# An inverse problem for the
porous medium equation
with partial data and a possibly singular absorption term

Cătălin I. Cârstea School of Mathematics, Sichuan University, Chengdu,
Sichuan, 610064, P.R.China; email<EMAIL_ADDRESS>Tuhin Ghosh
Department of Mathematics, Universität Bielefeld, 33615 Bielefeld, Germany;
email<EMAIL_ADDRESS>Gunther Uhlmann Department of Mathematics,
University of Washington, Seattle, WA 98195-4350, USA, and Institute for
Advanced Study of the Hong Kong University of Science and Technology, Hong
Kong; email<EMAIL_ADDRESS>
###### Abstract

In this paper we prove uniqueness in the inverse boundary value problem for
the three coefficient functions in the porous medium equation with an
absorption term $\epsilon\partial_{t}u-\nabla\cdot(\gamma\nabla u^{m})+\lambda
u^{q}=0$, with $m>1$, $m^{-1}<q<\sqrt{m}$, with the space dimension 2 or
higher. This is a degenerate parabolic type quasilinear PDE which has been
used as a model for phenomena in fields such as gas flow (through a porous
medium), plasma physics, and population dynamics. In the case when $\gamma=1$
a priori, we prove unique identifiability with data supported in an
arbitrarily small part of the boundary. Even for the global problem we improve
on previous work by obtaining uniqueness with a finite (rather than infinite)
time of observation and also by introducing the additional absorption term
$\lambda u^{q}$.

#### Keywords:

Inverse problems, porous medium equation, nonlinear parabolic equations.

##  1 Introduction

Let $\Omega\subset\mathbb{R}^{n}$, with $n\geq 2$, be a bounded, smooth
domain, and let $T\in(0,\infty)$. We would like to consider the following
equation in the space-time cylindrical domain $(0,T)\times\Omega$

$\left\\{\begin{array}[]{l}\epsilon(x)\partial_{t}u(t,x)-\nabla\cdot(\gamma(x)\nabla
u^{m}(t,x))+\lambda(x)u^{q}(t,x)=f(t,x),\\\\[5.0pt] u(0,x)=0,\quad
u|_{[0,T)\times\partial\Omega}=\phi(t,x),\quad u\geq 0.\end{array}\right.$ (1)

In the above $m$ and $q$ are fixed parameters that we assume to satisfy

$m>1,\quad m^{-1}<q<\sqrt{m}.$ (2)

The coefficients $\epsilon$, $\gamma$, and $\lambda$ are bounded nonnegative
functions. $\epsilon$ and $\gamma$ will be assumed to also have strictly
positive lower bounds. We will also assume that $\epsilon,\gamma,\lambda\in
C^{\infty}(\overline{\Omega})$, for the sake of convenience. Consistency
requires that $\phi(0,x)=0$, $\phi\geq 0$.

The equation (1) is usually refered to as a porous medium equation, with an
absorbtion term (i.e. the $\lambda u^{q}$ term). The name comes from its use
in modeling the flow of a gas through a porous medium (see [42],[49]), but
equations of this form are also used to model phenomena in other fields, such
as plasma physics (see [41]), or population dynamics (see [40]).

From the mathematical point of view, equation (1) is a degenerate parabolic
type quasilinear equation. Note that if $0<q<1$, then the absorption term
$\lambda u^{q}$ is not Lipschitz in $u$ and it is said to be singular in this
case. Furthermore, equation (1) is a particularly simple modification of the
classical heat equation (with a lower order term). This has made it a popular
object of study and the mathematical literature dedicated to the porous medium
equation is vast. A good survey of the field is the monograph [49].

In this paper we are interested in the inverse boundary value problem
associated to (1). Here this amounts to the following question: is the set of
pairs of Dirichlet and Neumann data corresponding to a suitably large class of
solutions of (1) sufficient for the reconstruction of the coefficients
$\epsilon$, $\gamma$, and $\lambda$? Historically, the first problem of this
kind was proposed by Calderón in [7], for the conductivity equation
$\nabla\cdot(\gamma\nabla u)=0$. The question of uniqueness in the original
Calderón inverse boundary value problem (i.e. does the boundary data uniquely
determine the coefficient $\gamma$) was answered affirmatively in [39] for
$n=2$ and in [48] for $n\geq 3$. Since then, there have been many results
covering analogous problems for linear and nonlinear equations. The full list
of even the most important of these is too extensive to include here. We
restrict ourselves to mentioning some of the important results known for
quasilinear and semilinear elliptic and parabolic equations. For semilinear
equations, see [8], [16], [19], [23], [25], [26], [31], [32], [33], [36],
[35], [46]. For quasilinear equations, not in divergence form, see [24]. For
quasilinear equations in divergence form see [11], [10], [12], [14], [15],
[18], [21], [27], [38], [43], [44], [45], [47], (also [9] for quasilinear
time-harmonic Maxwell systems).

Results for degenerate equations, such as equation (1), are quite few.
Examples would include the following works for the weighted p-Laplace
equation, $\nabla\cdot(a(x)|\nabla u|^{p-2}\nabla u)=0$, which is a degenerate
elliptic quasilinear PDE: [5], [2], [3], [20], [4], [28]. We note that a
uniqueness result without additional constraints, such as monotonicity, has
not yet been derived for the weighted p-Laplacian.

An interesting (and also practically useful) variation of the inverse boundary
value problem described above is the case of “partial data”. The goal remains
to prove uniqueness for the various coefficients appearing in the equation,
but with the known boundary data supported/restricted to a proper subset of
the boundary. In the original Calderón problem this was solved in [22] for
$n=2$. For $n\geq 3$, with some restrictions on the geometry of the subsets on
which the boundary data is known, a result can be found in [29]. We would also
like to mention here the work [17] on the linearized Calderón problem with
partial data. For elliptic semilinear and quasilinear problems partial data
results have been obtained in [30], [32], [33], [35].

Finally, we would like to mention one existing result for the porous medium
equation, namely [13], where the equation (1), without the absorption term, is
considered. Uniqueness in the inverse boundary value problem is obtained when
the boundary data are available for infinite time (i.e. $T=\infty$). In the
case of the heat equation this is sufficient to then obtain uniqueness for
data available on a finite time interval, since solutions are analytic in
time. This is not the case for the porous medium equation (see [49]), so
uniqueness with a finite time of observation must be obtained by other means,
which is one of the goals of this paper.

In this paper we consider the inverse boundary value problem for the equation
(1), with a finite time of observation (i.e. $T<\infty$), in both the partial
data and the global data cases.

###  1.1 Existence of solutions

Before stating the main result concering the inverse boundary value problem
for equation (1), we first need to outline the sense in which we consider a
function $u$ to be a solution of (1). The approach is adapted from [49].

Let $Q_{T}:=(0,T)\times\Omega$ and $S_{T}:=(0,T)\times\partial\Omega$. In
order to define a suitable space of Dirichlet boundary data. let

$C_{t}(Q_{T})=\left\\{\varphi\in
C^{\infty}(\overline{Q_{T}}):\mbox{supp\;}\varphi\cap\left[\\{T\\}\times\Omega\right]=\emptyset\right\\}$
(3)

and let $H^{1}_{t}(Q_{T})$ be the completion of this space in $H^{1}(Q_{T})$.
We will denote by $H^{\frac{1}{2}}_{t}(S_{T})$ the subspace of
$H^{\frac{1}{2}}(S_{T})$ that consists of traces of $H^{1}_{t}(Q_{T})$
functions. We also introduce

$C_{\diamond}(Q_{T})=\left\\{\varphi\in
C^{\infty}(\overline{Q_{T}}):\mbox{supp\;}\varphi\cap\left[S_{T}\cup\\{T\\}\times\Omega\right]=\emptyset\right\\},$
(4)

and its completion $H^{1}_{\diamond}(Q_{T})$ in $H^{1}(Q_{T})$.

