}x\wedge{\rm d}y$ (3.5)
$\displaystyle+\partial_{x}\Big{[}\Xi\big{(}|\nabla
u|\big{)}u_{x}u_{y}\Big{]}{\rm d}x\wedge{\rm d}y$ $\displaystyle=:$
$\displaystyle\big{[}-\partial_{y}\varphi_{1}+\partial_{x}\varphi_{2}\Big{]}{\rm
d}x\wedge{\rm d}y\,.$

Finally we have for ${\rm d}\gamma$

${\rm d}\gamma=-\partial_{y}\Big{[}\Xi\big{(}|\nabla u|\big{)}u_{y}\Big{]}{\rm
d}x\wedge{\rm d}y-\partial_{x}\Big{[}\Xi\big{(}|\nabla
u|\big{)}u_{x}\Big{]}{\rm d}x\wedge{\rm d}y\,.$ (3.6)

Let us first consider (3.6) by computing

$\displaystyle\partial_{y}\Big{[}\Xi\big{(}|\nabla
u|\big{)}u_{y}\Big{]}+\partial_{x}\Big{[}\Xi\big{(}|\nabla
u|\big{)}u_{x}\Big{]}$ $\displaystyle=$ $\displaystyle u_{yy}\Xi\big{(}|\nabla
u|\big{)}+u_{y}\Xi^{\prime}\big{(}|\nabla
u|\big{)}\frac{u_{x}u_{xy}+u_{y}u_{yy}}{|\nabla u|}$
$\displaystyle+u_{xx}\Xi\big{(}|\nabla
u|\big{)}+u_{x}\Xi^{\prime}\big{(}|\nabla
u|\big{)}\frac{u_{x}u_{xx}+u_{y}u_{xy}}{|\nabla u|}$ $\displaystyle=$
$\displaystyle u_{xx}\Bigg{[}\Xi\big{(}|\nabla
u|\big{)}+\frac{\Xi^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla
u|}u_{x}^{2}\Bigg{]}+u_{yy}\Bigg{[}\Xi\big{(}|\nabla
u|\big{)}+\frac{\Xi^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla
u|}u_{y}^{2}\Bigg{]}$
$\displaystyle+u_{xy}\Bigg{[}2u_{x}u_{y}\frac{\Xi^{\prime}\big{(}|\nabla
u|\big{)}}{|\nabla u|}\Bigg{]}\,,$

hence we have recalling (2.2)

${\rm d}\gamma=0\,.$ (3.7)

Next we discuss (3.5). Direct calculations show

$\displaystyle\partial_{y}\varphi_{1}$ $\displaystyle=$
$\displaystyle\frac{\Xi^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla
u|}(u_{x}u_{xy}+u_{y}u_{yy})\big{(}1+u_{x}^{2}\big{)}+\Xi\big{(}|\nabla
u|\big{)}2u_{x}u_{xy}$ $\displaystyle+\frac{\vartheta^{\prime}\big{(}|\nabla
u|\big{)}}{|\nabla u|}(u_{x}u_{xy}+u_{y}u_{yy})$

and in addition

$-\partial_{x}\varphi_{2}=-\frac{\Xi^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla
u|}(u_{x}u_{xx}+u_{y}u_{xy})u_{x}u_{y}-\Xi\big{(}|\nabla
u|\big{)}(u_{xx}u_{y}+u_{x}u_{xy})\,.$

Combining these equations one obtains

$\displaystyle\partial_{y}\varphi_{1}-\partial_{x}\varphi_{2}$
$\displaystyle=$ $\displaystyle-u_{y}u_{xx}\Bigg{[}\Xi\big{(}|\nabla
u|\big{)}+\frac{\Xi^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla
u|}u_{x}^{2}\Bigg{]}$ (3.8) $\displaystyle-
u_{y}u_{yy}\Bigg{[}-\frac{\Xi^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla
u|}\big{(}1+u_{x}^{2}\big{)}-\vartheta^{\prime}\big{(}|\nabla
u|\big{)}\Bigg{]}$ $\displaystyle-
u_{xy}\Bigg{[}-\frac{\Xi^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla
u|}u_{x}\big{(}1+u_{x}^{2}\big{)}-\Xi\big{(}|\nabla u|\big{)}u_{x}$
$\displaystyle-\frac{\vartheta^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla
u|}u_{x}+u_{x}u_{y}^{2}\frac{\Xi^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla
u|}\Bigg{]}$ $\displaystyle=:$ $\displaystyle-
u_{y}u_{xx}T_{1}-u_{y}u_{yy}T_{2}-u_{xy}T_{3}\,.$

Now we compare (3.8) with the equation (2.2) and observe that $T_{1}$ is the
first coefficient on the right-hand side of (2.2).

