# Fingerprints of Closed Trajectories of a Strebel Quadratic Differential

Gliaa Braek and Faouzi Thabet
University of Gabes, Tunisia

###### Abstract

In this note, we study the fingerprints of closed smooth curves that are
trajectories of a particular Strebel quadratic differentials. It is a
generalization of the cases of polynomials and rational lemniscates.

2010 Mathematics subject classification: 30C10, 30C15, 34E05.

Keywords and phrases: Stebel Quadratic differentials.Lemniscates. Conformal
Maps.Fingerprints.

##  1
A Strebel quadratic differential

We consider the quadratic differential on the Riemann sphere
$\widehat{\mathbb{C}}$ :

$\varpi\left(z\right)=-\left(\sum_{i=1}^{n}\frac{\alpha_{i}}{z-a_{i}}\right)^{2}dz^{2},$

where the $a_{i}$ ’s are pairwise distinct complex numbers, the $\alpha_{i}$
’s are real numbers with non-negative sum $\alpha.$

_Finite critical points_ and _infinite critical points_ of the quadratic
differential $\varpi$ are respectively its zero’s and poles
($\left\\{\infty\right\\}\cup\left\\{a_{j}\right\\}_{1\leq j\leq n}$); all
other points of $\widehat{\mathbb{C}}$ are called _regular points_ of
$\varpi.$

_Horizontal trajectories_ (or just trajectories) of the quadratic differential
$\varpi$ are the zero loci of the equation

$\varpi\left(z\right)>0,$

or equivalently

$\prod_{i=1}^{n}\left|z-a_{i}\right|^{\alpha_{i}}=\text{\emph{const}}.$ (1)

For a given non-negative real $\lambda,$ we define the _lemniscate_
$\Gamma_{\lambda}$ as the trajectory of the quadratic differential $\varpi$
with $\emph{const}=\lambda$ in (1) : __

$\Gamma_{\lambda}=\left\\{z\in\mathbb{C};\left|f\left(z\right)\right|=\prod_{i=1}^{n}\left|z-a_{i}\right|^{\alpha_{i}}=\lambda\right\\},$

where $f\left(z\right)$ is the multi-valued function defined by

$f\left(z\right)=\prod_{i=1}^{n}\left(z-a_{i}\right)^{\alpha_{i}}.$

Any connected component of $\Gamma_{\lambda}$ will be called a _sub-
lemniscate._

The _vertical_ (or, _orthogonal_) trajectories are obtained by replacing $\Im$
by $\Re$ in equation (1). The horizontal and vertical trajectories of the
quadratic differential $\varpi$ produce two pairwise orthogonal foliations of
the Riemann sphere $\widehat{\mathbb{C}}$.

A trajectory passing through a critical point of $\varpi$ is called _critical
trajectory_. In particular, if it starts and ends at a finite critical point,
it is called _finite critical trajectory_ , otherwise, we call it an _infinite
critical trajectory_.

If two different trajectories are not disjoint, then their intersection must
be a zero of the quadratic differential.

The closure of the set of finite and infinite critical trajectories is called
the _critical graph_ of $\varpi,$ we denote it by $\Gamma;$ see example in
Figure 1.

Figure 1: Critical graph of
$-\left(\frac{1}{z-1}-\frac{1}{z+1}+\frac{\sqrt{2}}{z}\right)^{2}dz^{2}.$

The local and global structures of the trajectories is well known (more
details about the theory of quadratic differentials can be found in [4],[3],
or [5]). The set $\widehat{\mathbb{C}}\setminus\Gamma_{f}$ consists of a
finite number of domains called the _Configurations Domain_ of $\varpi.$ For a
general quadratic differential on a $\widehat{\mathbb{C}}$, there are five
kind of Configuration Domains, see [3, Theorem3.5]. Since all the infinite
critical points of $\varpi$ are poles of order $2$ with negative residues,
then there are three possible Domain Configurations:

  * •

the _Circle Domain_ : It is swept by closed trajectories and contains exactly
one double pole that we will call the center of the domain. Its boundary is a
closed critical trajectory. For a suitably chosen real constant $c$ and some
real number $r>0,$ the function $z\longmapsto
r\exp\left(cf\left(t\right)\right)$ is a conformal map from the circle domain
$D$ onto the unit disk; it extends continuously to the boundary $\partial D,$
and sends the double pole to the origin.

  * •

the _Ring Domain_ : It is swept by closed trajectories. Its boundary consists
of two connected components. For a suitably chosen real constant $c$ and some
real numbers $0<r_{1}<r_{2},$ the function
$z\longmapsto\exp\left(cf\left(t\right)\right)$ is a conformal map from the
circle domain $D$ onto the annulus
$\left\\{z:r_{1}<\left|z\right|<r_{2}\right\\}$ and it extends continuously to
the boundary $\partial D.$

  * •

the _Dense domain :_ It is swept by recurrent critical trajectory i.e., the
interior of its closure is non-empty.

###### Definition 1

A quadratic differential is _Strebel_ if the set of its closed trajectories
covers $\widehat{\mathbb{C}}$ up to a zero Lebesgue measure set.

###### Proposition 2

The quadratic differential $\varpi$ is Strebel.

Proof. Since the infinite critical points of the quadratic differential
$\varpi$ are only double poles with negative residues, it suffices to show
that it has no recurrent trajectories. Indeed,If $\varpi$ has a recurrent
trajectory, then, its Domain Configuration contains a dense domain $D$ :

$\prod_{i=1}^{n}\left|z-a_{i}\right|^{\alpha_{i}}=\text{\emph{const} on }D.$

The function

$z\longmapsto\Re\left(\sum_{i=1}^{n}\alpha_{i}\log\left(z-a_{i}\right)\right)$

is continuous and constant on the open subset $D$ of $\mathbb{C},$ then it is
constant everywhere in $\mathbb{C}$, which is clearly impossible by
harmonicity.

Let $p\left(z\right)$ the complex polynomial satisfying

$\sum_{i=1}^{n}\frac{\alpha_{i}}{z-a_{i}}=\frac{p\left(z\right)}{\prod_{i=1}^{n}\left(z-a_{i}\right)}.$

By assumption, the leading coefficient $\alpha=\sum_{i=1}^{n}\alpha_{i}\neq 0$
of $p\left(z\right),$ then $\deg p=n-1.$ Let $z_{1},...,z_{n-1}$ be the zeros
(counted with multiplicities) of $p\left(z\right),$

$w_{1}=\prod_{i=1}^{n}\left|z_{1}-a_{i}\right|^{\alpha_{i}},...,w_{n-1}=\prod_{i=1}^{n}\left|z_{n-1}-a_{i}\right|^{\alpha_{i}}.$

###### Proposition 3

If for some $1$ $\leq i<j\leq n-1,$ we have

$\left|w_{i}\right|=\left|w_{j}\right|=\max\left\\{\left|w_{k}\right|;k=1,...,n-1\right\\},$

then, there exists a finite critical trajectory connecting $z_{i}$ and
$z_{j}.$ In particular, the critical graph $\Gamma_{f}$ is connected, if and
only if $\left|w_{1}\right|=\cdot\cdot\cdot=\left|w_{n-1}\right|.$

Proof. If no finite critical trajectory connecting $z_{i}$ and $z_{j},$ then a
lemniscate $\Gamma_{c},$ for some $c>\left|w_{i}\right|,$ is not connected :
$\Gamma_{c}$ is a disjoint union of $s\geq 2$ closed curves (sub-lemniscates)
$L_{1},...,L_{s},$ each of them encircles a part of the critical graph
$\Gamma_{f}.$ Looking at each of these curves as a $\varpi$-polygon and
applying the so-called Teichmüller Lemma (see [4, Thm.14.1]), we get for
$k=1,...,s$ :

$0=2+\sum n_{k}.$

Making the sum of all equalities, we obtain

$0=2s+2\left(n-1-n\right)=2s-2;$

a contradiction.

##  2 Fingerprints of sub-lemniscates

Let $\gamma$ be a $\mathcal{C}^{\infty}$ Jordan curve in $\mathbb{C};$ by a
Jordan theorem, $\gamma$ splits $\widehat{\mathbb{C}}$ into a bounded and an
unbounded simply connected components $D_{-}$ and $D_{+}.$ The Riemann mapping
theorem asserts that there exist conformal maps
$\phi_{-}:\Delta\longrightarrow$ $D_{-},$ and
$\phi_{+}:\widehat{\mathbb{C}}\setminus\overline{\Delta}\longrightarrow$
$D_{+},$ where $\Delta$ is the unit disk. The map $\phi_{+}$ is uniquely
determined by the normalization $\phi_{+}\left(\infty\right)=\infty$ and
$\phi_{+}^{\prime}\left(\infty\right)>0.$ It is well-known that $\phi_{-}$ and
$\phi_{+}$ extend to $\mathcal{C}^{\infty}$-diffeomorphisms on the closure of
their respective domain. The fingerprint of $\gamma$ is the the map $k$ $:=$
$\phi_{+}^{-1}\circ\phi_{-}:S^{1}\longrightarrow S^{1}$ from the unit circle
$S^{1}$ to itself. Note that $k$ is uniquely determined by up to post-
composition with an automorphism of $D$ onto itself. Moreover, the fingerprint
$k$ is invariant under translations and scalings of the curve $\gamma.$
Fingerprints of polynomial and rational ”proper” lemniscates have been studied
in [1],[2], [6], and [7]. Recall that a lemniscate is called ”proper” if it is
smooth and connected. In the this paper, we investigate the case of smooth and
not necessary connected lemniscates.

Let $\lambda>0,\lambda\notin\left\\{w_{k},k=1,...,n-1\right\\}.$ Since the
quadratic differential $\varpi$ is Strebel, the lemniscate $\Gamma_{\lambda}$
is the union of a finite number of disjoint sub-lemniscates in $\mathbb{C},$
each of them is entirely included either in a Circle or a Ring Domain of
$\varpi$. More precisely, if $\gamma$ is a sub-lemniscate of $\varpi,$ then

$\displaystyle D_{-}\cap\left\\{a_{j}\right\\}_{1\leq j\leq n}$
$\displaystyle\neq\emptyset,$ $\displaystyle
D_{+}\cap\left(\left\\{\infty\right\\}\cup\left\\{a_{j}\right\\}_{1\leq j\leq
n}\right)$ $\displaystyle\neq\emptyset.$

It’s obvious (by definition of the Circle Domains) that, $\gamma$ is entirely
included in a Circle Domain of $\varpi,$ if and only if

$card\left(D_{-}\cap\left\\{a_{j}\right\\}_{1\leq j\leq n}\right)=1,\text{or
}D_{+}\cap\left(\left\\{\infty\right\\}\cup\left\\{a_{j}\right\\}_{1\leq j\leq
n}\right)=\left\\{\infty\right\\}.$

Observe that a lemniscate is proper if and only if it is contained in the
circle domain of $\varpi$ containing the infinity $\infty.$

###  2.1 Lemniscates in a Circle Domain

###### Theorem 4

Let be $a$ a pole $\varpi$ (
$a\in\left\\{\infty\right\\}\cup\left\\{a_{j}\right\\}_{1\leq j\leq n}$ ) and
$\gamma_{a}$ a sub-lemniscate of $\varpi.$ Then, the fingerprint $k$
$:S^{1}\longrightarrow S^{1}$ of $\gamma_{a}$ is given by