Let by $\tau_{\Omega}$ the boundary trace operator for the domain $\Omega$. It
is bounded between $H^{1}(\Omega)$ and $H^{\frac{1}{2}}(\partial\Omega)$. Let
$\tau_{Q_{T}}$ be the boundary trace operator for $Q_{T}$ to the side boundary
$S_{T}$. It is not hard to check that $\tau_{Q_{T}}$ is bounded from
$L^{2}((0,T);H^{1}(\Omega))$ to
$L^{2}((0,T);H^{\frac{1}{2}}(\partial\Omega))$.

######  Definition 1.

We say that $u\in L^{\infty}(Q_{T})$, $u\geq 0$, is a weak solution of (1) in
$Q_{T}$ if it satisfies the following conditions:

  1. 1.

$\nabla(u^{m})$ exists in the sense of distributions and $\nabla(u^{m})\in
L^{2}(Q_{T})$;

  2. 2.

for any test function $\varphi\in C_{\diamond}(Q_{T})$ (or, equivalently, any
$\varphi\in H^{1}_{\diamond}(Q_{T})$)

$\int_{Q_{T}}\gamma\nabla\varphi\cdot\nabla(u^{m})\,\text{d}t\,\text{d}x+\int_{Q_{T}}\lambda\varphi
u^{q}\,\text{d}t\,\text{d}x-\int_{Q_{T}}\epsilon\partial_{t}\varphi\,u\,\text{d}t\,\text{d}x=\int_{Q_{T}}\varphi
f\,\text{d}t\,\text{d}x;$ (5)

  3. 3.

$\tau_{Q_{T}}(u^{m})=\phi^{m}$.

We will show in section 2 that

######  Theorem 1.

If $\phi\in C(\overline{S_{T}})$, $\phi^{m}\in C(\overline{S_{T}})\cap
C^{0,1}(S_{T})$, $\phi\geq 0$, and $\phi(0,x)=0$ for all $x\in\partial\Omega$,
and if $f\in L^{\infty}(Q_{T})$, $f\geq 0$, then there exists a unique weak
solution $u$ of (1) in $Q_{T}$. This solution satisfies the energy estimate

$||u^{m}||_{L^{2}((0,T);H^{1}(\Omega))}\leq
C(1+T)^{\frac{1}{2}}\left(||\phi^{m+q}||_{C^{0,1}(\Omega)}+||\phi||_{C(\Omega)}+||f||_{L^{\infty}(Q_{T})}\right),$
(6)

with a constant $C>0$ that depends on $\Omega$, $m$, $q$, and the upper and
lower bounds of $\epsilon$, $\gamma$, and $\lambda$. The solution $u$
satisfies the maximum principle

$0\leq\sup_{Q_{T}}u\leq\sup_{S_{T}}\phi.$ (7)

If $\phi_{1}\leq\phi_{2}$, $f_{1}\leq f_{2}$ are as above and give rise to
weak solutions $u_{1}$ and $u_{2}$, then

$u_{1}\leq u_{2}.$ (8)

We have been unable to find a proof of this exact result in the available
literature on the porous medium equation. For this reason, and also for the
convenience of the reader, we provide our own proof. The methods needed are
not new. We have followed the argument in [49], with some additional
techniques from [1].

If $\psi$ is as in the above theorem and $u$ is the corresponding weak
solution, the Neumann data $\gamma\partial_{\nu}u^{m}|_{S_{T}}$ can be defined
by

$\langle\gamma\partial_{\nu}u^{m}|_{S_{T}},\psi|_{S_{T}}\rangle=\int_{Q_{T}}\left(\gamma\nabla\psi\cdot\nabla(u^{m})+\lambda\psi
u^{q}-\epsilon\partial_{t}\psi u\right)\,\text{d}t\,\text{d}x.$ (9)

###  1.2 Main results

We can now define the Dirichlet-to-Neumann map

$\Lambda_{\epsilon,\gamma,\lambda}^{PM}(\phi)=\gamma\partial_{\nu}u^{m}|_{S_{T}}\in\left(H^{\frac{1}{2}}_{t}(S_{T})\right)^{\prime}.$
(10)

Suppose that we have two sets of coefficients, $\epsilon^{(i)}$,
$\gamma^{(i)}$, $\lambda^{(i)}$, and $\epsilon^{(ii)}$, $\gamma^{(ii)}$,
$\lambda^{(ii)}$, that are as above. The first main result of our paper is the
following.

######  Theorem 2.

If
$\Lambda_{\epsilon^{(i)},\gamma^{(i)},\lambda^{(i)}}^{PM}(\phi)=\Lambda_{\epsilon^{(ii)},\gamma^{(ii)},\lambda^{(ii)}}^{PM}(\phi)$,
for all $\phi\in C(\overline{S_{T}})$, such that $\phi^{m}\in
C(\overline{S_{T}})\cap C^{0,1}_{loc}(S_{T})$, $\phi\geq 0$, and $\phi(0,x)=0$
for all $x\in\partial\Omega$, then $\gamma^{(i)}=\gamma^{(ii)}$,
$\epsilon^{(i)}=\epsilon^{(ii)}$, and $\lambda^{(i)}=\lambda^{(ii)}$.

In the case when the $\gamma$ coefficient is a priori known to be constant, we
also have the following partial data result.

######  Theorem 3.

Let $\Sigma\subset\partial\Omega$ be open. If
$\Lambda_{\epsilon^{(i)},1,\lambda^{(i)}}^{PM}(\phi)=\Lambda_{\epsilon^{(ii)},1,\lambda^{(ii)}}^{PM}(\phi)$,
for all $\phi\in C(\overline{S_{\infty}})$, such that
$\mbox{supp\;}\phi\subset[0,T]\times\Sigma$, $\phi^{m}\in
C(\overline{S_{\infty}})\cap C^{0,1}_{loc}(S_{\infty})$, $\phi\geq 0$, and
$\phi(0,x)=0$ for all $x\in\partial\Omega$, then
$\epsilon^{(i)}=\epsilon^{(ii)}$ and $\lambda^{(i)}=\lambda^{(ii)}$.

We give a proof of Theorem 2 in section 3. The main idea for the proof is to
use two sucessive transformations of equation (1). The first is the change of
function $v=u^{m}$, which gives us the equation

$\epsilon\partial_{t}v^{\frac{1}{m}}-\nabla\cdot(\gamma\nabla v)+\lambda
v^{\frac{q}{m}}=0.$ (11)

The second transformation involves the function

$V(x)=\int_{0}^{T}(T-t)^{\alpha}v(t,x)\,\text{d}t,$ (12)

where $\alpha,T>0$ will be chosen in the course of the proof. Since the
equation satisfied by $v$ is non-linear, we do not obtain a closed form PDE
for $V$. It is however possible to show that $V$ satisfies a differential
inequality of the form

$0\leq\nabla\cdot(\gamma(x)\nabla V(x))\leq
C_{1}\epsilon(x)(V(x))^{\frac{1}{m}}+C_{2}\lambda(x)(V(x))^{\frac{q}{m}}.$
(13)

If we choose Dirichlet boundary data so that $v|_{S_{T}}(t,x)=ht^{m}g(x)$,
with $h$ a large parameter, we can deduce the first few terms in the
asymptotic expansion of $V$ as $h\to\infty$. The Neumann data for each of
these terms is determined by the Dirichlet-to-Neumann map, and we can use this
information to separately show uniqueness for each of the $\epsilon$,
$\gamma$, and $\lambda$ coefficients.