If we can show in additon that

$\displaystyle T_{2}$ $\displaystyle=$ $\displaystyle\Xi\big{(}|\nabla
u|\big{)}+\frac{\Xi^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla u|}u_{y}^{2}\,,$
(3.9) $\displaystyle T_{3}$ $\displaystyle=$ $\displaystyle
u_{y}\Bigg{[}2u_{x}u_{y}\frac{\Xi^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla
u|}\Bigg{]}\,.$ (3.10)

then we obtain

$\partial_{y}\varphi_{1}-\partial_{x}\varphi_{2}=0\,,\quad\mbox{hence}\quad{\rm
d}\beta=0\,.$ (3.11)

For discussing (3.9), i.e. the validity of the equation

$-\frac{\Xi^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla
u|}\big{(}1+u_{x}^{2}\big{)}-\frac{\vartheta^{\prime}\big{(}|\nabla
u|\big{)}}{|\nabla u|}=\Xi\big{(}|\nabla
u|\big{)}+\frac{\Xi^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla u|}u_{y}^{2}\,,$

we observe that the latter identity is equivalent to

$-\frac{\vartheta^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla
u|}=\Xi\big{(}|\nabla u|\big{)}+\frac{\Xi^{\prime}\big{(}|\nabla
u|\big{)}}{|\nabla u|}\big{(}1+|\nabla u|^{2}\big{)}\,.$ (3.12)

The definition of $\vartheta$ (recall (1.6)) now yields

$\displaystyle\vartheta^{\prime}(t)$ $\displaystyle=$
$\displaystyle-\Big{[}t\Xi(t)+\Xi^{\prime}(t)\big{(}1+t^{2}\big{)}\Big{]}\,.$
(3.13)

which immediately gives (3.12).

Equation (3.10) takes the form

$\displaystyle 2u_{y}^{2}u_{x}\frac{\Xi^{\prime}\big{(}|\nabla
u|\big{)}}{|\nabla u|}$ $\displaystyle=$
$\displaystyle-\frac{\Xi^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla
u|}u_{x}\big{(}1+u_{x}^{2}\big{)}-\Xi\big{(}|\nabla u|\big{)}u_{x}$
$\displaystyle-\frac{\vartheta^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla
u|}u_{x}+u_{x}u_{y}^{2}\frac{\Xi^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla
u|}\,,$

i.e.

$u_{y}^{2}u_{x}\frac{\Xi^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla
u|}=-\frac{\Xi^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla
u|}u_{x}\big{(}1+u_{x}^{2}\big{)}-\Xi\big{(}|\nabla
u|\big{)}u_{x}-\frac{\vartheta^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla
u|}u_{x}$

and this relation is equivalent to

$u_{x}\Bigg{[}\frac{\Xi^{\prime}\big{(}|\nabla u|\big{)}}{|\nabla
u|}\big{(}1+|\nabla u|^{2}\big{)}+\Xi\big{(}|\nabla
u|^{2}\big{)}\Bigg{]}=-\frac{\vartheta^{\prime}\big{(}|\nabla
u|\big{)}}{|\nabla u|}u_{x}\,.$

Again we end up with (3.13), hence we also have (3.10) and finally (3.11).

It remains to consider (3.4) and to show

${\rm d}\alpha=0\,.$ (3.14)

However this can be done using the same arguments leading to (3.11).

With (3.7), (3.11) and (3.14) it is shown that $\hat{N}\wedge{\rm d}X$ is a
closed differential form and we have (1.10) on the simply connected domain
$\Omega$, thus Theorem 1.2 is established.∎

##  4 A parametrization generated by $X^{*}$

In the minimal surface case, Theorem 1.2 implies the conformal representation
and the analyticity of nonparametric minimal surfaces as outlined, e.g., in
Section 2.3 of [6]. The approach given there can be done by just varying the
second surface-parameter.