$\left\\{\begin{array}[c]{c}k(z)=B_{\infty}(z)^{1/\alpha},\text{ \emph{if
}}a=\infty\\\ k^{-1}(z)=z^{\alpha/\alpha_{j}}B_{j}(z)^{1/\alpha_{j}},\text{
\emph{if} }a=a_{j}.\end{array}\right.$

with

$B_{\infty}\left(z\right)=e^{i\theta_{\infty}}\prod_{i=1}^{n}\left(\frac{z-\phi_{-}^{-1}\left(a_{i}\right)}{1-\overline{\phi_{-}^{-1}\left(a_{i}\right)}z}\right)^{\alpha_{i}},$

and

$B_{j}\left(z\right)=e^{i\theta_{j}}\prod_{i\neq
j}\left(\frac{z-\phi_{+}^{-1}\left(a_{i}\right)}{1-\overline{\phi_{+}^{-1}\left(a_{i}\right)}z}\right)^{\alpha_{i}}.$

for some real numbers $\theta_{\infty}$ and $\theta_{j}.$

Proof. Jenkins Theorem on the Configuration Domains of the quadratic
differential $\varpi$ asserts that there exists a connected neighborhood
$\mathcal{U}_{a}$ of $a$ (a Circle Domain of $\varpi$) bounded by finite
critical trajectories of $\varpi,$ such that all trajectories of $\varpi$
(sub-lemniscates) inside $\mathcal{U}_{a}$ are closed smooth curves encircling
$a.$ Moreover, for a suitably chosen non-vanishing real constant $c,$ the
function

$\psi_{a}:z\longmapsto\exp\left(c\sum_{i=1}^{n}\int^{z}\frac{\alpha_{i}}{z-a_{i}}dt\right)$

is a conformal map from $\mathcal{U}_{a}$ onto a certain disk centered in
$z=0.$

Figure 2: Critical graph of
$-\left(\frac{1}{z-1}-\frac{1}{z+1}+\frac{\sqrt{2}}{z}\right)^{2}dz^{2}.$ and
lemniscates in Circle Domains: $a=\infty$, and $a=-1$.

A more obvious form of it, is

$\psi_{a}\left(z\right)=\beta\prod_{i=1}^{n}\left(z-a_{i}\right)^{c\alpha_{i}}$

for some complex number $\beta.$ Baring in mind that $\psi$ is univalent near
$a$, we get

$c=\left\\{\begin{array}[c]{c}\frac{1}{\alpha},\text{ if }a=\infty\\\
\frac{1}{\alpha_{j}},\text{ if }a=a_{j}.\end{array}\right.$

It follows that the function

$z\longmapsto\left\\{\begin{array}[c]{c}\prod_{i=1}^{n}\left(z-a_{i}\right)^{\alpha_{i}/\alpha},\text{
if }a=\infty,\\\
\prod_{i=1}^{n}\left(z-a_{i}\right)^{\alpha_{i}/\alpha_{j}},\text{ if
}a=a_{j}.\end{array}\right.$

is a conformal map from $\mathcal{U}_{a}$ onto a certain disk $\Delta_{a}$
centered in $z=0.$ For the sake of simplicity, we may assume that $\Delta_{a}$
with a radius $R>1.$ For the given sub-lemniscate $\gamma_{a}$ in
$\mathcal{U}_{a}$ (see Figure 2), it is straightforward that the previous
function maps $\Omega_{-}$ conformally onto the unit disk $\Delta.$ Thus,

$\left\\{\begin{array}[c]{c}\phi_{+}^{-1}\left(z\right)=f\left(z\right)^{1/\alpha},\text{
\emph{if} }a=\infty,\\\
\phi_{-}^{-1}\left(z\right)=f\left(z\right)^{1/\alpha_{j}},\text{ \emph{if}
}a=a_{j}.\end{array}\right..$ (2)

In the case $a=\infty,$ the functions

$z\longmapsto\
\frac{\phi_{-}\left(z\right)-a_{i}}{\frac{z-\phi_{-}^{-1}\left(a_{i}\right)}{1-\overline{\phi_{-}^{-1}\left(a_{i}\right)}z}};\left|z\right|\leq
1,i=1,...,n$

are holomorphic and non vanishing in the simply connected unit disk
$\overline{\Delta}.$It follows that the function

$z\longmapsto\
\frac{f\circ\phi_{-}\left(z\right)}{z^{\alpha}\prod_{i=1}^{n}\left(\frac{z-\phi_{-}^{-1}\left(a_{i}\right)}{1-\overline{\phi_{-}^{-1}\left(a_{i}\right)}z}\right)^{\alpha_{i}}};\left|z\right|\leq
1$

is well defined, holomorphic and non vanishing in $\overline{\Delta},$ and has
modulus one in $S^{1}.$ Thus,

$\frac{f\circ\phi_{-}\left(z\right)}{\prod_{i=1}^{n}\left(\frac{z-\phi_{-}^{-1}\left(a_{i}\right)}{1-\overline{\phi_{-}^{-1}\left(a_{i}\right)}z}\right)^{\alpha_{i}}}=e^{i\theta};\left|z\right|\leq
1,$

for some real $\theta.$ From (2), we get

$\phi_{+}^{-1}\circ\phi_{-}\left(z\right)=B_{\infty}\left(z\right)^{1/\alpha};\left|z\right|=1.$

In the case $a=a_{j},$ with the normalization
$\frac{\phi_{+}\left(z\right)}{z}\sim b>0,$ as $z\rightarrow\infty,$ the
functions

$z\longmapsto\left\\{\begin{array}[c]{c}\frac{\phi_{+}\left(z\right)-a_{i}}{z\frac{z-\phi_{+}^{-1}\left(a_{i}\right)}{1-\overline{\phi_{+}^{-1}\left(a_{i}\right)}z}},\text{
\emph{if} }i\neq j\\\ \frac{\phi_{+}\left(z\right)-a_{j}}{z},\text{ \emph{if}
}i=j\end{array}\right.;\left|z\right|\geq 1$

are holomorphic and non vanishing in the simply connected
$\widehat{\mathbb{C}}\setminus\overline{\Delta},$ it follows that the function

$z\longmapsto\ \frac{f\circ\phi_{+}\left(z\right)}{z^{\alpha}\prod_{i\neq
j}\left(\frac{z-\phi_{+}^{-1}\left(a_{i}\right)}{1-\overline{\phi_{+}^{-1}\left(a_{i}\right)}z}\right)^{\alpha_{i}}};\left|z\right|\geq
1$

is well defined and holomorphic in $\mathbb{C}\setminus\overline{\Delta},$
does not vanish there, is continuous in $\mathbb{C}\setminus\Delta,$ and has
modulus one on $\partial\Delta=S^{1}$. We deduce the existence of
$\theta^{\prime}\in\mathbb{R}$ such that

$f\circ\phi_{+}\left(z\right)=e^{i\theta^{\prime}}z^{\alpha}\prod_{i\neq
j}\left(\frac{z-\phi_{+}^{-1}\left(a_{i}\right)}{1-\overline{\phi_{+}^{-1}\left(a_{i}\right)}z}\right)^{\alpha_{i}};\left|z\right|\geq
1.$

Combining with (2), we get

$\phi_{-}^{-1}\circ\phi_{+}\left(z\right)=z^{\alpha/\alpha_{j}}B_{j}\left(z\right)^{1/\alpha_{j}};\left|z\right|=1.$

###  2.2 Lemniscates in a Ring Domain

In the following, let $\mathcal{U}$ be a Ring Domain of the quadratic
differential $\varpi.$ It is bounded by two sub-lemniscates $\Gamma_{r}$ and
$\Gamma_{R}.$ We may assume that

$0<r<1<R.$

Figure 3: Critical graph of
$-\left(\frac{1}{z-1}-\frac{1}{z+1}+\frac{\sqrt{2}}{z}\right)^{2}dz^{2}$ with
a lemniscate in a Ring Domain ( $a=1,b=0$ ).

Suppose that $a_{1},...,a_{s}$ $\left(1<s<n\right)$ are in the bounded domain
of $\mathbb{C}$ with boundary $\Gamma_{r}.$ We consider the sub-lemniscate
$\Gamma_{1}$ of $\varpi$ in $\mathcal{U}$ (see Figure 3).

Since the function

$z\longmapsto\frac{f\circ\phi_{-}\left(z\right)}{\prod_{i=1}^{s}\left(\frac{z-\phi_{-}^{-1}\left(a_{i}\right)}{1-\overline{\phi_{-}^{-1}\left(a_{i}\right)}z}\right)^{\alpha_{i}}}$

is holomorphic in $\Delta$, is continuous in $\overline{\Delta},$ does not
vanish in $\Delta,$ and has modulus one on $\partial\Delta,$ we deduce that
there exists $\theta_{1}\in\mathbb{R}$ such that

$\frac{f\circ\phi_{-}\left(z\right)}{\prod_{i=1}^{s}\left(\frac{z-\phi_{-}^{-1}\left(a_{i}\right)}{1-\overline{\phi_{-}^{-1}\left(a_{i}\right)}z}\right)^{\alpha_{i}}}=\vskip
12.0pt plus 4.0pt minus 4.0pte^{i\theta_{1}};\left|z\right|\leq 1,$

and then

$f\circ\phi_{-}\left(z\right)=\vskip 12.0pt plus 4.0pt minus
4.0pte^{i\theta_{1}}\prod_{i=1}^{s}\left(\frac{z-\phi_{-}^{-1}\left(a_{i}\right)}{1-\overline{\phi_{-}^{-1}\left(a_{i}\right)}z}\right)^{\alpha_{i}};\left|z\right|\leq
1.$

Reasoning like in the previous subsection on $\phi_{+}\left(z\right)$ in the
case $a=a_{j},$ we get for some $\theta_{2}\in\mathbb{R}$

$f\circ\phi_{+}\left(z\right)=\vskip 12.0pt plus 4.0pt minus
4.0pte^{i\theta_{2}}z^{\alpha}\prod_{i=s+1}^{n}\frac{z-\phi_{+}^{-1}\left(a_{i}\right)}{1-\overline{\phi_{+}^{-1}\left(a_{i}\right)}z};\left|z\right|\geq
1.$

Combining the last two equalities for $\left|z\right|=1,$ with
$\theta_{1}-\theta_{2}=\theta,$ we obtain the following

###### Theorem 5

Let $\Gamma_{1}$ be a sub-lemniscate of $\varpi$ such that $\Omega_{-}$
contains exactly two different zeros $a$ and $b$ of $p$ with respective
multiplicities $\alpha$ and $\beta.$The fingerprint $k$ $:S^{1}\longrightarrow
S^{1}$ of $\Gamma_{1}$ satisfies the functional equation

$\left(B\circ
k\right)\left(z\right)=e^{i\theta}A\left(z\right);\left|z\right|=1.$

where $A$ and $B$ are given by

$\vskip 12.0pt plus 4.0pt minus 4.0ptB\left(z\right)=\vskip 12.0pt plus 4.0pt
minus
4.0ptz^{\alpha}\prod_{i=s+1}^{n}\frac{z-\phi_{+}^{-1}\left(a_{i}\right)}{1-\overline{\phi_{+}^{-1}\left(a_{i}\right)}z},$
$A\left(z\right)=\prod_{i=1}^{s}\left(\frac{z-\phi_{-}^{-1}\left(a_{i}\right)}{1-\overline{\phi_{-}^{-1}\left(a_{i}\right)}z}\right)^{\alpha_{i}}.$

###### Acknowledgement 6

Part of this work was partially supported by the Laboratory of Mathematics and
Applications ”LR17ES11” from the Faculty of Sciences of Gabès, Tunisia.