We prove Theorem 3 in section 4. The starting point for the argument is an
integral identity derived in section 3. We reduce this integral identity to a
problem similar to the linearized local Calderón problem considered in [17],
but with an $L^{1}$ coefficient function rather than the $L^{\infty}$ one. The
result follows by adapting the original argument to this new situation.

Finally, we would like to remark on the importance of the partial data case.
Of course, one reason to consider this case is that in possible real life
applications only partial data may be practically available. Here however
there is an additional theoretical reason, namely that in the partial data
case the equation remains in a degenerate regime for the entire time of
observation, since the solution must remain zero on a part of the boundary.
This shows that our method does not work by moving the equation to a non-
degenerate regime, but rather that it can handle the equation even in
situations that are not covered by previous works.

##  2 The forward problem

The main idea for the proof of existence of weak solutions to (1) comes from
the intuition that the maximum principle should hold for such solutions. If
the initial data, the Dirichlet boundary data, and the source term are
strictly positive (an bounded), and the maximum/minimum principle holds, then
the solution would not take values close to zero and infinity. When this is
the case, we can modify the equation so as to remove its singularities, while
keeping the same solution. This is the same method used in the proof of [49,
Theorem 5.14].

###  2.1 Existence of solutions

We will now make the above rigorous. Let $a_{k}(x,z)$, $b_{k}(x,z)$,
$k=1,2,3,\ldots$, be smooth, bounded functions, with the $a_{k}$ also having
strictly positive lower bounds, and such that when

$\frac{1}{k}\leq z\leq\sup_{S_{T}}\phi+T\sup_{Q_{T}}f+\frac{1+T}{k},$ (14)

we have

$a_{k}(x,z)=m\gamma(x)z^{m-1},\quad b_{k}(x,z)=\lambda(x)z^{q}.$ (15)

We also choose functions $f_{k}\in C^{\infty}(Q_{T})$ such that

$f(t,x)\leq f_{k+1}(t,x)\leq f_{k}(t,x)\leq f(t,x)+\frac{1}{k}.$ (16)

Let now $u_{k}$ be the solutions of

$\left\\{\begin{array}[]{l}\epsilon\partial_{t}u_{k}-\nabla\cdot\left(a_{k}(x,u_{k})\nabla
u_{k})\right)+b_{k}(x,u_{k})=f_{k},\\\\[5.0pt] u_{k}(0,x)=\frac{1}{k},\quad
u_{k}|_{[0,T)\times\partial\Omega}=\phi+\frac{1}{k}.\end{array}\right.$ (17)

The problem (17) has a solution $u_{k}\in C^{1,2}(Q_{T})\cap
C(\overline{Q_{T}})$ (see [34], or [37]), which furthermore satisfies the
maximum principle

$\frac{1}{k}\leq
u_{k}(t,x)\leq\frac{1}{k}+\sup_{S_{T}}\phi+t\sup_{Q_{T}}f_{k}.$ (18)

It follows that $u_{k}$ is also a solution to

$\left\\{\begin{array}[]{l}\epsilon\partial_{t}u_{k}-\nabla\cdot(\gamma\nabla
u^{m}_{k})+\lambda u_{k}^{q}=f_{k},\\\\[5.0pt] u_{k}(0,x)=\frac{1}{k},\quad
u_{k}|_{[0,T)\times\partial\Omega}=\phi+\frac{1}{k}.\end{array}\right.$ (19)

By (18), $u_{k}$ is also a solution to
$\epsilon\partial_{t}u_{k}-\nabla\cdot\left(a_{k+1}(x,u_{k},\nabla
u_{k})\right)=f_{k}$. By the comparison principle (see [37, Theorem 9.7]), we
have that

$0\leq u_{k+1}\leq u_{k},\quad\forall k=1,2\ldots.$ (20)

Let $u$ be the pointwise limit

$u(t,x)=\lim_{k\to\infty}u_{k}(t,x).$ (21)

It is clear that $u\in L^{\infty}(\Omega)$ and

$0\leq u(t,x)\leq\sup_{S_{T}}\phi+t\sup_{Q_{T}}f.$ (22)

A simple application of the monotone convergence theorem gives that in
$L^{2}(Q_{T})$ norm we have $u_{k}\to u$, $u_{k}^{m}\to u^{m}$, and
$u_{k}^{q}\to u^{q}$.

Let $\tilde{\phi}:Q_{T}\to[0,\infty)$ be a smooth extension of the Dirichlet
data, such that we still have $\tilde{\phi}(0,x)=0$ for all $x\in\Omega$ and

$||\tilde{\phi}||_{W^{1,\infty}(Q_{T})}\leq C||\phi||_{C^{0,1}(S_{T})},$ (23)

with a constant $C>0$. Let

$\eta_{k}=u_{k}^{m}-\left(\tilde{\phi}+\frac{1}{k}\right)^{m},$ (24)

which is zero on $S_{T}$. We will multiply (19) by $\eta_{k}$ and integrate
over $Q_{T}$. Note that

$-\int_{Q_{T}}\eta_{k}\nabla(\gamma\nabla
u_{k}^{m})\,\text{d}t\,\text{d}x\\\\[5.0pt] =\int_{Q_{T}}\gamma|\nabla
u_{k}^{m}|^{2}\,\text{d}t\,\text{d}x-\int_{Q_{T}}\gamma\nabla\left(\tilde{\phi}+\frac{1}{k}\right)^{m}\cdot\nabla
u_{k}^{m}\,\text{d}t\,\text{d}x,$ (25)

and

$\int_{Q_{T}}\epsilon\partial_{t}u_{k}\eta_{k}\,\text{d}t\,\text{d}x=\int_{\Omega}\frac{\epsilon}{m+1}u_{k}^{m+1}(T)\,\text{d}x\\\\[5.0pt]
-\int_{\Omega}\epsilon
u_{k}(T)\left(\tilde{\phi}(T)+\frac{1}{k}\right)^{m}\,\text{d}x+\int_{Q_{T}}\epsilon
u_{k}\partial_{t}\left(\tilde{\phi}+\frac{1}{k}\right)^{m}\,\text{d}t\,\text{d}x.$
(26)

It follows that

$\int_{Q_{T}}\gamma|\nabla
u_{k}^{m}|^{2}\,\text{d}t\,\text{d}x+\int_{Q_{T}}\lambda
u_{k}^{m+q}\,\text{d}t\,\text{d}x+\int_{\Omega}\frac{\epsilon}{m+1}u_{k}^{m+1}(T)\,\text{d}x\\\\[5.0pt]
=\int_{Q_{T}}\gamma\nabla\left(\tilde{\phi}+\frac{1}{k}\right)^{m}\cdot\nabla
u_{k}^{m}\,\text{d}t\,\text{d}x+\int_{\Omega}\epsilon
u_{k}(T)\left(\tilde{\phi}(T)+\frac{1}{k}\right)^{m}\,\text{d}x\\\\[5.0pt]
-\int_{Q_{T}}\epsilon
u_{k}\partial_{t}\left(\tilde{\phi}+\frac{1}{k}\right)^{m}\,\text{d}t\,\text{d}x+\int_{Q_{T}}f_{k}\left[u_{k}^{m}-\left(\tilde{\phi}+\frac{1}{k}\right)^{m}\right]\\\\[5.0pt]
+\int_{Q_{T}}\lambda
u_{k}^{q}\left(\tilde{\phi}+\frac{1}{k}\right)^{m}\,\text{d}t\,\text{d}x.$
(27)

After straightforward estimates we obtain that

$\int_{Q_{T}}|\nabla
u_{k}^{m}|^{2}\,\text{d}t\,\text{d}x<C^{\prime}(1+T)\Bigg{(}\left\|\left(\phi+\frac{1}{k}\right)^{m}\right\|_{C^{0,1}(S_{T})}^{2}+\left\|\left(\phi+\frac{1}{k}\right)^{m}\right\|_{C(S_{T})}\\\\[5.0pt]
+\left\|\left(\phi+\frac{1}{k}\right)^{m+q}\right\|_{C(S_{T})}+\left\|\phi+\frac{1}{k}\right\|_{C(S_{T})}^{2}+\left\|f+\frac{1}{k}\right\|_{L^{\infty}(Q_{T})}^{2}\Bigg{)},$
(28)

where $C^{\prime}>0$ is a constant independent of $k$ and $T$.