Here we do not expect analytic solutions in general and we prefer to follow a
variant given in, e.g., [3], which yields Lemma 4.1 in its symmetric
formulation.

Given $b(x,y)$, $a(x,y)$ according to Theroem 1.2 we consider the differential
form

$\omega:=b(x,y){\rm d}x-a(x,y){\rm d}y\,.$

Then we have on account of $-{\rm d}a=\alpha$, $-{\rm d}b=\beta$ and recalling
(3.2), (3.3)

${\rm d}\omega=-b_{y}(x,y){\rm d}x\wedge{\rm d}y-a_{x}(x,y){\rm d}x\wedge{\rm
d}y=0\,,$

which means that the form is closed and we may define the line integral

$E(x,y):=\int_{(x_{0},y_{0})}^{(x,y)}\omega\,,$ (4.1)

where we have

$\nabla E(x,y)=\left(\begin{array}[]{c}b(x,y)\\\\[4.30554pt]
-a(x,y)\end{array}\right)\,.$ (4.2)

With the notation of (3.4) and (3.5) one obtains

$\left(\begin{array}[]{c}\varphi_{1}\\\\[4.30554pt]
\varphi_{2}\end{array}\right)=\nabla
b\,,\quad\left(\begin{array}[]{c}\psi_{1}\\\\[4.30554pt]
\psi_{2}\end{array}\right)=-\nabla a\,.$ (4.3)

Combining (4.2) and (4.3) finally gives

$D^{2}E=\left(\begin{array}[]{cc}\varphi_{1}&\varphi_{2}\\\\[4.30554pt]
\psi_{1}&\psi_{2}\end{array}\right)\,.$ (4.4)

Discussing $D^{2}E$ the relevance of condition (1.11) becomes obvious.

######  Proposition 4.1.

Consider the function $E$: $\Omega\to\mathbb{R}$ given in (4.1) and suppose
that (1.11) holds.

Then for any $(x,y)\in\Omega$ the bilinear form $D^{2}E(x,y)$ is positive
definite.

_Proof of Propositon 4.1._ Fix some $(x,y)\in\Omega$ and abbreviate
$D^{2}E(x,y)$ through $D^{2}E$. Observe that (4.4) implies for all
$\eta=(\eta_{1},\eta_{2})\in\mathbb{R}^{2}-\\{0\\}$:

$\displaystyle D^{2}E(\eta,\eta)$ $\displaystyle=$
$\displaystyle\Xi\big{(}|\nabla
u|\big{)}\Big{[}\eta_{1}^{2}\big{(}1+u_{x}^{2}\big{)}+2\eta_{1}\eta_{2}u_{x}u_{y}+\eta_{2}^{2}\big{(}1+u_{y}^{2}\big{)}\Big{]}$
(4.5) $\displaystyle+\vartheta\big{(}|\nabla
u|\big{)}\big{(}\eta_{1}^{2}+\eta_{2}^{2}\big{)}\,.$

Considering (4.5) we note

$|2\eta_{1}\eta_{2}u_{x}u_{y}|\leq\eta_{1}^{2}u_{x}^{2}+\eta_{2}^{2}u_{y}^{2}\,,$

hence the definition (1.6) of the function $\vartheta$ shows

$\displaystyle D^{2}E(\eta,\eta)$ $\displaystyle\geq$
$\displaystyle\Big{[}\Xi\big{(}|\nabla u|\big{)}+\vartheta\big{(}|\nabla
u|\big{)}\Big{]}|\eta|^{2}$ $\displaystyle=$ $\displaystyle\Bigg{[}g(|\nabla
u|)-|\nabla u|g^{\prime}\big{(}|\nabla u|\big{)}\Bigg{]}|\eta|^{2}\,.$

Thus we can apply hypothesis (1.11) to see that $D^{2}E$ is positive definite.
∎

In the next step we introduce a diffeomorphism generated by the gradient field
$\nabla E$, i.e. by $X^{*}$ (recall (1.17)) :

$\Lambda(x,y)=\left(\begin{array}[]{c}x\\\ y\end{array}\right)+\nabla
E(x,y)\,,\quad(x,y)\in\Omega\,.$ (4.6)

As shortly outlined in the appendix, well known arguments show that $\Lambda$
in fact is a diffeomorphism onto its image.