## References

  * [1] Ebenfelt,P., Khavinson,D., and Shapiro,H.S.: Two-dimensional shapes and lemniscates, Complex analysis and dynamical systems IV. Part 1, 553 (2011), 45-59.
  * [2] Frolova A., Khavinson D., Vasil’ev A. (2018) Polynomial Lemniscates and Their Fingerprints: From Geometry to Topology. In: Agranovsky M., Golberg A., Jacobzon F., Shoikhet D., Zalcman L. (eds) Complex Analysis and Dynamical Systems. Trends in Mathematics. Birkhäuser, Cham.
  * [3] Jenkins,J.A.: Univalent functions and conformal mapping, Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge, Heft18. Reihe: Moderne Funktionentheorie, Springer-Verlag, Berlin-Gottingen-Heidelberg, 1958.
  * [4] Strebel,K.: Quadratic differentials, Vol. 5 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer-Verlag, Berlin, 1984.
  * [5] Thabet, F.: On The Existence of Finite Critical Trajectories in a Family of Quadratic Differentials. Bulletin of the Australian Mathematical Society, 94(1), 80-91.
  * [6] Younsi, M.: Shapes, Fingerprints, and Rational Functions. Proc. Amer. Math. Soc. 144 (2016), 1087-1093.
  * [7] Fingerprints, Lemniscates, and Quadratic Differentials.:Sbornik Mathematics,2019, Vol 210.

Higher Institute of Applied Sciences and Technology of Gabes,

Avenue Omar Ibn El Khattab, 6029. Tunisia.

E-mail adresses:

<EMAIL_ADDRESS>
<EMAIL_ADDRESS>


#  Boundary determination of coefficients appearing in a perturbed weighted
$p$-Laplace equation

Nitesh Kumar, Tanmay Sarkar and Manmohan Vashisth  Nitesh Kumar, Department of
Mathematics, Indian Institute of Technology Jammu, NH-44 Bypass Road, Nagrota
PO, Jagti, Jammu - 181 221, India<EMAIL_ADDRESS>Tanmay Sarkar,
Department of Mathematics, Indian Institute of Technology Jammu, NH-44 Bypass
Road, Nagrota PO, Jagti, Jammu - 181 221, India<EMAIL_ADDRESS>Manmohan Vashisth, Department of Mathematics, Indian Institute of Technology
Ropar, Rupnagar, Punjab - 140001, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We study an inverse boundary value problem associated with $p$-Laplacian which
is further perturbed by a linear second order term, defined on a bounded set
$\Omega$ in $\mathbb{R}^{n},n\geq 2$. We recover the coefficients at the
boundary from the boundary measurements which are given by the Dirichlet to
Neumann map. Our approach relies on the appropriate asymptotic expansion of
the solution and it allows one to recover the coefficients pointwise.
Furthermore, by considering the localized Dirichlet-to-Neumann map around a
boundary point, we provide a procedure to reconstruct the normal derivative of
the coefficients at that boundary point.

###### Key words and phrases:

Inverse problems, $p$-Laplace operator, Dirichlet-to-Neumann map, boundary
determination.

###### 2021 Mathematics Subject Classification:

Primary 35R30, 35J60.

##  1\. Introduction

In this paper, we consider the following weighted $p$-Laplace equation which
is perturbed by a linear term

$\begin{cases}\nabla\cdot(\sigma(x)\nabla u(x)+\gamma(x)|\nabla
u(x)|^{p-2}\nabla u(x))=0,&\quad x\in~{}\Omega,\\\ u(x)=f(x),&\quad
x\in\partial\Omega,\end{cases}$ (1.1)

where the domain $\Omega\subseteq\mathbb{R}^{n},n\geq 2$ is a bounded open set
with smooth boundary $\partial\Omega$. Throughout the paper, we assume that
the exponent $p$ satisfies $1<p<\infty$. Furthermore, we consider the
coefficients $\sigma$ and $\gamma$ are positive functions with the regularity
$\sigma\in C^{\infty}(\overline{\Omega})$ and $\gamma\in L^{\infty}(\Omega)$
respectively. Moreover, the coefficients $\sigma$ and $\gamma$ are assumed to
satisfy the following conditions:

$\begin{cases}0<\lambda<\sigma(x)<\lambda^{-1},\\\
0<m_{1}<\gamma(x)<m_{1}^{-1},\end{cases}$

where $\lambda$, and $m_{1}$ are positive constants. The Dirichlet data $f$ is
considered as follows: $f\in W^{1/2,2}(\partial\Omega)$ whenever $p\in(1,2)$,
and $f\in W^{1-1/p,p}(\partial\Omega)$ whenever $p\in(2,\infty)$.

The nonlinear Dirichlet to Neumann map (DN-map) is defined by

$\Lambda_{\sigma,\gamma}(f)=(\sigma(x)+\gamma(x)|\nabla
u|^{p-2})\partial_{\nu}u|_{\partial\Omega},$ (1.2)

where $u$ is the unique solution of equation (1.1) with the boundary value
$f$. Throughout the paper, we assume that $\nu$ is the outer unit normal to
$\partial\Omega$.

In case of $p=2$, the equation(1.1) becomes a linear conductivity equation and
there are several results related to boundary determination of conductivity
coefficients appearing in the linear equation. For instance, in [15, 26] the
smooth conductivity and all its derivatives at the boundary are recovered from
the boundary measurements of the solution. The authors in [15, 26] proved the
uniqueness of conductivity together with its derivatives at the boundary of
domain. Later Nachman in [20] recovered the coefficient at the boundary by
assuming that $\gamma\in W^{1,m}(\Omega),~{}m>n$. Under the assumption
$m>n/2$, Nachman in [20] also recovered the first normal derivative of
$\gamma\in W^{2,m}(\Omega)$. Furthermore, Brown [3] described a procedure for
recovering $\gamma|_{\partial\Omega}$ in a pointwise manner by assuming
$\gamma\in L^{\infty}(\Omega)$. For the conductivity equation, Nakamura and
Tanuma [21, 22] provided a formula for reconstructing the conductivity and its
higher order derivatives from the localized DN-map. Moreover, there are
several local boundary determination results, for instance, one can refer to
[1, 13, 23] and references therein.

Motivated by the applications in image processing, fluid mechanics and
modeling of sand-piles, Calderón-type inverse problems for the $p$-Laplace
equation are studied by several authors. For instance, Salo and Zhong [24]
studied the boundary uniqueness problem associated with $p$-Laplace equation
for $n\geq 2$. Brander in [2] recovered the gradient of $\gamma$ at the
boundary using the Rellich’s type identity. Furthermore, Cârstea and Kar in[4]
considered the weighted perturbed $p$-Laplace equation and provided a
procedure for reconstruction of the coefficients in the interior of the domain
using the complex geometric optics solution for $n\geq 3$. However, up to our
knowledge, the boundary determination problem for weighted perturbed
$p$-Laplace equation concerning the coefficient $\gamma$ has not been studied
in the literature.

In this paper, we will give a procedure for recovery of the coefficients at
boundary of the domain. To do this, we use the $\epsilon$-expansion of
solution which varies according to the choice of $p$. The approach of using
the $\epsilon$-expansion or linearization for solving the inverse problems for
nonlinear elliptic equations have been used by several authors, see for
example [4, 5, 6, 7, 9, 10, 11, 12, 14, 16, 17, 19]. Motivated by [4, 5, 12]
our idea is to find an appropriate integral identity, from which
reconstruction of the coefficient $\gamma$ at the boundary of the domain is
possible. Using the approach used in [5] we derived the integral identity by
constructing a sequence of explicit functions which are solutions of
conductivity equation. Such solutions can be used to extract the Taylor series
of the coefficient $\gamma$ at the boundary point from the knowledge of
$\Lambda_{\sigma,\gamma}$.

We can now state our boundary determination results.

######  Theorem 1.1.

(Reconstruction of coefficients)
Suppose $p>1$ with $p\neq 2$ and $\Omega\subseteq\mathbb{R}^{n},n\geq 2$ is a
bounded domain with smooth boundary $\partial\Omega$. Furthermore, assume that
the coefficients are continuous on $\overline{\Omega}$ and bounded below by a
positive constant. Then the coefficients $\sigma$ and $\gamma$ in (1.1) can be
reconstructed at the boundary from the knowledge of given DN-map
$\Lambda_{\sigma,\gamma}$.

######  Theorem 1.2.

(Reconstruction of normal derivatives of coefficients at the boundary)
Assume that $p\in(1,\infty)\backslash\\{2\\}$ and for $n\geq 2$,
$\Omega\subseteq\mathbb{R}^{n}$ is a bounded domain with smooth boundary
$\partial\Omega$. Suppose $\sigma\in C^{\infty}(\overline{\Omega})$ and
$D^{\alpha^{\prime}}_{x^{\prime}}D^{\alpha_{n}}_{x_{n}}\gamma$ is continuous
around $x_{0}$ for any multi-index $(\alpha^{\prime},\alpha_{n})$ such that
$|\alpha^{\prime}|+2\alpha_{n}\leq 2$, where
$x=(x^{\prime},x_{n}),~{}x^{\prime}\in\mathbb{R}^{n-1}$. Then the normal
derivative of the coefficients can be evaluated at the boundary point of
$\Omega$ from the knowledge of given DN-map $\Lambda_{\sigma,\gamma}$
associated to (1.1).

The paper is organized as follows: we state the well-posedness results of
(1.1) along with corresponding $\epsilon-$expansion of the solution in section
2 which further includes the epsilon expansion of Dirichlet to Neumann map. In
section 3, we provide a detailed proof for the reconstruction of the
coefficient at the boundary of the domain (cf. Theorem 1.1). We construct a
few approximate solutions using the localized Dirichlet data to extract the
normal derivative of $\gamma$ and obtain a reconstruction formula in Section
4.

##  2\. $\varepsilon$-expansion of the solution for perturbed $p$-Laplacian
problem

In this section, we state the existence of the solution and corresponding
energy estimates for the boundary value problem (BVP) (1.1). Based on the
range of $p$, we have weak solution and strong solution of the BVP (1.1)
corresponding to sufficiently small boundary data. We remark that such
boundary data are considered since we require the resulting estimates while
analyzing the boundary determination problem. More precisely, we shall use the
weak solution and strong solution of BVP 1.1 while reconstructing the scalar
coefficient $\gamma$ at the boundary. The following Proposition 2.1 can be
proved by defining suitable energy functionals and subsequently, considering
the minimizing sequence (for details, we refer to [5, Section 2]).

######  Proposition 2.1 (Weak solutions).

  1. (i)

Assume that $1<p<2$ and $f\in W^{1/2,2}(\partial\Omega)$. Then the BVP (1.1)
has a unique solution

$u\in W^{1/2,2}(\Omega)\cap W^{1,p}(\Omega),$

and it satisfies the following estimate

$\|u\|_{W^{1,2}(\Omega)}\leq C\|f\|_{W^{1/2,2}(\partial\Omega)},$

where positive constant $C$ does not depend on the boundary data $f$.

  2. (ii)

Assume that $2<p<\infty$ and $f\in W^{1-{1/p},p}(\partial\Omega)$. Then the
BVP (1.1) has a unique solution

$u\in W^{1/2,2}(\Omega)\cap W^{1,p}(\Omega),$

along with the stability estimate

$\|u\|_{W^{1,p}(\Omega)}\leq C\|f\|_{W^{1-{1/p},p}(\partial\Omega)},$

where $C=C(\Omega)>0$.