It follows that (a subsequence of) $\nabla u_{k}^{m}$ converges weakly in
$L^{2}(Q_{T})$ to a limit $U$. It is easy to see that $U=\nabla u^{m}$ in the
sense of distributions. Since $\tau_{Q_{T}}(u_{k}^{m})(t)\to\phi^{m}$ in
$L^{2}((0,T);H^{\frac{1}{2}}(\partial\Omega))$ by construction, and at the
same time $\tau_{Q_{T}}(u_{k}^{m})\rightharpoonup\tau_{Q_{T}}(u^{m})$, it
follows that

$\tau_{Q_{T}}(u^{m})=\phi^{m}.$ (29)

Therefore $u$ is a weak solution of (1) in $Q_{T}$.

###  2.2 Uniqueness of solutions

Before proving the energy estimate and the comparison principle part of
Theorem 1, we first need to prove the uniqueness of weak solutions. Once
uniqueness has been established, we can deduce the desired properties from the
corresponding ones that hold for $u_{k}$, since we would then know that each
weak solution can be obtained as a limit of such approximate soultions. The
following argument uses ideas from the proofs of [49, Theorem 6.5, Theorem
6.6] and from [1, Section 3].

Suppose $u_{1}$, $u_{2}$ are both weak solutions of (1) in $Q_{T}$, with
possibly different boundary data $\phi_{1}\leq\phi_{2}$ and source terms
$f_{1}$ and $f_{2}$. Then for any $\varphi\in C_{\diamond}(Q_{T})\cap
C^{\infty}(Q_{T})$ we have

$-\int_{Q_{T}}\epsilon\partial_{t}\varphi(u_{1}-u_{2})\,\text{d}t\,\text{d}x+\int_{Q_{T}}\lambda\varphi(u_{1}^{q}-u_{2}^{q})\,\text{d}t\,\text{d}x\\\\[5.0pt]
\leq\int_{Q_{T}}\nabla\cdot(\gamma\nabla\varphi)(u_{1}^{m}-u_{2}^{m})\,\text{d}t\,\text{d}x+\int_{Q_{T}}\varphi(f_{1}-f_{2})\,\text{d}t\,\text{d}x.$
(30)

Let

$a(t,x)=\left\\{\begin{array}[]{l}\frac{u_{1}^{m}(t,x)-u_{2}^{m}(t,x)}{u_{1}-u_{2}},\quad\text{if
}u_{1}(t,x)\neq u_{2}(t,x),\\\\[5.0pt] 0,\quad\text{if
}u_{1}(t,x)=u_{2}(t,x).\end{array}\right.$ (31)

The function $a$ is continuous, but may not be smooth. For
$k=(k_{1},k_{2})\in\mathbb{N}^{2}$, let $a_{k}\in C^{\infty}(Q_{T})$ be such
that

$\frac{1}{k_{1}}\leq a_{k}\leq k_{2},$ (32)

and $a_{k}\to a$.

Let $\theta\in C^{\infty}_{0}(Q_{T})$, $\theta\geq 0$ be an arbitrary function
and $\varphi_{k}$ be the unique smooth solution of the backwards in time
linear parabolic problem in $Q_{T}$

$\left\\{\begin{array}[]{l}\epsilon\partial_{t}\varphi_{k}+a_{k}\nabla\cdot(\gamma\nabla\varphi_{k})+\theta=0,\\\\[5.0pt]
\varphi_{k}(T,x)=0,\;\forall
x\in\Omega,\quad\varphi_{k}|_{S_{T}}=0.\end{array}\right.$ (33)

By the maximum principle we have that $\varphi_{k}\geq 0$. Using $\varphi_{k}$
as a test function in (30) we get

$\int_{Q_{T}}\theta(u_{1}-u_{2})\,\text{d}t\,\text{d}x+\int_{Q_{T}}\lambda\varphi_{k}(u_{1}^{q}-u_{2}^{q})\,\text{d}t\,\text{d}x\\\\[5.0pt]
\leq\int_{Q_{T}}(a-a_{k})\nabla\cdot(\gamma\nabla\varphi_{k})(u_{1}-u_{2})\,\text{d}t\,\text{d}x+\int_{Q_{T}}\varphi_{k}(f_{1}-f_{2})\,\text{d}t\,\text{d}x.$
(34)

Let

$J_{k}=\int_{Q_{T}}|u_{1}-u_{2}|\,|a-a_{k}|\,|\nabla\cdot(\gamma\nabla\varphi_{k})|\,\text{d}t\,\text{d}x$
(35)

and note immediately that

$J_{k}\leq\left(\int_{Q_{T}}a_{k}[\nabla\cdot(\gamma\nabla\varphi_{k})]^{2}\,\text{d}t\,\text{d}x\right)^{\frac{1}{2}}\left(\int_{Q_{T}}|u_{1}-u_{2}|^{2}\frac{|a-a_{k}|^{2}}{a_{k}}\,\text{d}t\,\text{d}x\right)^{\frac{1}{2}}.$
(36)

We will control each of the two factors on the right hand side separately.

Let $\zeta:[0,T]\to[\frac{1}{2},1]$ be a smooth function such that
$\zeta^{\prime}(t)\geq c>0$. We multiply (33) by
$\frac{\zeta}{\epsilon}\nabla\cdot(\gamma\nabla\varphi_{k})$ and integrate to
obtain

$\int_{Q_{T}}\partial_{t}\varphi_{k}\zeta\nabla\cdot(\gamma\nabla\varphi_{k})\,\text{d}t\,\text{d}x+\int_{Q_{T}}\frac{\zeta}{\epsilon}a_{k}[\nabla\cdot(\gamma\nabla\varphi_{k})]^{2}\,\text{d}t\,\text{d}x+\int_{Q_{T}}\zeta\frac{\theta}{\epsilon}\nabla\cdot(\gamma\nabla\varphi_{k})\,\text{d}t\,\text{d}x=0.$
(37)

We have that

$\int_{Q_{T}}\partial_{t}\varphi_{k}\zeta\nabla\cdot(\gamma\nabla\varphi_{k})\,\text{d}t\,\text{d}x=-\int_{Q_{T}}\zeta\gamma\nabla\varphi_{k}\cdot\nabla(\partial_{t}\varphi_{k})\,\text{d}t\,\text{d}x\\\\[5.0pt]
=-\frac{1}{2}\int_{Q_{T}}\zeta\gamma\partial_{t}(\nabla\varphi_{k})^{2}\,\text{d}t\,\text{d}x\geq\frac{1}{2}\int_{Q_{T}}\zeta^{\prime}\gamma(\nabla\varphi_{k})^{2}\,\text{d}t\,\text{d}x.$
(38)