A more refined analysis of $\Lambda$ of course depends on the underlying
function $X^{*}$. In the classical minimal surface case we are directly lead
to a conformal parametrization without referring to Lichtenstein’s mapping
theorem (see, e.g., [6], Section 2.3 for further comments).

Here we expect the asymptotic correspondence to this method as a natural
consequence of our main theorem 1.2.

We start with an estimate for the Jacobian which proves the claim (1.19) of
Corollary 1.1.

######  Proposition 4.2.

Suppose that we have Assumption 1.1 and consider the diffeomorhism $\Lambda$
defined in (4.6).

Then we have for all $(x,y)\in\Omega$

${\rm det}\,D\Lambda\geq 1+\Xi\big{(}|\nabla u|\big{)}\big{(}1+|\nabla
u|^{2}\big{)}\,.$

_Proof of Proposition 4.2._ The Jacobian of $\Lambda$ is given by

$\displaystyle\operatorname{det}D\Lambda$ $\displaystyle=$
$\displaystyle\partial_{x}\Lambda_{1}\partial_{y}\Lambda_{2}-\partial_{x}\Lambda_{2}\partial_{y}\Lambda_{1}$
(4.7) $\displaystyle=$ $\displaystyle\Big{[}1+\big{[}\Xi\big{(}|\nabla
u|\big{)}(1+u_{x}^{2})+\vartheta\big{(}|\nabla u|\big{)}\big{]}\Big{]}$
$\displaystyle\quad\cdot\Big{[}1+\big{[}\Xi\big{(}|\nabla
u|\big{)}(1+u_{y}^{2})+\vartheta\big{(}|\nabla u|\big{)}\big{]}\Big{]}$
$\displaystyle-\Xi^{2}\big{(}|\nabla u|\big{)}\big{(}|\nabla
u|\big{)}u^{2}_{x}u^{2}_{y}$ $\displaystyle=$ $\displaystyle
1+\Xi\big{(}|\nabla u|\big{)}\big{(}2+|\nabla
u|^{2}\big{)}+\Xi^{2}\big{(}|\nabla u|\big{)}\big{(}1+|\nabla u|^{2}\big{)}$
$\displaystyle+\vartheta\big{(}|\nabla u|\big{)}+\vartheta\big{(}|\nabla
u|\big{)}\Xi\big{(}|\nabla u|\big{)}\big{(}2+|\nabla u|^{2}\big{)}$
$\displaystyle+\vartheta^{2}\big{(}|\nabla u|\big{)}\,.$

Moreover, the definition (1.6) of $\vartheta$ shows

$\displaystyle\vartheta\big{(}|\nabla u|\big{)}+\vartheta\big{(}|\nabla
u|\big{)}\Xi\big{(}|\nabla u|\big{)}\big{(}2+|\nabla
u|^{2}\big{)}+\vartheta^{2}\big{(}|\nabla u|\big{)}$ (4.8) $\displaystyle=$
$\displaystyle\Big{[}g\big{(}|\nabla u|\big{)}-|\nabla
u|g^{\prime}\big{(}|\nabla u|\big{)}\Big{]}-\Xi\big{(}|\nabla u|\big{)}$
$\displaystyle+\Big{[}g\big{(}|\nabla u|\big{)}-|\nabla
u|g^{\prime}\big{(}|\nabla u|\big{)}\Big{]}\Xi\big{(}|\nabla
u|\big{)}\big{(}2+|\nabla u|^{2}\big{)}$ $\displaystyle-\Xi^{2}\big{(}|\nabla
u|\big{)}\big{(}2+|\nabla u|^{2}\big{)}+\Big{[}g\big{(}|\nabla
u|\big{)}-|\nabla u|g^{\prime}\big{(}|\nabla u|\big{)}\Big{]}^{2}$
$\displaystyle-2\Xi\big{(}|\nabla u|\big{)}\Big{[}g\big{(}|\nabla
u|\big{)}-|\nabla u|g^{\prime}\big{(}|\nabla
u|\big{)}\Big{]}+\Xi^{2}\big{(}|\nabla u|\big{)}\,.$