Proposition 2.1 can be further improved in case of $p>2$. In fact, there
exists a strong solution for the BVP (1.1). By defining an appropriate
operator and using the Sobolev embedding results and finally, demonstrating
the fixed point of contraction mapping, the existence of the strong solution
can be established. For detailed proof of the following Proposition 2.2, we
refer to [5, Theorem 2.2].

######  Proposition 2.2 (Strong solution).

Assume that $p>2$ and $m\in\mathbb{R}$ such that $m>n$. Furthermore, let $f\in
W^{2-{1/m},m}(\partial\Omega)$ and there holds

$\displaystyle\|f\|_{W^{2-{1/m},m}(\partial\Omega)}\leq M,$

for some $M>0$. Then there exists a unique solution for BVP (1.1)

$u\in W^{2,m}(\Omega).$

Moreover, the solution $u$ satisfies

$\|u\|_{W^{2,m}(\Omega)}\leq C\|f\|_{W^{2-{1/m},m}(\partial\Omega)},$ (2.1)

where constant $C>0$ is depending on the choice of $p$, domain $\Omega$ and
coefficients $\sigma$, and $\gamma$.

Based on the well-posedness results, described in Proposition 2.1 and
Proposition 2.2, we define the nonlinear DN-map in a weak sense as follows:

$\displaystyle\langle\Lambda_{\sigma,\gamma}(f),w|_{\partial\Omega}\rangle=\int_{\Omega}(\sigma(x)+\gamma(x)|\nabla
u|^{p-2})\nabla u\cdot\overline{\nabla w}~{}dx,$ (2.2)

where $w\in W^{2,m}(\Omega)\cap W^{1,2}(\Omega)$. Afterwards, we look into the
$\varepsilon$-expansion of the solution for the BVP (1.1). Since the expansion
depends on the range of $p$, we carry out the analysis via two cases.

###  2.1. Case-1: $p>2$

We look for $\varepsilon$-expansion of the solution of (1.1) for $p>2$ based
on the existence of strong solution mentioned in Proposition 2.2.

Let us consider the following BVP

$\displaystyle\begin{cases}\nabla\cdot(\sigma(x)\nabla
u_{\varepsilon}(x)+\gamma(x)|\nabla u_{\varepsilon}(x)|^{p-2}\nabla
u_{\varepsilon}(x))=0,&\quad x\in~{}\Omega,\\\ u_{\varepsilon}(x)=\varepsilon
f(x),&\quad x\in\partial\Omega.\\\ \end{cases}$ (2.3)

From the stability estimate (2.1), we obtain

$\displaystyle\|\varepsilon^{-1}u_{\varepsilon}\|_{W^{2,m}(\Omega)}\leq
C\|f\|_{W^{2-1/m,m}(\partial\Omega)}.$ (2.4)

As a consequence, we consider the following $\varepsilon$-expansion

$\displaystyle u_{\varepsilon}=\varepsilon u_{0}+\varepsilon^{p-1}v,$ (2.5)

where $u_{0}\in W^{2,m}(\Omega)$ and $v\in W^{2,m}(\Omega)$ will be determined
by substituting (2.5) in the BVP (2.3). We obtain that $u_{0}$ satisfies the
following BVP

$\displaystyle\begin{cases}\nabla\cdot(\sigma(x)\nabla u_{0}(x))=0,&\quad
x\in\Omega,\\\ u_{0}(x)=f(x),&\quad x\in\partial\Omega,\end{cases}$ (2.6)

and eventually, $v$ satisfies the BVP

$\displaystyle\begin{cases}\nabla\cdot(\sigma(x)\nabla
v)+\varepsilon^{1-p}\nabla\cdot(\gamma(x)|\nabla u_{\varepsilon}|^{p-2}\nabla
u_{\varepsilon})=0,&\quad x\in~{}\Omega,\\\ v(x)=0&\quad
x\in\partial\Omega.\\\ \end{cases}$ (2.7)

###  2.2. Case-2: $1<p<2$.

For $p\in(1,2)$, $\varepsilon$-expansion of the solution of BVP (1.1) is a
consequence of existence of a weak solution of (1.1). Since $p\in(1,2)$, then
we get a sublinear term instead of a nonlinear term. Therefore, for arbitrary
small $\varepsilon>0$, we will take the Dirichlet boundary data as
$\varepsilon^{-1}f$ in (1.1).

Let us consider

$\displaystyle\begin{cases}\nabla\cdot(\sigma(x)\nabla
u_{\varepsilon}+\gamma(x)|\nabla u_{\varepsilon}|^{p-2}\nabla
u_{\varepsilon})=0,&\quad x\in~{}\Omega,\\\
u_{\varepsilon}(x)=\varepsilon^{-1}f(x),&\quad x\in\partial\Omega.\\\
\end{cases}$ (2.8)

The well-posedness of above BVP (2.8) is assured by the Proposition 2.1 along
with the stability estimate

$\|\varepsilon u_{\varepsilon}\|_{W^{1,2}(\Omega)}\leq
C\|f\|_{W^{1/2,2}(\partial\Omega)}.$ (2.9)

By taking into account the estimate (2.9), we consider the following
expansion:

$\displaystyle u_{\varepsilon}=\varepsilon^{-1}u_{0}+\varepsilon^{1-p}v,$
(2.10)

where $u_{0}\in W^{1,2}(\Omega)$ and $v\in W^{1,2}(\Omega)$ will be determined
by substituting (2.10) in the BVP (2.8). It is straightforward to observe that
$u_{0}$ satisfies the following BVP

$\displaystyle\begin{cases}\nabla\cdot(\sigma\nabla u_{0})=0,&\quad
x\in\Omega,\\\ u_{0}(x)=f(x),&\quad x\in\partial\Omega,\end{cases}$ (2.11)

and $v$ satisfies the BVP

$\displaystyle\begin{cases}\nabla\cdot(\sigma(x)\nabla
v)+\varepsilon^{p-1}\nabla\cdot(\gamma(x)|\nabla u_{\varepsilon}|^{p-2}\nabla
u_{\varepsilon})=0,&\quad x\in~{}\Omega,\\\ v(x)=0,&\quad
x\in\partial\Omega.\\\ \end{cases}$ (2.12)

###  2.3. $\varepsilon$-expansion of DN-map

In order to recover the parameter $\gamma$ at the boundary, we require the
$\varepsilon$-expansion of the DN-map described in (2.2). We carry out the
following analysis motivated by Cârstea et al. [5] in which the reconstruction
of $\gamma$ in the interior of domain in $\mathbb{R}^{n}$ $(n\geq 3)$ is
described. The following estimate will be instrumental for our analysis: for
$p>2$,

$\displaystyle\big{|}|\eta|^{p-2}\eta-|\beta|^{p-2}\beta\big{|}\leq
C(|\eta|+|\beta|)^{p-2}|\eta-\beta|,$ (2.13)

where $C>0$ and $\eta,\beta\in\mathbb{C}^{n}$ (for detailed proof, one can
look into [24] ). Let us consider $w\in W^{2,m}(\Omega)\cap W^{1,2}(\Omega)$.
Using the estimate (2.13), we obtain

$\begin{split}\Big{|}\int_{\Omega}&\gamma(x)(|\nabla
u_{\varepsilon}|^{p-2}\nabla u_{\varepsilon}-\varepsilon^{p-1}|\nabla
u_{0}|^{p-2}\nabla u_{0})\cdot\overline{\nabla w}~{}dx\Big{|}\\\ &\leq
C\varepsilon^{2p-3}\int_{\Omega}(|\varepsilon^{-1}\nabla
u_{\varepsilon}|^{p-2}+|\nabla u_{0}|^{p-2})|\nabla v||\nabla
w|~{}dx=\mathcal{O}(\varepsilon^{2p-3}).\end{split}$ (2.14)

With the help of expansion (2.5) and above estimate (2.14), we get

$\displaystyle\langle\Lambda_{\sigma,\gamma}(\varepsilon
f),w\big{|}_{\partial\Omega}\rangle$
$\displaystyle=\int_{\Omega}(\sigma(x)+\gamma(x)|\nabla
u_{\varepsilon}|^{p-2})\nabla u_{\varepsilon}\cdot\overline{\nabla w}~{}dx$
$\displaystyle=\varepsilon\int_{\Omega}\sigma(x)\nabla
u_{0}\cdot\overline{\nabla
w}~{}dx~{}+~{}\varepsilon^{p-1}\int_{\Omega}\sigma(x)\nabla
v\cdot\overline{\nabla w}~{}dx$
$\displaystyle\qquad+\varepsilon^{p-1}\int_{\Omega}\gamma(x)|\nabla
u_{0}|^{p-2}\nabla u_{0}\cdot\overline{\nabla
w}~{}dx~{}+~{}\mathcal{O}(\varepsilon^{2p-3}).$

Subsequently, by considering the limit $\varepsilon$ tending to zero, we have

$\begin{split}\lim_{\varepsilon\rightarrow
0}\frac{1}{\varepsilon}\langle\Lambda_{\sigma,\gamma}(\varepsilon
f),w\big{|}_{\partial\Omega}\rangle&=\int_{\Omega}\sigma(x)\nabla
u_{0}\cdot\overline{\nabla w}~{}dx\\\
&=\langle\Lambda_{\sigma}(f),w\big{|}_{\partial\Omega}\rangle,\end{split}$
(2.15)

where $\Lambda_{\sigma}(f)$ is the DN-map associated to the Dirichlet BVP
(2.6).

For $p\in(1,2)$, in order to recover the parameter $\gamma$ at the boundary,
we need to consider the $\varepsilon$-expansion of the DN-map using
$u_{\varepsilon}$ described in (2.10). As in the earlier case, we start with
$w\in W^{1,2}(\Omega)$. Using the estimate (2.13), we obtain

$\begin{split}\Big{|}\int_{\Omega}&\gamma(x)(|\nabla
u_{\varepsilon}|^{p-2}\nabla u_{\varepsilon}-\varepsilon^{1-p}|\nabla
u_{0}|^{p-2}\nabla u_{0})\cdot\overline{\nabla w}~{}dx\Big{|}\\\ &\leq
C\varepsilon^{3-2p}\int_{\Omega}(|\varepsilon\nabla
u_{\varepsilon}|^{p-2}+|\nabla u_{0}|^{p-2})|\nabla v||\nabla
w|~{}dx=\mathcal{O}(\varepsilon^{3-2p}).\end{split}$ (2.16)

With the help of expansion (2.10) and above estimate (2.16), we get

$\displaystyle\langle\Lambda_{\sigma,\gamma}(\varepsilon^{-1}f),w\big{|}_{\partial\Omega}\rangle$
$\displaystyle=\int_{\Omega}(\sigma(x)+\gamma(x)|\nabla
u_{\varepsilon}|^{p-2})\nabla u_{\varepsilon}\cdot\overline{\nabla w}~{}dx$
$\displaystyle=\varepsilon^{-1}\int_{\Omega}\sigma(x)\nabla
u_{0}\cdot\overline{\nabla
w}~{}dx~{}+~{}\varepsilon^{1-p}\int_{\Omega}\sigma(x)\nabla
v\cdot\overline{\nabla w}~{}dx$
$\displaystyle\qquad+\varepsilon^{1-p}\int_{\Omega}\gamma(x)|\nabla
u_{0}|^{p-2}\nabla u_{0}\cdot\overline{\nabla
w}~{}dx~{}+~{}\mathcal{O}(\varepsilon^{3-2p}).$

By considering the DN-map $\Lambda_{\sigma}(f)$ associated with the Dirichlet
BVP (2.11), we have

$\begin{split}\lim_{\varepsilon\rightarrow
0}\varepsilon\langle\Lambda_{\sigma,\gamma}(\varepsilon^{-1}f),w\big{|}_{\partial\Omega}\rangle&=\int_{\Omega}\sigma(x)\nabla
u_{0}\cdot\overline{\nabla w}~{}dx\\\
&=\langle\Lambda_{\sigma}(f),w\big{|}_{\partial\Omega}\rangle.\end{split}$
(2.17)

Hence we conclude that the DN-map $\Lambda_{\sigma}(f)$ is known from the
knowledge of given DN-map $\Lambda_{\sigma,\gamma}$, for all values of $p$
under consideration.