Then

$\frac{1}{2}\int_{Q_{T}}\zeta^{\prime}\gamma|\nabla\varphi_{k}|^{2}\,\text{d}t\,\text{d}x+\int_{Q_{T}}\frac{\zeta}{\epsilon}a_{k}[\nabla\cdot(\gamma\nabla\varphi_{k})]^{2}\,\text{d}t\,\text{d}x\leq\int_{Q_{T}}\zeta\gamma\nabla\frac{\theta}{\epsilon}\cdot\nabla\varphi_{k}\,\text{d}t\,\text{d}x.$
(39)

Estimating

$\int_{Q_{T}}\zeta\gamma\nabla\frac{\theta}{\epsilon}\cdot\nabla\varphi_{k}\,\text{d}t\,\text{d}x\leq\frac{c}{4}\int_{Q_{T}}\gamma|\nabla\varphi_{k}|^{2}\,\text{d}t\,\text{d}x+\frac{1}{c}\int_{Q_{T}}\gamma|\nabla\frac{\theta}{\epsilon}|^{2}\,\text{d}t\,\text{d}x,$
(40)

we conclude that

$\int_{Q_{T}}\gamma|\nabla\varphi_{k}|^{2}\,\text{d}t\,\text{d}x+\int_{Q_{T}}a_{k}[\nabla\cdot(\gamma\nabla\varphi_{k})]^{2}\,\text{d}t\,\text{d}x\leq
C\int_{Q_{T}}\gamma|\nabla\frac{\theta}{\epsilon}|^{2}\,\text{d}t\,\text{d}x,$
(41)

with a constant $C>0$ which do not depend on $k$.

The estimate for the second factor on the right hand side of (36) relies only
on the strategy for approximating $a$ by $a_{k}$. There is no difference at
all between our case here and the argument in [49], so we quote the result

$\int_{Q_{T}}|u_{1}-u_{2}|^{2}\frac{|a-a_{k}|^{2}}{a_{k}}\,\text{d}t\,\text{d}x\leq\frac{C}{k_{1}}.$
(42)

It follows that in the limit $J_{k}\to 0$.

Let $t_{0}\in(0,T)$, $0\leq\psi(x)\leq 1$ be a smooth function, and
$\rho_{\eta}$, $\eta>0$, be a standard mollifier. We set

$\theta(t,x)=\epsilon(x)\psi(x)\rho_{\eta}(t-t_{0}).$ (43)

Then

$\varphi(t,x)=\int_{t}^{T}\rho_{\eta}(s-t_{0})\,\text{d}s$ (44)

is a supersolution for (33), so for all $k$ we have that
$\varphi_{k}\leq\varphi$.

Taking the limit in (34), we obtain

$\int_{Q_{T}}\epsilon\psi\rho_{\eta}(u_{1}-u_{2})\,\text{d}t\,\text{d}x\leq\int_{Q_{T}}\lambda(u_{2}^{q}-u_{1}^{q})_{+}\left(\int_{t}^{T}\rho_{\eta}(s-t_{0})\,\text{d}s\right)\,\text{d}t\,\text{d}x\\\\[5.0pt]
+\int_{Q_{T}}(f_{1}-f_{2})_{+}\left(\int_{t}^{T}\rho_{\eta}(s-t_{0})\,\text{d}s\right)\,\text{d}t\,\text{d}x.$
(45)

Taking the $\eta\to 0$ limit, we get

$\int_{\Omega}\epsilon\psi(u_{1}(t_{0})-u_{2}(t_{0}))_{+}\,\text{d}x\leq\int_{0}^{t_{0}}\int_{\Omega}\lambda(u_{2}^{q}-u_{1}^{q})_{+}\,\text{d}t\,\text{d}x+\int_{0}^{t_{0}}\int_{\Omega}(f_{1}-f_{2})_{+}\,\text{d}t\,\text{d}x,$
(46)

which easily gives that

$\int_{\Omega}(u_{1}(t_{0})-u_{2}(t_{0}))_{+}\,\text{d}x\leq
C\left(\int_{0}^{t_{0}}\int_{\Omega}(u_{2}^{q}-u_{1}^{q})_{+}\,\text{d}t\,\text{d}x+\int_{0}^{t_{0}}\int_{\Omega}(f_{1}-f_{2})_{+}\,\text{d}t\,\text{d}x\right).$
(47)

We need to distinguish two cases. The first is when $q\geq 1$. In this case we
can from the start assume that $\phi_{1}=\phi_{2}$ and $f_{1}=f_{2}$. Note
that both $u_{1}$ and $u_{2}$ are bounded, so there exists $M>0$ such that

$(u_{2}^{q}-u_{1}^{q})_{+}\leq M(u_{2}-u_{1})_{+}.$ (48)

Then

$\int_{\Omega}(u_{1}(t_{0})-u_{2}(t_{0}))_{+}\,\text{d}x\leq
C\int_{0}^{t_{0}}\int_{\Omega}(u_{2}(t)-u_{1}(t))_{+}\,\text{d}t\,\text{d}x\\\\[5.0pt]
\leq
C\int_{0}^{t_{0}}\int_{0}^{t}\int_{\Omega}(u_{1}(s)-u_{2}(s))_{+}\,\text{d}s\,\text{d}t\,\text{d}x\\\\[5.0pt]
\leq
Ct_{0}\int_{0}^{t_{0}}\int_{\Omega}(u_{1}(s)-u_{2}(s))_{+}\,\text{d}s\,\text{d}x.$
(49)

Gronwall’s inequality implies that

$\int_{\Omega}(u_{1}(t_{0})-u_{2}(t_{0}))_{+}\,\text{d}x=0,\quad\forall
t_{0}\in(0,T),$ (50)

and by exchanging the roles of $u_{1}$ and $u_{2}$ we conclude that
$u_{1}=u_{2}$ everywhere.

The second case is when $0<q<1$. Here we would like to take $u_{1}=\hat{u}$ to
be any weak solution of (1) in $Q_{T}$, with boundary data $\phi$ and source
term $f$, and $u_{2}$ to be the approximate solution $u_{k}$ constructed above
in the existence step. There is a slight difference to the assumtions above
since $u_{k}$ has positive initial data $\frac{1}{k}$. Going over the argument
with this in mind quickly shows that an additional term of the form

$\int_{\Omega}(-u_{k}(0,x))_{+}\,\text{d}x$ (51)

should appear on the right hand side of (47). Since this term is actually
zero, we may continue without modifying anything.

There is another modification we wish to make. Let $\omega>0$ and let
$u_{1}=e^{\omega t}\hat{u}$, $u_{2}=e^{\omega t}u_{k}$. It is easy to see that
$u_{1}$ satisfies the equation

$\partial_{t}u_{1}-\nabla\cdot(\gamma\nabla u_{1})+\lambda u_{1}^{q}-\omega
u_{1}=e^{\omega t}f$ (52)

in the weak sense, with a similar situation holding for $u_{2}$. If we apply
(47) to these choices of $u_{1}$ and $u_{2}$ we have

$e^{\omega t_{0}}\int_{\Omega}(\hat{u}(t_{0})-u_{k}(t_{0}))_{+}\leq
C\left(\int_{0}^{t_{0}}\int_{\Omega}e^{\omega
t}(u_{k}^{q}-\hat{u}^{q}-\omega(u_{k}-\hat{u}))_{+}\,\text{d}t\,\text{d}x\right.\\\\[5.0pt]
\left.+\int_{0}^{t_{0}}\int_{\Omega}e^{\omega
t}(f-f_{k})_{+}\,\text{d}t\,\text{d}x\right).$ (53)

The last term is zero. If

$u_{k}^{q}(t,x)-\hat{u}^{q}(t,x)-\omega(u_{k}(t,x)-\hat{u}(t,x))\geq 0,$ (54)

and $u_{k}(t,x)\geq\hat{u}(t,x)$, then we must have that

$\omega\leq\frac{u_{k}^{q}(t,x)-\hat{u}^{q}(t,x)}{u_{k}(t,x)-\hat{u}(t,x)}\leq(u_{k}(t,x)-\hat{u}(t,x))^{q-1}\leq
k^{1-q}.$ (55)

If we choose $\omega>k^{1-q}$, then we see that (54) can only hold when
$\hat{u}(t,x)\geq u_{k}(t,x)$. In this case we have

$e^{\omega t_{0}}\int_{\Omega}(\hat{u}(t_{0})-u_{k}(t_{0}))_{+}\leq
C\omega\int_{0}^{t_{0}}\int_{\Omega}e^{\omega
t}(\hat{u}-u_{k})_{+}\,\text{d}t\,\text{d}x.$ (56)

By Gronwall’s inequality we conclude that $\hat{u}\leq u_{k}$, and therefore
$\hat{u}\leq u$, where $u$ is the solution constructed in the previous
subsection.