Combining (4.7) and (4.8) yields

$\displaystyle\operatorname{det}D\Lambda$ $\displaystyle=$ $\displaystyle
1+\Xi\big{(}|\nabla u|\big{)}\big{(}1+|\nabla u|^{2}\big{)}$ (4.9)
$\displaystyle+\Big{[}g\big{(}|\nabla u|\big{)}-|\nabla
u|g^{\prime}\big{(}|\nabla u|\big{)}\Big{]}$
$\displaystyle\cdot\Bigg{[}1+\Xi\big{(}|\nabla u|\big{)}|\nabla
u|^{2}+\Big{[}g\big{(}|\nabla u|\big{)}-|\nabla u|g^{\prime}\big{(}|\nabla
u|\big{)}\Big{]}\Bigg{]}\,.$

With (4.9) the proof of Proposition 4.2 again follows from (1.11). ∎

Let us now rewrite the functions $\varphi_{1}$ and $\psi_{2}$ in the following
form

$\displaystyle\varphi_{1}$ $\displaystyle=$ $\displaystyle\Xi\big{(}|\nabla
u|\big{)}\big{(}1+u_{x}^{2}\big{)}+g\big{(}|\nabla u|\big{)}-\Xi\big{(}|\nabla
u|\big{)}|\nabla u|^{2}-\Xi\big{(}|\nabla u|\big{)}$ $\displaystyle=$
$\displaystyle g\big{(}|\nabla u|\big{)}-\Xi\big{(}|\nabla
u|\big{)}u_{y}^{2}\,,$ $\displaystyle\psi_{2}$ $\displaystyle=$ $\displaystyle
g\big{(}|\nabla u|\big{)}-\Xi\big{(}|\nabla u|\big{)}u_{x}^{2}\,.$ (4.10)

Using (4) we recall the definition of $\Lambda$ and (4.4), hence

$D\Lambda=\left(\begin{array}[]{cc}1+\varphi_{1}&\varphi_{2}\\\\[17.22217pt]
\psi_{1}&1+\psi_{2}\end{array}\right)\,,$

and we calculate for all $(\hat{x},\hat{y})\in\hat{\Omega}$
($(x,y):=\lambda^{-1}(\hat{x},\hat{y})$)

$\displaystyle
D\big{(}\Lambda^{-1}\big{)}(\hat{x},\hat{y})=\big{(}D\Lambda(x,y)\big{)}^{-1}$
(4.13) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}1+g\big{(}|\nabla
u|\big{)}-\Xi\big{(}|\nabla u|\big{)}u_{y}^{2}&\Xi\big{(}|\nabla
u|\big{)}u_{x}u_{y}\\\\[17.22217pt] \Xi\big{(}|\nabla
u|\big{)}u_{x}u_{y}&1+g\big{(}|\nabla u|\big{)}-\Xi\big{(}|\nabla
u|\big{)}u_{x}^{2}\end{array}\right)^{-1}$ $\displaystyle=$
$\displaystyle\frac{1}{\operatorname{det}D\Lambda}\left(\begin{array}[]{cc}1+g\big{(}|\nabla
u|\big{)}-\Xi\big{(}|\nabla u|\big{)}u_{x}^{2}&-\Xi\big{(}|\nabla
u|\big{)}u_{x}u_{y}\\\\[17.22217pt] -\Xi\big{(}|\nabla
u|\big{)}u_{x}u_{y}&1+g\big{(}|\nabla u|\big{)}-\Xi\big{(}|\nabla
u|\big{)}u_{y}^{2}\end{array}\right)$ (4.15) $\displaystyle=:$
$\displaystyle\frac{1}{\operatorname{det}D\Lambda}\,\Pi(x,y)\,.$ (4.16)

In addition we compute

$\displaystyle D\Big{(}u\circ\Lambda^{-1}\Big{)}(\hat{x},\hat{y})$
$\displaystyle=$ $\displaystyle
Du(x,y)\,D\big{(}\Lambda^{-1}\big{)}(\hat{x},\hat{y})$ (4.17) $\displaystyle=$
$\displaystyle\frac{1}{\operatorname{det}D\Lambda}\Pi(x,y)\,\nabla u(x,y)$
$\displaystyle=$
$\displaystyle\frac{1}{\operatorname{det}D\Lambda}\,\big{[}1+g-|\nabla
u|^{2}\Xi\big{]}\nabla u\big{]}$ $\displaystyle=:$
$\displaystyle\frac{1}{\operatorname{det}D\Lambda}\,\pi(x,y)\,.$