##  3\. Proof for the Theorem 1.1

In this section, we provide the reconstruction procedure of the coefficients
$\sigma$, and $\gamma$. In order to prove the Theorem 1.1, we shall consider
two separate cases based on the range of $p$.

###  3.1. Reconstruction of $\sigma$

For the reconstruction of $\sigma$ from the knowledge of the DN-map
$\Lambda_{\sigma}(f)$ associated to BVP (2.6), one can refer to [20, Theorem
5.1].

###  3.2. Reconstruction of $\gamma$

Using the $\varepsilon$-expansion of the DN-map, we will give a procedure for
the recovery of the parameter $\gamma$ at the boundary $\partial\Omega$. For
this, our idea is to find a suitable integral identity which is known from the
knowledge of the DN-map. By the asymptotic expansion of the solution along
with appropriate choice of sequence of solutions of conductivity equation, the
desired reconstruction of $\gamma$ at the boundary of $\Omega$ is carried out.

####  3.2.1. Case: $p>2$

Let us consider $w\in{W}^{2,m}(\Omega)$ such that it satisfies

$\displaystyle\nabla\cdot(\sigma\nabla w)=0\quad\text{ on }\Omega.$ (3.1)

By multiplying equation (2.3) with $w$ and integrating the resulting equation
over $\Omega$, we get

$\int_{\Omega}w(\nabla\cdot(\sigma\nabla u_{\varepsilon}+\gamma|\nabla
u_{\varepsilon}|^{p-2}\nabla u_{\varepsilon}))~{}~{}dx=0$

and it further yields the following equation by using the integration by parts

$\int_{\Omega}\nabla w\cdot(\sigma\nabla u_{\varepsilon}+\gamma|\nabla
u_{\varepsilon}|^{p-2}\nabla
u_{\varepsilon})~{}dx=\int_{\partial\Omega}w(\sigma+\gamma|\nabla
u_{\varepsilon}|^{p-2})\partial_{\nu}u_{\varepsilon}~{}dS.$

Using (3.1), the above expression becomes

$\int_{\Omega}\nabla w\cdot(\gamma|\nabla u_{\varepsilon}|^{p-2}\nabla
u_{\varepsilon})~{}dx=\int_{\partial\Omega}w(\sigma+\gamma|\nabla
u_{\varepsilon}|^{p-2})\partial_{\nu}u_{\varepsilon}~{}dS-\varepsilon\langle\Lambda_{\sigma}(f),w|_{\partial\Omega}\rangle.$
(3.2)

We observe that the right hand side terms of (3.2) are known by the prescribed
DN-maps $\Lambda_{\sigma,\gamma}$ and $\Lambda_{\sigma}$ respectively. As a
result, the left hand side of (3.2) is known to us. Let us define

$\displaystyle K^{*}$ $\displaystyle:=\int_{\Omega}\nabla w\cdot(\gamma|\nabla
u_{\varepsilon}|^{p-2}\nabla u_{\varepsilon})~{}dx.$

The term $K^{*}$ is known from the knowledge of the DN-map associated with
perturbed $p$-Laplacian problem.

We can further rewrite $K^{*}$ as

$\displaystyle K^{*}=\int_{\Omega}\nabla w\cdot(\gamma|\nabla
u_{\varepsilon}|^{p-2}\nabla u_{\varepsilon})~{}dx$
$\displaystyle-\varepsilon^{p-1}~{}\int_{\Omega}\nabla w\cdot(\gamma|\nabla
u_{0}|^{p-2}\nabla u_{0})~{}dx$
$\displaystyle+\varepsilon^{p-1}\int_{\Omega}\nabla w\cdot(\gamma|\nabla
u_{0}|^{p-2}\nabla u_{0})~{}dx.$

By taking into account the estimate (2.14), we deduce that

$\displaystyle K^{*}$ $\displaystyle=\varepsilon^{p-1}\int_{\Omega}\nabla
w\cdot(\gamma|\nabla u_{0}|^{p-2}\nabla
u_{0})~{}dx+\mathcal{O}(\varepsilon^{2p-3}).$

Further, we define $I(u_{0},w)$ as follows:

$\displaystyle I(u_{0},w):=\int_{\Omega}\nabla w\cdot(\gamma|\nabla
u_{0}|^{p-2}\nabla u_{0})~{}dx.$ (3.3)

As a consequence, we have

$\displaystyle
K^{*}=\varepsilon^{p-1}I(u_{0},w)+\mathcal{O}(\varepsilon^{2p-3}),$ (3.4)

where $w\in W^{2,m}(\Omega)$ and $u_{0}$ satisfies the BVP (2.6). Furthermore,
we observe that

$\displaystyle I(u_{0},w)=\lim_{\varepsilon\rightarrow
0^{+}}\varepsilon^{1-p}\Big{(}K^{*}-\mathcal{O}(\varepsilon^{2p-3})\Big{)}$

and subsequently, $I(u_{0},w)$ is known to us.

Let us choose the functions $u_{1},u_{2},u_{3}\in{C}^{\infty}(\Omega)$ such
that

$\displaystyle\nabla\cdot(\sigma\nabla u_{j})=0,\quad j=1,2,3.$ (3.5)

In addition, we assume that $u_{1}$ is real valued and $\nabla u_{1}$ does not
have any zeros. Since $I(u_{0},w)$ is known for any $w\in W^{2,m}(\Omega)$. In
particular $I(u_{0},u_{3})$ is known. By defining $J_{1}$ and $J_{2}$ as
follows:

$\displaystyle J_{1}(u_{1},u_{2},u_{3}):$
$\displaystyle=\frac{2}{p-2}\partial_{\bar{z}}[I(u_{1}+z\overline{u_{2}},u_{3})]\Big{|}_{z=0}$
$\displaystyle=\int_{\Omega}\gamma|\nabla u_{1}|^{p-4}(\nabla u_{1}\cdot\nabla
u_{2})(\nabla u_{1}\cdot\nabla u_{3})~{}dx,$ $\displaystyle
J_{2}(u_{1},u_{2},u_{3}):$
$\displaystyle=\frac{2}{p-2}\partial_{z}[I(u_{1}+zu_{2},u_{3})]\Big{|}_{z=0}$
$\displaystyle=\int_{\Omega}\gamma|\nabla u_{1}|^{p-4}[(\overline{\nabla
u_{1}}\cdot\nabla u_{2})(\nabla u_{1}\cdot\nabla u_{3})+\frac{2}{p-2}|\nabla
u_{1}|^{2}(\nabla u_{2}\cdot\nabla u_{3})]~{}dx,$

we conclude that the functionals $J_{1}(u_{1},u_{2},u_{3})$ and
$J_{2}(u_{1},u_{2},u_{3})$ will be known to us. Furthermore, we define

$\displaystyle\beta_{u_{1}}(x):=\gamma|\nabla u_{1}(x)|^{p-2}$ (3.6)

and

$\displaystyle
K(u_{1},u_{2},u_{3}):=\frac{p-2}{2}[J_{2}(u_{1},u_{2},u_{3})-J_{1}(u_{1},u_{2},u_{3})].$

Hence $K$ can be represented as

$K(u_{1},u_{2},u_{3})=\int_{\Omega}\beta_{u_{1}}(\nabla u_{2}\cdot\nabla
u_{3})~{}dx.$

For each choice of $u_{2}$ and $u_{3}$ satisfying (3.5), the term
$K(u_{1},u_{2},u_{3})$ is known. Since, $\sigma$ is real valued, therefore by
choosing $u_{3}=\overline{u_{2}}$, we find that

$K(u_{1},u_{2},\overline{u_{2}})=\int_{\Omega}\beta_{u_{1}}\nabla
u_{2}\cdot\overline{\nabla u_{2}}~{}dx,$ (3.7)

is known to us.

Let $x_{0}\in\partial\Omega$ be arbitrary, but fixed. From the knowledge of
$K(u_{1},u_{2},\overline{u_{2}})$ in (3.7), we will recover $\beta_{u_{1}}$ at
$x_{0}$. By using (3.6), we recover $\gamma(x_{0})$ from
$\beta_{u_{1}}(x_{0})$. Following the similar approach presented in [24, 3],
we can construct a sequence of solutions $\\{u_{M}\\}$ satisfying the
conductivity equation (3.5) with coefficient frozen at the boundary point
$x_{0}$. Suppose $\rho$ is a $C^{1}$ function and we choose $\zeta$ as a unit
tangent vector to $\partial\Omega$ at $x_{0}$ such that

$\displaystyle\zeta\cdot\nabla\rho(x_{0})=0,\qquad|\zeta|=|\nabla\rho(x_{0})|.$
(3.8)

Furthermore, we choose a cut-off function $\eta\in C^{\infty}(\mathbb{R}^{n})$
with $0\leq\eta(x-x_{0})\leq 1$ such that

$\displaystyle\eta(x-x_{0})=\begin{cases}1,&\quad|x-x_{0}|\leq\frac{1}{2},\\\
0,&\quad|x-x_{0}|\geq 1.\end{cases}$

For large positive numbers $M$ and $N$, we define $\eta_{M}(x)=\eta(Mx)$ and
$N=N(M)$ with
$\quad\frac{M}{N}=\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.7}{$\scriptscriptstyle\mathcal{O}$}}(1)$
as $M\rightarrow\infty$. Subsequently, we can deduce the following:

$\displaystyle
M^{n-1}N\int_{\Omega}\eta(Mx)\exp({-2N\rho(x)})~{}dx\longrightarrow\frac{1}{2}\int_{\mathbb{R}^{n-1}}\eta(x^{\prime},x_{0})~{}dx^{\prime}\quad\text{
as }M\rightarrow\infty$ (3.9)

since $\eta$ is supported in the ball $B(x_{0},1)$.