Returning to (47) we get that

$\int_{\Omega}(u(t_{0})-\hat{u}(t_{0}))\,\text{d}x\leq
C\int_{0}^{t_{0}}\int_{\Omega}(\hat{u}^{q}-u^{q})_{+}\,\text{d}t\,\text{d}x=0,$
(57)

so $u=\hat{u}$.

###  2.3 Energy inequality and maximum principle

Once uniqueness of solutions has been established, we can derive properties of
the weak solutions of (1) from properties of the approximate solutions
$u_{k}$.

The energy inequality follows easily once we note that

$||\nabla u^{m}||_{L^{2}(Q_{T})}\leq\liminf_{k\to\infty}||\nabla
u_{k}^{m}||_{L^{2}(Q_{T})}\\\\[5.0pt] \leq
C(1+T)^{\frac{1}{2}}\left(||\phi^{m+q}||_{C^{0,1}(S_{T})}+||\phi||_{C(S_{T})}+||f||_{L^{\infty}(Q_{T})}\right).$
(58)

Regarding the comparison principle, suppose we have boundary data and sources
$\phi_{1}\leq\phi_{2}$, $f_{1}\leq f_{2}$, with corresponding solutions
$u_{1}$ and $u_{2}$. The approximate solutions, by the comparison principle
for non-degenerate quasilinear elliptic equations (see [37, Theorem 9.7]) must
satisfy

$u_{1,k}\leq u_{2,k},$ (59)

and this property is preserved in the limit. This concludes the proof of
Theorem 1.

##  3 The inverse problem with full data

In this section we give a proof of Theorem 2. We first reformulate the problem
by performing the change of function $v(t,x)=u^{m}(t,x)$. The new function $v$
satisfies

$\left\\{\begin{array}[]{l}\epsilon(x)\partial_{t}v(t,x)^{\frac{1}{m}}-\nabla\cdot(\gamma(x)\nabla
v(t,x))+\lambda v^{\frac{q}{m}}(t,x)=0,\\\\[5.0pt] v(0,x)=0,\quad
v|_{S_{T}}=f(t,x),\quad v\geq 0,\end{array}\right.$ (60)

where $f=\phi^{m}$.

The notion of weak solutions for (1) we have introduced above naturally
transforms into a notion of weak solutions for (60), in the space-time domain
$Q_{T}$. The Dirichlet-to-Neumann map $\Lambda^{PM}_{\epsilon,\gamma,\lambda}$
uniquely determines the Dirichlet-to-Neumann map

$\Lambda^{v}_{\epsilon,\gamma,\lambda}(f)=\gamma\partial_{\nu}v|_{S_{T}},$
(61)

associated to the equation (60). Here $f\in C(\overline{S_{T}})\cap
C^{0,1}(S_{T})$, $f\geq 0$, and $f(0,x)=0$ for all $x\in\partial\Omega$.

###  3.1 Time-integral transform and basic estimates

Let $T,\alpha>0$, $h>1$. We choose the boundary data to be of the form

$f(t,x)=t^{m}\,h\,g(x),\quad g\geq 0.$ (62)

Later we will fix $\alpha$ and $T$ to particular values, so we will not
emphasize in the following the dependence of various quantities on these. Let

$V(x)=\int_{0}^{T}(T-t)^{\alpha}v(t,x)\,\text{d}t.$ (63)

Then

$\nabla\cdot(\gamma\nabla V(x))={\mathcal{N}}_{t}(x)+{\mathcal{N}}_{a}(x),$
(64)

where

${\mathcal{N}}_{t}(x)=\epsilon(x)\alpha\int_{0}^{T}(T-t)^{\alpha-1}v^{\frac{1}{m}}(t,x)\,\text{d}t,$
(65)

and

${\mathcal{N}}_{a}(x)=\lambda(x)\int_{0}^{T}(T-t)^{\alpha}v^{\frac{q}{m}}(t,x)\,\text{d}t.$
(66)

Since

$\int_{0}^{T}(T-t)^{\alpha}t^{m}\,\text{d}t=T^{1+\alpha+m}\frac{\Gamma(1+\alpha)\Gamma(1+m)}{\Gamma(2+\alpha+m)},$
(67)

we have that

$\left\\{\begin{array}[]{l}\nabla\cdot(\gamma\nabla
V)={\mathcal{N}}_{t}+{\mathcal{N}}_{a},\\\\[5.0pt]
V|_{\partial\Omega}=hT^{1+\alpha+m}\frac{\Gamma(1+\alpha)\Gamma(1+m)}{\Gamma(2+\alpha+m)}g.\end{array}\right.$
(68)

In connection to this equation we introduce the Dirichlet-to-Neumann map

$\Lambda^{h}_{\epsilon,\gamma,\lambda}(g)=h^{-1}\gamma\partial_{\nu}V|_{\partial\Omega},$
(69)

which is determined by $\Lambda^{v}_{\epsilon,\gamma,\lambda}$ and hence also
by $\Lambda^{PM}_{\epsilon,\gamma,\lambda}$.

By Hölder’s inequality we have

$0\leq{\mathcal{N}}_{t}(x)=\epsilon(x)\alpha\int_{0}^{T}(T-t)^{\frac{\alpha-m^{\prime}}{m^{\prime}}}\left[(T-t)^{\alpha}v\right]^{\frac{1}{m}}\,\text{d}t\\\\[5.0pt]
\leq\epsilon(x)\alpha\left(\int_{0}^{T}(T-t)^{\alpha-m^{\prime}}\,\text{d}t\right)^{\frac{1}{m^{\prime}}}\left(V(x)\right)^{\frac{1}{m}}\\\\[5.0pt]
=T^{\frac{\alpha}{m^{\prime}}-\frac{1}{m}}\frac{\alpha}{(\alpha-m^{\prime}+1)^{\frac{1}{m^{\prime}}}}\epsilon(x)\left(V(x)\right)^{\frac{1}{m}}.$
(70)

Similarly

$0\leq{\mathcal{N}}_{a}(x)\leq
T^{\alpha+\frac{m}{m-q}}\left(\alpha\left(1-\frac{q}{m}\right)+1\right)^{-\frac{m}{m-q}}\lambda(x)(V(x))^{\frac{q}{m}}.$
(71)

Note that the above can only work if $\alpha>m^{\prime}-1=\frac{1}{m-1}$,
which we will assume from now on.