After these preparations we consider the parametrization of the surface
$\operatorname{graph}(u)$ already introduced in (1.18):

$\chi:\,(\hat{x},\hat{y})\mapsto\Big{(}\Lambda^{-1}(\hat{x},\hat{y}),u\circ\Lambda^{-1}(\hat{x},\hat{y})\Big{)}\,,\quad(\hat{x},\hat{y})\in\hat{\Omega}\,.$

Then (4.13) and (4.17) yield

$D\chi(\hat{x},\hat{y})=:\frac{1}{\operatorname{det}D\Lambda}\Big{(}X\,\,Y\big{)}=\frac{1}{\operatorname{det}D\Lambda}\left(\begin{array}[]{c}\Pi(x,y)\\\\[4.30554pt]
\pi(x,y)\end{array}\right)\,,$ (4.18)

where (4.13), (4.17) and the definition (4.18) of $X$, $Y$ imply

$\displaystyle X=\left(\begin{array}[]{c}1+g\big{(}|\nabla
u|\big{)}-\Xi\big{(}|\nabla u|\big{)}u_{x}^{2}\\\\[8.61108pt]
-\Xi\big{(}|\nabla u|\big{)}u_{x}u_{y}\\\\[8.61108pt]
u_{x}\big{[}1+g\big{(}|\nabla u|\big{)}-\Xi\big{(}|\nabla u|\big{)}|\nabla
u|^{2}\big{]}\end{array}\right)\,,$ $\displaystyle
Y=\left(\begin{array}[]{c}-\Xi\big{(}|\nabla
u|\big{)}u_{x}u_{y}\\\\[8.61108pt] 1+g\big{(}|\nabla
u|\big{)}-\Xi\big{(}|\nabla u|\big{)}u_{y}^{2}\\\\[8.61108pt]
u_{y}\big{[}1+g\big{(}|\nabla u|\big{)}-\Xi\big{(}|\nabla u|\big{)}|\nabla
u|^{2}\big{]}\end{array}\right)\,.$

Now we come to the last part of Corollary 1.1:

######  Lemma 4.1.

If $X$ and $Y$ are given as above, then we have the equations

$\displaystyle X\cdot Y$ $\displaystyle=$ $\displaystyle
u_{x}u_{y}\Theta\big{(}|\nabla u|\big{)}\,,$ $\displaystyle|X|^{2}-|Y|^{2}$
$\displaystyle=$
$\displaystyle\big{[}u_{x}^{2}-u_{y}^{2}\big{]}\Theta\big{(}|\nabla
u|\big{)}\,,$

where the function $\Theta$: $[0,\infty)\to\mathbb{R}$ is given by

$\Theta(t)=\Bigg{[}1-\frac{g(t)g^{\prime}(t)}{t}\Bigg{]}+\Bigg{[}\big{(}g(t)-tg^{\prime}(t)\big{)}-\frac{g^{\prime}(t)}{t}\Bigg{]}\Big{[}2+\big{(}g(t)-tg^{\prime}(t)\big{)}\Big{]}\,.$

_Proof of Lemma 4.1._ By elementary calculations we obtain

$\displaystyle X\cdot Y$ $\displaystyle=$ $\displaystyle
u_{x}u_{y}\Bigg{[}-\Xi\big{(}|\nabla u|\big{)}\Big{[}2\big{(}1+g\big{(}|\nabla
u|\big{)}-\Xi\big{(}|\nabla u|\big{)}|\nabla u|^{2}\Big{]}$ (4.21)
$\displaystyle+\Big{[}1+g\big{(}|\nabla u|\big{)}-\Xi\big{(}|\nabla
u|\big{)}|\nabla u|^{2}\Big{]}^{2}\Bigg{]}$ $\displaystyle=:$ $\displaystyle
u_{x}u_{y}\,\tilde{\Theta}\big{(}|\nabla u|\big{)}\,.$

as well as

$\displaystyle|X|^{2}-|Y|^{2}$ $\displaystyle=$
$\displaystyle\big{[}u_{x}^{2}-u_{y}^{2}\big{]}\,\tilde{\Theta}\big{(}|\nabla
u|\big{)}$ (4.22)

with the same function $\tilde{\Theta}$.