For each $M\in\mathbb{N}$ and $M\geq 2$, we define a solution of (3.5) as

$\displaystyle u_{M}:=C_{M,N}(w_{0}+w_{1}),$ (3.10)

where $C_{M,N}$ is a scaling constant and it is described by

$C_{M,N}=\sqrt{\frac{M^{n-1}N^{-1}}{c_{\eta}}}$

in which $c_{\eta}$ is given by

$\qquad
c_{\eta}:=\frac{1}{2}\int_{\mathbb{R}^{n-1}}\eta(x^{\prime},x_{0})~{}dx^{\prime}.$

In (3.10), we consider $w_{0}$ as an approximate solution of (3.5) and it
assumes the following form

$w_{0}(x)=\eta(M(x-x_{0}))\exp\big{(}{N(i\zeta\cdot x-\rho(x))}\big{)},\qquad
x\in\mathbb{R}^{n}.$

The above form of $w_{0}$ is justified thanks to (3.8). As a consequence, we
observe that $w_{1}$ satisfies

$\displaystyle\begin{cases}\nabla\cdot(\sigma(x)\nabla
w_{1}(x))=-\nabla\cdot(\sigma(x)\nabla w_{0}(x)),&\quad x\in\Omega,\\\
w_{1}(x)=0,&\quad x\in\partial\Omega,\end{cases}$

in addition with [24]

$\displaystyle\int_{\Omega}|\nabla
w_{0}|^{2}~{}dx=\mathcal{O}(M^{1-n}N),\qquad\int_{\Omega}|\nabla
w_{1}|^{2}~{}dx=\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.7}{$\scriptscriptstyle\mathcal{O}$}}(M^{1-n}N)\quad\text{
as }M\rightarrow\infty.$ (3.11)

Note that $w_{0}\in C^{1}_{c}(\mathbb{R}^{n})$ is vanishing outside a small
ball $B(x_{0},1/M)$. For this choice of sequence $\\{u_{M}\\}$ in (3.10) we
get

$\displaystyle\int_{\Omega}\beta_{u_{1}}\nabla u_{M}\cdot\overline{\nabla
u_{M}}~{}dx=\frac{M^{n-1}N^{-1}}{c_{\eta}}\int_{\Omega}\beta_{u_{1}}(|\nabla
w_{0}|^{2}+\nabla w_{0}\cdot\overline{\nabla w_{1}}+\overline{\nabla
w_{0}}\cdot\nabla w_{1}+|\nabla w_{1}|^{2})~{}dx.$ (3.12)

We observe that

$\displaystyle\bigg{|}\int_{\Omega}\beta_{u_{1}}\big{(}\nabla
w_{0}\cdot\overline{\nabla w_{1}}$ $\displaystyle+\overline{\nabla
w_{0}}\cdot\nabla w_{1}+|\nabla w_{1}|^{2}\big{)}~{}dx\bigg{|}$
$\displaystyle\leq C\big{(}\|\nabla w_{0}\|_{L^{2}(\Omega)}+\|\nabla
w_{1}\|_{L^{2}(\Omega)}\big{)}\|\nabla
w_{1}\|_{L^{2}(\Omega)}=\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.7}{$\scriptscriptstyle\mathcal{O}$}}(M^{1-n}N),$

where we have used the Cauchy-Schwarz inequality and estimates (3.11).

For the first term on right hand side of (3.12), we have

$\displaystyle\nabla w_{0}$
$\displaystyle=N(i\zeta-\nabla\rho)\eta_{M}\exp\big{(}{N(i\zeta\cdot
x-\rho(x))}\big{)}+M(\nabla\eta)(M\cdot)\exp\big{(}{N(i\zeta\cdot
x-\rho(x))}\big{)}$ $\displaystyle=:T_{1}+T_{2},$

and consequently,

$\displaystyle\int_{\Omega}\beta_{u_{1}}(x)|\nabla w_{0}|^{2}~{}dx$
$\displaystyle=\beta_{u_{1}}(x_{0})\int_{\Omega}|\nabla
w_{0}|^{2}~{}dx+\int_{\Omega}\big{(}\beta_{u_{1}}(x)-\beta_{u_{1}}(x_{0})\big{)}|\nabla
w_{0}|^{2}~{}dx$
$\displaystyle=\beta_{u_{1}}(x_{0})\int_{\Omega}|T_{1}|^{2}~{}dx+\beta_{u_{1}}(x_{0})\int_{\Omega}(T_{1}\cdot\overline{T_{2}}+\overline{T_{1}}\cdot
T_{2}+|T_{2}|^{2})~{}dx$
$\displaystyle\qquad+\int_{\Omega}\big{(}\beta_{u_{1}}(x)-\beta_{u_{1}}(x_{0})\big{)}|\nabla
w_{0}|^{2}~{}dx$
$\displaystyle=\beta_{u_{1}}(x_{0})\int_{\Omega}|T_{1}|^{2}~{}dx+\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.7}{$\scriptscriptstyle\mathcal{O}$}}(M^{1-n}N),$

where we have used the fact that
$\|T_{1}\|^{2}_{L^{2}(\Omega)}=\mathcal{O}(M^{1-n}N)$, and
$\|T_{2}\|^{2}_{L^{2}(\Omega)}=\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.7}{$\scriptscriptstyle\mathcal{O}$}}(M^{1-n}N)$
as $M\rightarrow\infty$, and the continuity of $\beta_{u_{1}}$ at $x_{0}$.
Moreover, we have

$\displaystyle M^{n-1}N^{-1}\int_{\Omega}|T_{1}|^{2}~{}dx$
$\displaystyle=M^{1-n}N\int_{\Omega}\eta(M(x-x_{0}))^{2}\exp({-2N\rho})(1+|\nabla\rho|^{2})~{}dx$
$\displaystyle=M^{1-n}N\bigg{(}2\int_{\Omega}\eta(M(x-x_{0}))^{2}\exp({-2n\rho})~{}dx$
$\displaystyle\qquad+\int_{\Omega}\eta(M(x-x_{0}))^{2}\exp({-2N\rho})(|\nabla\rho|^{2}-|\nabla\rho(x_{0})|^{2})~{}dx\bigg{)},$

where the first term and second term in the right hand side converges to
$c_{\eta}$ (due to (3.9)) and
$\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.7}{$\scriptscriptstyle\mathcal{O}$}}(1)$
(since $x\mapsto|\nabla\rho|^{2}$ is continuous at $x_{0}$) respectively as
$M\rightarrow\infty$. Hence we obtain

$\displaystyle\frac{M^{n-1}N^{-1}}{c_{\eta}}\int_{\Omega}\beta_{u_{1}}(x)|\nabla
w_{0}|^{2}~{}dx\longrightarrow\beta_{u_{1}}(x_{0})\quad\text{ as
}M\rightarrow\infty.$ (3.13)

As a consequence, we conclude that from (3.12)

$\displaystyle\int_{\Omega}\beta_{u_{1}}(x)\nabla u_{M}\cdot\overline{\nabla
u_{M}}~{}dx\longrightarrow\beta_{u_{1}}(x_{0})\quad\text{ as
}M\rightarrow\infty.$

In (3.7) replacing $u_{2}$ by $u_{M}$, we are able to find
$\beta_{u_{1}}(x_{0})$. Hence $\gamma(x_{0})$ is known from (3.6). Since
$x_{0}\in\partial\Omega$ is arbitrary, one can recover $\gamma$ at the
boundary of the domain.

####  3.2.2. Case: $1<p<2$

Let $w$ be any function such that $w\in{W}^{1,2}(\Omega)$ and it satisfies

$\displaystyle\nabla\cdot(\sigma\nabla w)=0\quad\text{ in }\Omega.$ (3.14)

We multiply the equation (2.8) by $w$ and integrate the resulting equation
over $\Omega$ to obtain

$\int_{\Omega}w\Big{(}\nabla\cdot(\sigma\nabla u_{\varepsilon}+\gamma|\nabla
u_{\varepsilon}|^{p-2}\nabla u_{\varepsilon})\Big{)}~{}~{}dx=0.$ (3.15)

The above equation reduces to

$\int_{\Omega}\nabla w\cdot(\sigma\nabla u_{\varepsilon}+\gamma|\nabla
u_{\varepsilon}|^{p-2}\nabla
u_{\varepsilon})~{}dx=\int_{\partial\Omega}w(\sigma+\gamma|\nabla
u_{\varepsilon}|^{p-2})\partial_{\nu}u_{\varepsilon}~{}dS,$

where we have applied integration by parts. With the help of (3.14) we get

$\int_{\Omega}\nabla w\cdot(\gamma|\nabla u_{\varepsilon}|^{p-2}\nabla
u_{\varepsilon})~{}dx=\int_{\partial\Omega}w(\sigma+\gamma|\nabla
u_{\varepsilon}|^{p-2})\partial_{\nu}u_{\varepsilon}~{}dS-\varepsilon^{-1}\langle\Lambda_{\sigma}(f),w|_{\partial\Omega}\rangle.$
(3.16)

Since the DN-maps $\Lambda_{\sigma,\gamma,p}$ and $\Lambda_{\sigma}$ defined
in a weak sense in (2.2) are given to us, therefore the right hand side of
(3.16) is also known.

Let us define

$\displaystyle L^{*}$ $\displaystyle:=\int_{\Omega}\nabla w\cdot(\sigma\nabla
u_{\varepsilon}+\gamma|\nabla u_{\varepsilon}|^{p-2}\nabla
u_{\varepsilon})~{}dx.$

From the knowledge of the DN-map associated with perturbed problem, we know
the integral $L^{*}$. Moreover, $L^{*}$ can be represented as follows

$\displaystyle L^{*}$ $\displaystyle=\int_{\Omega}\nabla w\cdot(\gamma|\nabla
u_{\varepsilon}|^{p-2}\nabla
u_{\varepsilon})~{}dx-\varepsilon^{1-p}~{}\int_{\Omega}\nabla
w\cdot(\gamma|\nabla u_{0}|^{p-2}\nabla u_{0})~{}dx$
$\displaystyle\qquad\qquad\qquad+\varepsilon^{1-p}\int_{\Omega}\nabla
w\cdot(\gamma|\nabla u_{0}|^{p-2}\nabla u_{0})~{}dx$
$\displaystyle=\varepsilon^{1-p}\int_{\Omega}\nabla w\cdot(\gamma|\nabla
u_{0}|^{p-2}\nabla u_{0})~{}dx+\mathcal{O}(\varepsilon^{3-2p})$
$\displaystyle=\varepsilon^{1-p}I(u_{0},w)+\mathcal{O}(\varepsilon^{3-2p}),$

where the estimate (2.16) is used and $I(u_{0},w)$ is defined in (3.3). Hence

$\displaystyle I(u_{0},w)=\int_{\Omega}\nabla w\cdot(\gamma|\nabla
u_{0}|^{p-2}\nabla u_{0})~{}dx$ (3.17)

is known to us. As we have described a reconstruction procedure of the
coefficient at the boundary using $I(u_{0},w)$ in case of $p>2$, in a similar
way, we can recover the coefficient $\gamma$ at the boundary $\partial\Omega$
from (3.17) for $1<p<2$.

##  4\. Proof for the Theorem 1.2

In this section, we present a procedure for the reconstruction of normal
derivative of $\gamma$ at the boundary point $x_{0}$ of $\Omega$ under certain
regularity assumption on $\gamma$. Unlike in [20, 22], we do not require the
values of $\gamma$ in the neighbourhood of $x_{0}$. However, we require
certain regularity assumption on $\gamma$ around $x_{0}$. We also use the fact
that $\gamma$ is known at $x_{0}$ as described in section 3.

###  4.1. Reconstruction of normal derivative of $\sigma$

Let $\alpha\in\mathbb{R}^{n}\backslash\\{0\\}$. The reconstruction of
$\partial_{\alpha}\sigma$ from the knowledge of the DN-map
$\Lambda_{\sigma}(f)$ associated to BVP (2.6), can be done using [21, 22].

###  4.2. Reconstruction of normal derivative of $\gamma$

For every $x\in\mathbb{R}^{n}$, we write $x=(x^{\prime},x_{n})$ with
$x^{\prime}\in\mathbb{R}^{n-1}$. Without loss of generality, we assume that
$x_{0}=0$ and $\partial\Omega$ is flat around $x_{0}$. More precisely, there
exists $\delta>0$ such that

$\displaystyle\Omega\cap B_{\delta}(x_{0})=\\{x\in
B_{\delta}(x_{0}):x_{n}>0\\}.$ (4.1)

In fact, using the boundary normal coordinates [18], we can locally transform
the coefficients to the form (4.1). Let $\zeta=(\zeta^{\prime},0)$ be an unit
tangent vector to $\partial\Omega$ at $x_{0}$.