Then (choose $p>n$) we get

$||{\mathcal{N}}_{t}||_{L^{\infty}(\Omega)}\leq
CT^{\frac{\alpha}{m^{\prime}}-\frac{1}{m}}||V||_{W^{1,p}(\Omega)}^{\frac{1}{m}}$
(72)

and

$||{\mathcal{N}}_{a}||_{L^{\infty}(\Omega)}\leq
CT^{\alpha+\frac{m}{m-q}}||V||_{W^{1,p}(\Omega)}^{\frac{q}{m}}.$ (73)

By elliptic estimates we have

$||V||_{W^{2,p}(\Omega)}\leq
C\bigg{(}hT^{1+\alpha+m}||g||_{W^{2-\frac{1}{p},p}(\Omega)}+T^{\frac{\alpha}{m^{\prime}}-\frac{1}{m}}||V||_{W^{1,p}(\Omega)}^{\frac{1}{m}}\\\\[5.0pt]
+T^{\alpha+\frac{m}{m-q}}||V||_{W^{1,p}(\Omega)}^{\frac{q}{m}}\bigg{)},$ (74)

so, since $h>1$,

$\max(||V||_{W^{2,p}(\Omega)},h)\\\\[5.0pt] \leq
C\bigg{(}hT^{1+\alpha+m}||g||_{W^{2-\frac{1}{p},p}(\Omega)}+h+T^{\frac{\alpha}{m^{\prime}}-\frac{1}{m}}\left[\max(||V||_{W^{1,p}(\Omega)},h)\right]^{\frac{1}{m}}\\\\[5.0pt]
+T^{\alpha+\frac{m}{m-q}}\left[\max(||V||_{W^{1,p}(\Omega)},h)\right]^{\frac{q}{m}}\bigg{)}\\\\[5.0pt]
\leq
C\bigg{(}hT^{1+\alpha+m}||g||_{W^{2-\frac{1}{p},p}(\Omega)}+h+(T^{\frac{\alpha}{m^{\prime}}-\frac{1}{m}}+T^{\alpha+\frac{m}{m-q}})\left[\max(||V||_{W^{1,p}(\Omega)},h)\right]\bigg{)}.$
(75)

Since $\frac{\alpha}{m^{\prime}}-\frac{1}{m},\alpha+\frac{m}{m-q}>0$, we will
choose $T$ small enough to be able to absorb the
$T^{\frac{\alpha}{m^{\prime}}-\frac{1}{m}}+T^{\alpha+\frac{m}{m-q}}$ term into
the left hand side. Then, supressing explicit dependence on $T$, we have

$||V||_{W^{2,p}(\Omega)}\leq
hC\left(||g||_{W^{2-\frac{1}{p},p}(\Omega)}+1\right).$ (76)

###  3.2 Asymptotic expansion to first order in $h$

We make the Ansatz

$V(x)=h\frac{T^{1+\alpha+m}\Gamma(1+\alpha)\Gamma(1+m)}{\Gamma(2+\alpha+m)}V_{0}(x)+R_{1}(x),$
(77)

where

$\left\\{\begin{array}[]{l}\nabla\cdot(\gamma\nabla V_{0})=0,\\\\[5.0pt]
V_{0}|_{\partial\Omega}=g.\end{array}\right.$ (78)

Then $R_{1}$ must satisfy

$\left\\{\begin{array}[]{l}\nabla\cdot(\gamma\nabla
R_{1})={\mathcal{N}}_{t}+{\mathcal{N}}_{a},\\\\[5.0pt]
R_{1}|_{\partial\Omega}=0.\end{array}\right.$ (79)

Let $\sigma=\frac{1}{m}\max(1,q)<1$. We have the estimate

$||R_{1}||_{W^{2,p}(\Omega)}\leq
Ch^{\sigma}\left(||g||_{W^{2-\frac{1}{p},p}(\Omega)}+1\right)^{\sigma}.$ (80)

The consequence for the Dirichlet-to-Neumann map is that as $h\to\infty$

$\Lambda^{h}_{\epsilon,\gamma,\lambda}(g)=\frac{T^{1+\alpha+m}\Gamma(1+\alpha)\Gamma(1+m)}{\Gamma(2+\alpha+m)}\gamma\partial_{\nu}V_{0}|_{\partial\Omega}+\mathscr{O}(h^{1-\sigma}).$
(81)

It follows that the DN map

$\Lambda_{\gamma}(g)=\gamma\partial_{\nu}V_{0}|_{\partial\Omega},$ (82)

which is the DN map for the original Calderón problem, is determined by
$\Lambda^{h}_{\epsilon,\gamma}$. By [39] in the $n=2$ case or [48] in the
$n\geq 3$ case, we now have uniqueness for $\gamma$.

Before moving on, note that by the maximum/minimum principle we have
$V_{0}\geq 0$ and $R_{1}\leq 0$. For our convenience below we introduce here

$v_{0}(t,x)=ht^{m}V_{0}(x),$ (83)

which is such that

$\int_{0}^{T}(T-t)^{\alpha}v_{0}(t,x)\,\text{d}t=h\frac{T^{1+\alpha+m}\Gamma(1+\alpha)\Gamma(1+m)}{\Gamma(2+\alpha+m)}V_{0}(x).$
(84)

###  3.3 Asymptotic expansion to second order in $h$

Here we refine the Ansatz for $V$ to

$V(x)=hV_{0}(x)+h^{\frac{1}{m}}V_{t}(x)+h^{\frac{q}{m}}V_{a}(x)+R_{2}(x),$
(85)

where $V_{0}$ is as above and

$\left\\{\begin{array}[]{l}\nabla\cdot(\gamma\nabla
V_{t})=h^{-\frac{1}{m}}{\mathcal{N}}_{0t},\\\\[5.0pt]
V_{t}|_{\partial\Omega}=0,\end{array}\right.$ (86)
$\left\\{\begin{array}[]{l}\nabla\cdot(\gamma\nabla
V_{a})=h^{-\frac{1}{m}}{\mathcal{N}}_{0a},\\\\[5.0pt]
V_{a}|_{\partial\Omega}=0,\end{array}\right.$ (87)

with

${\mathcal{N}}_{0t}(x)=\alpha\epsilon(x)\int_{0}^{T}(T-t)^{\alpha-1}v_{0}^{\frac{1}{m}}(t,x)\,\text{d}t=h^{\frac{1}{m}}\epsilon(x)\frac{T^{\alpha+1}}{\alpha+1}V_{0}^{\frac{1}{m}}(x)$
(88)

and

${\mathcal{N}}_{0a}(x)=\lambda(x)\int_{0}^{T}(T-t)^{\alpha}v_{0}^{\frac{q}{m}}(t,x)\,\text{d}t=h^{\frac{q}{m}}\lambda(x)\frac{T^{\alpha+1+q}}{\alpha+1+q}V_{0}^{\frac{q}{m}}(x).$
(89)

Clearly, $V_{t}$ and $V_{a}$ are independent of $h$ and, by the maximum
principle, $V_{t},V_{a}\leq 0$.