Let us write the function $\tilde{\Theta}(t)$ from (4.21) and (4.22) as a more
convenient form. We have

$2\big{(}1+g(t)\big{)}-\Xi(t)t^{2}=\big{(}2+g(t)\big{)}+\big{(}g(t)-tg^{\prime}(t)\big{)}\,,$

hence

$\displaystyle-\Xi(t)\Bigg{[}2\big{(}1+g(t)\big{)}-\Xi(t)t^{2}\Bigg{]}$ (4.23)
$\displaystyle=$
$\displaystyle-\frac{\big{(}2+g(t)\big{)}g^{\prime}(t)}{t}-\frac{g^{\prime}(t)}{t}\big{(}g(t)-tg^{\prime}(t)\big{)}\,.$

Moreover, we note

$\Big{[}1+g(t)-tg^{\prime}(t)\Big{]}^{2}=1+2\big{(}g(t)-tg^{\prime}(t)\big{)}+\big{(}g(t)-tg^{\prime}(t)\big{)}^{2}\,.$
(4.24)

Adding (4.23) and (4.24) we obtain

$\displaystyle\tilde{\Theta}(t)$ $\displaystyle=$
$\displaystyle\Bigg{[}1-\frac{g(t)g^{\prime}(t)}{t}\Bigg{]}-\frac{2g^{\prime}(t)}{t}+\Bigg{[}2-\frac{g^{\prime}(t)}{t}\Bigg{]}\big{(}g(t)-tg^{\prime}(t)\big{)}$
$\displaystyle+\big{(}g(t)-tg^{\prime}(t)\big{)}^{2}$ $\displaystyle=$
$\displaystyle\Bigg{[}1-\frac{g(t)g^{\prime}(t)}{t}\Bigg{]}$
$\displaystyle+\Bigg{[}\big{(}g(t)-tg^{\prime}(t)\big{)}-\frac{g^{\prime}(t)}{t}\Bigg{]}\Big{[}2+\big{(}g(t)-tg^{\prime}(t)\big{)}\Big{]}=\Theta(t)\,,$

hence we have proved Lemma 4.1. ∎

######  Example 4.1.

We recall that in the minimal surface case we have for all $t\geq 0$

$g(t)-tg^{\prime}(t)=\frac{g^{\prime}(t)}{t}\quad\mbox{and}\quad\frac{g(t)g^{\prime}(t)}{t}\equiv
1\,.$

As a consequence of Lemma 4.1 we obtain an explicit asymptotic expansion for
the function $\Theta$.

######  Corollary 4.1.

Suppose that we have the assumptions of Corollary 1.1. Then we have (1.21),
i.e. for all $t>0$

$|\Theta(t)|\leq d_{1}|t|^{2-\mu}+d_{2}|t|^{-1}$

with some real numbers $d_{1}$, $d_{2}>0$.

_Proof of Corollary 4.1._ As in Remark 1.2, $ii$), we now have using (1.20)

$g^{\prime}(t)\Big{[}g(t)-tg^{\prime}(t)\Big{]}=tR(t)\,,\quad\mbox{i.e.}\quad
R(t)=O\big{(}t^{1-\mu}\big{)}\,.$

By Remark 1.2, $ii$), we obtain in addition

$\displaystyle 1-\frac{g(t)g^{\prime}(t)}{t}$ $\displaystyle=$
$\displaystyle\Big{[}1-\big{(}g^{\prime}(t)\big{)}^{2}\Big{]}+O\big{(}t^{1-\mu}\big{)}$
$\displaystyle=$ $\displaystyle
h(t)\big{(}1+g^{\prime}(t)\big{)}+O\big{(}t^{1-\mu}\big{)}\,.$

This, together with the boundedness of $g^{\prime}$ and once more applying
(1.20) gives the corollary. ∎

We finally note that Proposition 4.2, Lemma 4.1 and Corollay 4.1 together
yield the proof of Corollary 1.1, ∎

##  5 Appendix

For the sake of completeness we append Proposition 5.1 which guarantees the
fact that the function $\Lambda$ defined in (4.6) is a diffeomorhism onto its
image. The relation (5.1) is of particular interest if
$\Omega=\mathbb{R}^{2}$. The short proof is outlined below following
Osserman’s book [3].