Motivated by [24, 22], we wish to choose the approximate solutions $\Phi_{N}$
and $\Psi_{N}$ having oscillating boundary data supported in a neighbourhood
of $x_{0}$. For $N\in\mathbb{N}$ and $x\in\partial\Omega$, we consider

$\begin{split}\Phi_{N}(x)&:=\exp\big{(}{iNx^{\prime}\cdot\zeta^{\prime}-Nx_{n}}\big{)}\eta(\sqrt{N}x^{\prime}),\\\
\Psi_{N}(x)&:=\exp\Big{(}{i\frac{N}{2}x^{\prime}\cdot\zeta^{\prime}-\frac{N}{2}x_{n}}\Big{)}\eta(\sqrt{N}x^{\prime});\end{split}$
(4.2)

corresponding to the boundary data

$\displaystyle f_{N}(x^{\prime})$
$\displaystyle=\exp\big{(}{iNx^{\prime}\cdot\zeta^{\prime}}\big{)}\eta(\sqrt{N}x^{\prime}),\qquad
g_{N}(x^{\prime})=\exp\big{(}{i\frac{N}{2}x^{\prime}\cdot\zeta^{\prime}}\big{)}\eta(\sqrt{N}x^{\prime});$
(4.3)

respectively, where $\eta\in C^{2}$ is a cutoff function supported in a unit
ball such that

$0\leq\eta(x^{\prime})\leq
1,\quad\int_{\mathbb{R}^{n-1}}\eta^{2}(x^{\prime})~{}dx^{\prime}=1.$ (4.4)

We have observed that (cf. (3.7))

$\displaystyle K(u_{1},u_{2},u_{3})=\int\beta_{u_{1}}(x)\nabla
u_{2}\cdot\nabla u_{3}~{}dx$

is known for any $u_{1},u_{2}$ and $u_{3}$ satisfying the conductivity
equation (3.5). Let us denote $h:=\beta_{u_{1}}$. To prove the Theorem 1.2, we
use the following proposition:

######  Proposition 4.1.

Let $\partial\Omega$ be locally $C^{2}$ at $x_{0}$. Suppose
$D^{\alpha^{\prime}}_{x^{\prime}}D^{\alpha_{n}}_{x_{n}}\gamma$ is continuous
around $x_{0}$ for any multi-index $(\alpha^{\prime},\alpha_{n})$ such that
$|\alpha^{\prime}|+2\alpha_{n}\leq 2$. Then

$\displaystyle\lim_{N\rightarrow\infty}N^{\frac{n-1}{2}}\Big{(}2K(u_{1},u_{2},\overline{u_{2}})-K(u_{1},u_{3},\overline{u_{3}})\Big{)}$
$\displaystyle=\frac{1}{2}{\partial}_{x_{n}}h(x_{0})+\frac{3}{2}h(x_{0})\int_{\mathbb{R}^{n-1}}|\nabla_{x^{\prime}}\eta|^{2}~{}dx^{\prime}.$

###### Proof.

Let $\xi\in C^{\infty}([0,\infty))$ be a cutoff function which satisfies
$0\leq\xi\leq 1$ with

$\displaystyle\xi(x_{n})=\begin{cases}1,&\quad 0\leq x_{n}\leq\frac{1}{2},\\\
0,&\quad x_{n}\geq 1.\end{cases}$

For each $N\in\mathbb{N}$, we define $\xi_{N}(x_{n}):=\xi(\sqrt{N}x_{n})$. Our
computation is heavily relying on the choice of approximate solutions of
conductivity equation (3.5). As a consequence, we consider

$u_{2}=\xi_{N}\Psi_{N}+s_{N},\qquad u_{3}=\xi_{N}\Phi_{N}+r_{N},$ (4.5)

where $\Phi_{N}$ and $\Psi_{N}$ are defined in (4.2), and $s_{N}$ satisfies

$\begin{cases}\nabla\cdot(\sigma(x)\nabla
s_{N})=-\nabla\cdot\big{(}\sigma(x)\nabla(\xi_{N}\Psi_{N})\big{)},&\quad
x\in\Omega,\\\ s_{N}=0,&\quad x\in\partial\Omega,\end{cases}$ (4.6)

and similarly, $r_{N}$ satisfies

$\begin{cases}\nabla\cdot(\sigma(x)\nabla
r_{N})=-\nabla\cdot\big{(}\sigma(x)\nabla(\xi_{N}\Phi_{N})\big{)},&\quad
x\in\Omega,\\\ r_{N}=0,&\quad x\in\partial\Omega.\end{cases}$ (4.7)

Taking into account (4.5), we have

$\displaystyle 2$ $\displaystyle
K(u_{1},u_{2},\overline{u_{2}})-K(u_{1},u_{3},\overline{u_{3}})$
$\displaystyle=\int_{\Omega}h(x)\bigg{\\{}2\nabla(\xi_{N}\Psi_{N}+s_{N})\cdot\overline{\nabla(\xi_{N}\Psi_{N}+s_{N})}-\nabla(\xi_{N}\Phi_{N}+r_{N})\cdot\overline{\nabla(\xi_{N}\Phi_{N}+r_{N})}\bigg{\\}}~{}dx$
$\displaystyle=\int_{\Omega}h(x)\bigg{\\{}2\nabla(\xi_{N}\Psi_{N})\cdot\overline{\nabla(\xi_{N}\Psi_{N})}-\nabla(\xi_{N}\Phi_{N})\cdot\overline{\nabla(\xi_{N}\Phi_{N})}\bigg{\\}}~{}dx$
$\displaystyle\quad+2\int_{\Omega}h(x)\nabla
s_{N}\cdot\overline{\nabla(\xi_{N}\Psi_{N})}~{}dx-\int_{\Omega}h(x)\nabla
r_{N}\cdot\overline{\nabla(\xi_{N}\Phi_{N})}~{}dx$
$\displaystyle\quad+2\int_{\Omega}h(x)\nabla(\xi_{N}\Psi_{N})\cdot\overline{\nabla
s_{N}}~{}dx-\int_{\Omega}h(x)\nabla(\xi_{N}\Psi_{N})\cdot\overline{\nabla
r_{N}}~{}dx$ $\displaystyle\quad+2\int_{\Omega}h(x)\nabla
s_{N}\cdot\overline{\nabla s_{N}}~{}dx-\int_{\Omega}h(x)\nabla
r_{N}\cdot\overline{\nabla r_{N}}~{}dx$
$\displaystyle=:T_{1}+T_{2}+T_{3}+T_{4}+T_{5}+T_{6}+T_{7},$

where $T_{2}$ is given by

$\displaystyle T_{2}=2\int_{\Omega}h(x)\nabla
s_{N}\cdot\overline{\nabla(\xi_{N}\Psi_{N})}~{}dx,$

and using the arguments from [21, 22] we have
$T_{2}=\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.7}{$\scriptscriptstyle\mathcal{O}$}}(N^{-\frac{n-1}{2}})$
as $N\rightarrow\infty$. Following the similar argument in [13], we can
demonstrate that $N\rightarrow\infty$,

$\displaystyle T_{3}$ $\displaystyle=-\int_{\Omega}h(x)\nabla
r_{N}\cdot\overline{\nabla(\xi_{N}\Phi_{N})}~{}dx=\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.7}{$\scriptscriptstyle\mathcal{O}$}}(N^{-\frac{n-1}{2}}),$
$\displaystyle T_{4}$
$\displaystyle=2\int_{\Omega}h(x)\nabla(\xi_{N}\Psi_{N})\cdot\overline{\nabla
s_{N}}~{}dx=\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.7}{$\scriptscriptstyle\mathcal{O}$}}(N^{-\frac{n-1}{2}}),$
$\displaystyle T_{5}$
$\displaystyle=-\int_{\Omega}h(x)\nabla(\xi_{N}\Psi_{N})\cdot\overline{\nabla
r_{N}}~{}dx=\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.7}{$\scriptscriptstyle\mathcal{O}$}}(N^{-\frac{n-1}{2}}),$

and using the equation (4.6)-(4.7) and estimates given for $T_{2}$ and $T_{3}$
one can show that as $N\rightarrow\infty$,

$\displaystyle T_{6}$ $\displaystyle=2\int_{\Omega}h(x)\nabla
s_{N}\cdot\overline{\nabla
s_{N}}~{}dx=\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.7}{$\scriptscriptstyle\mathcal{O}$}}(N^{-\frac{n-1}{2}}),$
$\displaystyle T_{7}$ $\displaystyle=-\int_{\Omega}h(x)\nabla
r_{N}\cdot\overline{\nabla
r_{N}}~{}dx=\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.7}{$\scriptscriptstyle\mathcal{O}$}}(N^{-\frac{n-1}{2}}).$

We set the scaling transformation

$y_{i}=\sqrt{N}x_{i},\quad i=1,2,\cdots,n-1,\qquad y_{n}=Nx_{n}.$ (4.8)

Taking into account the supports of $\xi_{N}\Psi_{N}(y)$ and
$\xi_{N}\Phi_{N}(y)$, we define

$\displaystyle\Omega_{N}=\Big{\\{}x:|x^{\prime}|\leq\frac{1}{\sqrt{N}},0\leq
x_{n}\leq\frac{1}{2\sqrt{N}}\Big{\\}},\quad\Omega^{\prime}_{N}=\Big{\\{}x:|x^{\prime}|\leq\frac{1}{\sqrt{N}},\frac{1}{2\sqrt{N}}\leq
x_{n}\leq\frac{1}{\sqrt{N}}\Big{\\}}$

and hence, we can rewrite $T_{1}$ as

$\displaystyle T_{1}$
$\displaystyle=\int_{\Omega_{N}}h(x)\bigg{\\{}2\nabla\Psi_{N}\cdot\overline{\nabla\Psi_{N}}-\nabla\Phi_{N}\cdot\overline{\nabla\Phi_{N}}\bigg{\\}}~{}dx$
$\displaystyle\quad+\int_{\Omega_{N}^{\prime}}h(x)\bigg{\\{}2\nabla(\xi_{N}\Psi_{N})\cdot\overline{\nabla(\xi_{N}\Psi_{N})}-\nabla(\xi_{N}\Phi_{N})\cdot\overline{\nabla(\xi_{N}\Phi_{N})}\bigg{\\}}~{}dx$
$\displaystyle=T_{11}+T_{12},$

where $T_{11}$ and $T_{12}$ are given by

$\displaystyle
T_{11}:=\int_{\Omega_{N}}h(x)\bigg{\\{}2\nabla\Psi_{N}\cdot\overline{\nabla\Psi_{N}}-\nabla\phi_{N}\cdot\overline{\nabla\Phi_{N}}\bigg{\\}}~{}dx,$
$\displaystyle
T_{12}:=\int_{\Omega^{\prime}_{N}}h(x)\bigg{\\{}2\nabla(\xi_{N}\Psi_{N})\cdot\overline{\nabla(\xi_{N}\Psi_{N})}-\nabla(\xi_{N}\Phi_{N})\cdot\overline{\nabla(\xi_{N}\Phi_{N})}\bigg{\\}}~{}dx.$

$T_{11}$ can be further represented by

$\displaystyle T_{11}$
$\displaystyle=\int_{\Omega_{N}}h(x)\bigg{\\{}2\nabla\Psi_{N}\cdot\overline{\nabla\Psi_{N}}-\nabla\Phi_{N}\cdot\overline{\nabla\Phi_{N}}\bigg{\\}}~{}dx$
$\displaystyle=N^{2}\int_{\Omega_{N}}h(x^{\prime},x_{n})\eta^{2}(\sqrt{N}x^{\prime})~{}\big{(}\exp({-Nx_{n}})-2\exp({-2Nx_{n}})\big{)}~{}dx$
$\displaystyle\qquad+N\int_{\Omega_{N}}h(x^{\prime},x_{n})|(\nabla_{y^{\prime}}\eta)(\sqrt{N}x^{\prime})|^{2}~{}\big{(}2\exp({-Nx_{n}})-\exp({-2Nx_{n}})\big{)}~{}dx,$

and using the scaling transformation introduced in (4.8), we get

$\begin{split}N^{\frac{n-1}{2}}T_{11}&=N\int^{\sqrt{N}/2}_{0}\int_{|y^{\prime}|\leq
1}h\left(\frac{y^{\prime}}{\sqrt{N}},\frac{y_{n}}{N}\right)\eta^{2}(y^{\prime})~{}\big{(}\exp({-y_{n}})-2\exp({-2y_{n}})\big{)}~{}dy^{\prime}dy_{n}\\\
&\quad+\int^{\sqrt{N}/2}_{0}\int_{|y^{\prime}|\leq
1}h\left(\frac{y^{\prime}}{\sqrt{N}},\frac{y_{n}}{N}\right)|(\nabla_{y^{\prime}}\eta)(y^{\prime})|^{2}~{}\big{(}2\exp({-y_{n}})-\exp({-2y_{n}})\big{)}~{}~{}dy^{\prime}dy_{n}.\end{split}$
(4.9)

From the regularity assumption on $h$, we can expand $h$ around $x_{0}=0$.
More precisely, we can write

$\begin{split}h\left(\frac{y^{\prime}}{\sqrt{N}},\frac{y_{n}}{N}\right)=&h(x_{0})+\frac{1}{\sqrt{N}}\sum_{|\alpha^{\prime}|=1}{\partial}^{|\alpha^{\prime}|}_{x^{\prime\alpha^{\prime}}}h(x_{0})y^{\prime\alpha^{\prime}}+\frac{1}{N}\partial_{x_{n}}h(x_{0})y_{n}\\\
&+\frac{1}{N}\sum_{|\alpha^{\prime}|=2}\frac{1}{\alpha!}{\partial}^{|\alpha^{\prime}|}_{x^{\prime\alpha^{\prime}}}h(x_{0})y^{\prime\alpha^{\prime}}+\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.7}{$\scriptscriptstyle\mathcal{O}$}}(N^{-1})\qquad\text{as
}N\rightarrow\infty.\end{split}$ (4.10)

It is observed that as $N\rightarrow\infty$,

$\displaystyle\int^{\sqrt{N}/2}_{0}(\exp{-y_{n}}-2\exp{-2y_{n}})~{}dy_{n}=\mathcal{O}\big{(}\exp({-\sqrt{N}/{2}})\big{)},$
$\displaystyle\int^{\sqrt{N}/2}_{0}\big{(}2\exp({-y_{n}})-\exp({-2y_{n}})\big{)}~{}dy_{n}=\frac{3}{2}+\mathcal{O}\big{(}\exp({-\sqrt{N}/{2}})\big{)},$
$\displaystyle\int^{\sqrt{N}/2}_{0}y_{n}\big{(}\exp({-y_{n}})-2\exp({-2y_{n}})\big{)}~{}dy_{n}=\frac{1}{2}+\mathcal{O}(\sqrt{N}e^{-\sqrt{N}/{2}}).$

With the help of above estimates and expansion (4.10), and using the
properties of $\eta$ introduced in (4.4), we end up with

$\begin{split}N^{\frac{n-1}{2}}&\int_{\Omega_{N}}h(x)\bigg{\\{}2\nabla\Psi_{N}\cdot\overline{\nabla\Psi_{N}}-\nabla\Phi_{N}\cdot\overline{\nabla\Phi_{N}}\bigg{\\}}~{}dx\\\
&=\frac{1}{2}\partial_{x_{n}}h(x_{0})+\frac{3}{2}h(x_{0})\int_{\mathbb{R}^{n-1}}|(\nabla_{x^{\prime}}\eta)|^{2}~{}dx^{\prime}+\mathchoice{{\scriptstyle\mathcal{O}}}{{\scriptstyle\mathcal{O}}}{{\scriptscriptstyle\mathcal{O}}}{\scalebox{0.7}{$\scriptscriptstyle\mathcal{O}$}}(1)\qquad\text{as
}N\rightarrow\infty.\end{split}$ (4.11)

We see that

$\displaystyle\nabla(\xi_{N}\Psi_{N})=\exp\left(i\frac{N}{2}x^{\prime}\cdot\zeta^{\prime}-\frac{N}{2}x_{n}\right)\begin{bmatrix}\Big{\\{}i\frac{N}{2}\zeta^{\prime}\eta(\sqrt{N}x^{\prime})+\sqrt{N}(\nabla_{y^{\prime}}\eta)(\sqrt{N}x^{\prime})\Big{\\}}\xi(\sqrt{N}x_{n})\\\
\Big{\\{}-\frac{N}{2}\xi(\sqrt{N}x_{n})+\sqrt{N}(\nabla_{x_{n}}\xi)(\sqrt{N}x_{n})\Big{\\}}\eta(\sqrt{N}x^{\prime})\end{bmatrix},$

and

$\displaystyle\nabla(\xi_{N}\Phi_{N})=\exp\left(iNx^{\prime}\cdot\zeta^{\prime}-Nx_{n}\right)\begin{bmatrix}\Big{\\{}iN\zeta^{\prime}\eta(\sqrt{N}x^{\prime})+\sqrt{N}(\nabla_{y^{\prime}}\eta)(\sqrt{N}x^{\prime})\Big{\\}}\xi(\sqrt{N}x_{n})\\\
\Big{\\{}-N\xi(\sqrt{N}x_{n})+\sqrt{N}(\nabla_{x_{n}}\xi)(\sqrt{N}x_{n})\Big{\\}}\eta(\sqrt{N}x^{\prime})\end{bmatrix}.$

As a consequence, we have

$\displaystyle T_{12}$
$\displaystyle=\int_{\Omega^{\prime}_{N}}h(x)\bigg{\\{}2\nabla(\xi_{N}\Psi_{N})\cdot\overline{\nabla(\xi_{N}\Psi_{N})}-\nabla(\xi_{N}\Phi_{N})\cdot\overline{\nabla(\xi_{N}\Phi_{N})}\bigg{\\}}~{}dx$
$\displaystyle={N^{2}}\int_{\Omega^{\prime}_{N}}h(x^{\prime},x_{n})\xi^{2}(\sqrt{N}x_{n})\eta^{2}(\sqrt{N}x^{\prime})\\{\exp(-Nx_{n})-2\exp(-2Nx_{n})\\}~{}dx$
$\displaystyle+N\int_{\Omega^{\prime}_{N}}h(x^{\prime},x_{n})\xi^{2}(\sqrt{N}x_{n})|\nabla_{y^{\prime}}\eta|^{2}(\sqrt{N}x^{\prime})\\{2\exp(-Nx_{n})-\exp(-2Nx_{n})\\}~{}dx$
$\displaystyle-2N^{3/2}\int_{\Omega^{\prime}_{N}}h(x)\xi(\sqrt{N}x_{n})\eta(\sqrt{N}x^{\prime})(\nabla_{x_{n}}\xi)(\sqrt{N}x_{n})\\{\exp(-Nx_{n})-\exp(-2Nx_{n})\\}~{}dx$
$\displaystyle+N\int_{\Omega^{\prime}_{N}}h(x^{\prime},x_{n})\eta^{2}(\sqrt{N}x^{\prime})|(\nabla_{x_{n}}\xi)(\sqrt{N}x_{n})|^{2}\\{2\exp(-Nx_{n})-\exp(-2Nx_{n})\\}~{}dx$
$\displaystyle=:I+II+III+IV.$

Note that

$\displaystyle I$
$\displaystyle={N^{2}}\int_{\Omega^{\prime}_{N}}h(x^{\prime},x_{n})\xi^{2}(\sqrt{N}x_{n})\eta^{2}(\sqrt{N}x^{\prime})\\{\exp(-Nx_{n})-2\exp(-2Nx_{n})\\}~{}dx$
$\displaystyle=\frac{N}{N^{\frac{n-1}{2}}}\int_{\frac{\sqrt{N}}{2}}^{\sqrt{N}}\int_{|y^{\prime}|\leq
1}h\left(\frac{y^{\prime}}{\sqrt{N}},\frac{y_{n}}{N}\right)\xi^{2}\left(\frac{y_{n}}{\sqrt{N}}\right)\eta^{2}(y^{\prime})\\{\exp(-y_{n})-2\exp(-2y_{n})\\}~{}dy^{\prime}dy_{n},$

where we have used the scaling transformation (4.8). Taking into account the
inequality

$\displaystyle\int_{\frac{\sqrt{N}}{2}}^{\sqrt{N}}\\{\exp(-y_{n})-2\exp(-2y_{n})\\}~{}dy_{n}=\mathcal{O}\big{(}\exp(-\sqrt{N}/2)\big{)}\quad\text{as
}N\rightarrow\infty,$

we can deduce that

$\displaystyle|I|\leq C\exp(-\sqrt{N}/2)\quad\text{ as }N\rightarrow\infty,$

for some constant $C$. Following the similar argument as above, we see that

$\displaystyle|II|+|III|+|IV|\leq C\exp(-\sqrt{N}/2)\quad\text{as
}N\rightarrow\infty.$

Subsequently, we can conclude that

$\displaystyle
T_{12}=\mathcal{O}\big{(}\exp({-\sqrt{N}/{2}})\big{)}\quad\text{as
}N\rightarrow\infty.$ (4.12)

Hence combining the above estimates (4.11) and (4.12), (4.9) reduces to

$\begin{split}\lim_{N\rightarrow\infty}N^{\frac{n-1}{2}}\big{(}2K(u_{1},u_{2},\overline{u_{2}})&-K(u_{1},u_{3},\overline{u_{3}})\big{)}\\\
&=\frac{1}{2}\partial_{x_{n}}h(x_{0})+\frac{3}{2}h(x_{0})\int_{\mathbb{R}^{n-1}}|(\nabla_{x^{\prime}}\eta)|^{2}~{}dx^{\prime}.\end{split}$
(4.13)

Hence the Proposition 4.1 follows. ∎

In section 3, we have recovered $h$ at the boundary of $\Omega$, hence
$h(x_{0})$ is known to us. The cutoff function $\eta$ introduced in (4.4) is
arbitrary, the right hand side term

$\displaystyle
h(x_{0})\int_{\mathbb{R}^{n-1}}|\nabla_{x^{\prime}}\eta(x^{\prime})|^{2}~{}dx^{\prime}$

of (4.13) is known. Hence, the normal derivative $\partial_{x_{n}}h(x_{0})$ is
determined. As a consequence, $\partial_{x_{n}}\gamma(x_{0})$ can be
reconstructed and hence the Theorem 1.2 follows.

## Acknowledgments

MV is partially supported by Start-up Research Grant SRG/2021/001432 from the
Science and Engineering Research Board, Government of India.

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