######  Proposition 5.1.

With the hypotheses of Proposition 4.1 we suppose that $\Omega$ is a convex
open set and let $\Lambda$: $\Omega\to G:=\Lambda(\Omega)$,

$\Lambda(x,y)=\left(\begin{array}[]{c}x\\\ y\end{array}\right)+\nabla
E(x,y)\,,\quad(x,y)\in\Omega\,,$

with $E$ defined in (4.1).

Then $\Lambda$ is a diffeomorphism, moreover we have for any
$B_{r}(x_{0},y_{0})\subset\Omega$

$B_{r}\Big{(}\Lambda\big{(}(x_{0},y_{0})\big{)}\Big{)}\subset G\,.$ (5.1)

_Proof of Proposition 5.1._ On the convex set $B_{r}(0)$ and we let

$e(t):=E\big{(}tx+(1-t)y\big{)}\,,\quad t\in[0,1]\,.$

Then

$e^{\prime\prime}(t)=D^{2}E\big{(}tx+(1-t)y\big{)}\big{(}x-y,x-y\big{)}>0\quad\mbox{for
all $t\in(0,1)$}\,,$

which shows that $e^{\prime}(t)$ is an increasing function, i.e.

$0<e^{\prime}(1)-e^{\prime}(0)=\Big{[}\nabla E(x)-\nabla
E(y)\Big{]}\cdot(x-y)\,.$ (5.2)

By the definition of $\Lambda$ and by (5.1), (5.2) we obtain ($x\not=y\in
B_{r}(0)$)

$\displaystyle\big{[}\Lambda(x_{1},y_{1})-\Lambda(x_{2}.y_{2})\big{]}\cdot\big{[}(x_{1},y_{1})-(x_{2},x_{2})\big{]}$
$\displaystyle=$ $\displaystyle|(x_{1},y_{1})-(x_{2},y_{2})|^{2}$
$\displaystyle+\Big{[}\nabla E(x_{1},y_{1})-\nabla
E(x_{2},y_{2})\Big{]}\cdot\big{[}(x_{1},y_{1})-(x_{2},x_{2})\big{]}$
$\displaystyle>$ $\displaystyle|(x_{1},y_{1})-(x_{2},y_{2})|^{2}\,,$

and by the Cauchy-Schwarz inequality this implies

$\big{|}\Lambda(x_{1},y_{1})-\Lambda(x_{2},y_{2})\big{|}>|(x_{1},y_{1})-(x_{2},y_{2})|\,.$
(5.3)

Observe that (5.3) gives the injectivity of $\Lambda$ and since in addition

$\operatorname{det}D\Lambda(x,y)>0\quad\mbox{for}\quad(x,y)\in B_{r}(0)\,,$

$\Lambda$: $B_{r}(0)\to G:=\Lambda\big{(}B_{r}(0)\big{)}$ is a global
diffeomorphism and we have (5.1) if $G=\mathbb{R}^{2}$.

In the case $G\not=\mathbb{R}^{2}$ there exists
$(\hat{x},\hat{y})\in\partial\big{(}\mathbb{R}^{2}\setminus G\big{)}=\partial
G$, such that

$\Big{|}(\hat{x},\hat{y})-\Lambda\big{(}(x_{0},y_{0})\big{)}\Big{|}=\inf_{\eta\in\mathbb{R}^{2}\setminus
G}\Big{|}\eta-\Lambda\big{(}(x_{0},y_{0})\big{)}\Big{|}\,.$

This means that we find sequences $\\{(x,y)^{(k)}\\}$,
$\\{(\hat{x},\hat{y})^{(k)}=\Lambda\big{(}(x,y)^{(k)}\big{)}\\}$ such that as
$k\to\infty$

$(x,y)^{(k)}\to(x,y)\in\partial
B_{r}(x_{0},y_{0})\quad\mbox{and}\quad(\hat{x},\hat{y})^{(k)}\to(\hat{x},\hat{y})\,.$

The definition of $(\hat{x},\hat{y})$ finally yields

$\inf_{\eta\in\mathbb{R}^{2}\setminus
G}\Big{|}\eta-\Lambda\big{(}(x_{0},y_{0})\big{)}\Big{|}\geq r\,,$

hence (5.1) and the proof of Lemma 5.1 is complete. $\Box$

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