The remainder term $R_{2}$ must satisfy

$\left\\{\begin{array}[]{l}\nabla\cdot(\gamma\nabla
R_{2})=({\mathcal{N}}_{t}-{\mathcal{N}}_{0t})+({\mathcal{N}}_{a}-{\mathcal{N}}_{0a}),\\\\[5.0pt]
R_{2}|_{\partial\Omega}=0.\end{array}\right.$ (90)

Note that

$\epsilon\partial_{t}v_{0}^{\frac{1}{m}}-\nabla\cdot(\gamma\nabla
v_{0})=h^{\frac{1}{m}}\epsilon V_{0}^{\frac{1}{m}}(x)\geq 0,\quad
v_{0}|_{S_{T}}=v|_{S_{T}},$ (91)

so $v_{0}$ is a supersolution and therefore we have that $v_{0}\geq v$. It
follows then that

${\mathcal{N}}_{0t}\geq{\mathcal{N}}_{t},\quad{\mathcal{N}}_{0a}\geq{\mathcal{N}}_{a}.$
(92)

Using the same Hölder inequality trick applied above

$0\leq{\mathcal{N}}_{0t}-{\mathcal{N}}_{t}=\alpha\epsilon\int_{0}^{T}(T-t)^{\alpha-1}(v_{0}^{\frac{1}{m}}-v^{\frac{1}{m}})\,\text{d}t\\\\[5.0pt]
\leq\alpha\epsilon\int_{0}^{T}(T-t)^{\alpha-1}(v_{0}-v)^{\frac{1}{m}}\,\text{d}t\\\\[5.0pt]
\leq
T^{\frac{\alpha}{m^{\prime}}-\frac{1}{m}}\frac{\alpha}{(\alpha-m^{\prime}+1)^{\frac{1}{m^{\prime}}}}\epsilon(x)\left(h\frac{T^{1+\alpha+m}\Gamma(1+\alpha)\Gamma(1+m)}{\Gamma(2+\alpha+m)}V_{0}(x)-V(h,x)\right)^{\frac{1}{m}}\\\\[5.0pt]
=T^{\frac{\alpha}{m^{\prime}}-\frac{1}{m}}\frac{\alpha}{(\alpha-m^{\prime}+1)^{\frac{1}{m^{\prime}}}}\epsilon(x)(-R_{1}(x))^{\frac{1}{m}}.$
(93)

Similarly

$0\leq{\mathcal{N}}_{0a}-{\mathcal{N}}_{a}\leq
T^{\alpha+\frac{m}{m-q}}\left(\alpha\left(1-\frac{q}{m}\right)+1\right)^{-\frac{m}{m-q}}\lambda(x)(-R_{1}(x))^{\frac{q}{m}}$
(94)

It follows that

$||({\mathcal{N}}_{t}-{\mathcal{N}}_{0t})+({\mathcal{N}}_{a}-{\mathcal{N}}_{0a})||_{L^{\infty}(\Omega)}=\mathscr{O}(h^{\sigma^{2}}),$
(95)

so, by elliptic estimates we have

$||R_{2}||_{W^{2,p}(\Omega)}=\mathscr{O}(h^{\sigma^{2}}).$ (96)

The Dirichlet-to-Neumann map then has the following expansion as $h\to\infty$

$\Lambda^{h}_{\epsilon,\gamma,\lambda}(g)=\frac{T^{1+\alpha+m}\Gamma(1+\alpha)\Gamma(1+m)}{\Gamma(2+\alpha+m)}\gamma\partial_{\nu}V_{0}|_{\partial\Omega}\\\\[5.0pt]
+h^{\frac{1}{m}-1}\gamma\partial_{\nu}V_{t}|_{\partial\Omega}+h^{\frac{q}{m}-1}\gamma\partial_{\nu}V_{a}|_{\partial\Omega}+\mathscr{O}(h^{\sigma^{2}-1}).$
(97)

Assume for now that $q\neq 1$. From our assumptions of the range of allowed
values for $q$, we have that

$\frac{1}{m}-1,\frac{q}{m}-1>\sigma^{2}-1.$ (98)

In this case it follows that the Neumann data
$\gamma\partial_{\nu}V_{t}|_{\partial\Omega}$,
$\gamma\partial_{\nu}V_{a}|_{\partial\Omega}$ are each determined individually
by $\Lambda^{PM}_{\epsilon,\gamma,\lambda}$.

Let $W$ be any smooth solution to $\nabla\cdot(\gamma\nabla W)=0$. We have
that

$\langle\gamma\partial_{\nu}V_{t}|_{\partial\Omega},W|_{\partial\Omega}\rangle=\int_{\Omega}\gamma\nabla
V_{1}\cdot\nabla
W\,\text{d}x+\frac{T^{\alpha+1}}{\alpha+1}\int_{\Omega}\epsilon
V_{0}^{\frac{1}{m}}W\,\text{d}x\\\\[5.0pt]
=\frac{T^{\alpha+1}}{\alpha+1}\int_{\Omega}\epsilon
V_{0}^{\frac{1}{m}}W\,\text{d}x.$ (99)

We can take $V_{0}$ to be of the form

$V_{0}(x)=1+sH(x),\quad\nabla\cdot(\gamma\nabla H)=0,\;H>0.$ (100)

Then

$m\left.\frac{\,\text{d}}{\,\text{d}s}\right|_{s=0}\int_{\Omega}\epsilon\left(1+sH(x)\right)^{\frac{1}{m}}W\,\text{d}x=\int_{\Omega}\epsilon
HW\,\text{d}x$ (101)

is determined by $\Lambda^{PM}_{\epsilon,\gamma,\lambda}$. From here it is
easy to see that we must have

$\int_{\Omega}\left(\epsilon^{(i)}-\epsilon^{(ii)}\right)UW\,\text{d}x=0$
(102)

for all $U,W$ such that $\nabla\cdot(\gamma\nabla U)=\nabla\cdot(\gamma\nabla
W)=0$. As shown in [6] in the $n=2$ case and in [48] in the $n\geq 3$ case,
this implies that $\epsilon^{(i)}=\epsilon^{(ii)}$.

Similarly

$\langle\gamma\partial_{\nu}V_{t}|_{\partial\Omega},W|_{\partial\Omega}\rangle=\frac{T^{\alpha+1+q}}{\alpha+1+q}\int_{\Omega}\lambda
V_{0}^{\frac{q}{m}}W\,\text{d}x$ (103)

and an identical argument gives that $\lambda^{(i)}=\lambda^{(ii)}$.

It remains to consider the case $q=1$. In this situation, we have that
$\Lambda^{PM}_{\epsilon,\gamma,\lambda}$ determines all quantities of the form

$\langle\gamma\partial_{\nu}(V_{t}+V_{a})|_{\partial\Omega},W|_{\partial\Omega}\rangle=\int_{\Omega}V_{0}^{\frac{1}{m}}W\left(\frac{T^{\alpha+1}}{\alpha+1}\epsilon+\frac{T^{\alpha+2}}{\alpha+2}\lambda\right)\,\text{d}x.$
(104)

We can use our ability to continuously shrink the observation time $T$ in
order to determine the $\epsilon$ and $\lambda$ terms individually. This
concludes the proof.

##  4 The inverse problem with partial data

In this section we give a proof of Theorem 3. Most of the argument of the
previous section still holds. In particular, we still have the expansion (97),
which in the same way as above implies that

$\int_{\Omega}(\epsilon^{(i)}-\epsilon^{(ii)})V^{\frac{1}{m}}W\,\text{d}x=\int_{\Omega}(\lambda^{(i)}-\lambda^{(ii)})V^{\frac{q}{m}}W\,\text{d}x=0,$
(105)

for any $V$, $W$ which satisfy $\triangle V=\triangle W=0$,
$V|_{\Gamma}=W|_{\Gamma}=0$, where
$\Gamma=\partial\Omega\setminus\overline{\Sigma}$. In what follows we will
focus on the integral identity for the $\epsilon$ coefficients, the case of
$\lambda$ being nearly identical.

######  Lemma 1.

There exists $U_{0}\in C^{\infty}(\overline{\Omega})$ such that $U_{0}\geq 0$,
$\triangle U_{0}=0$, $U_{0}|_{\Gamma}=0$, and
$\Gamma\subset\mbox{supp\;}(\partial_{\nu}U_{0}|_{\partial\Omega})$.

###### Proof.

If $f\in C^{\infty}(\partial\Omega)$ is such that $f\geq 0$, $f|_{\Gamma}=0$,
let $u_{f}\in C^{\infty}(\overline{\Omega})$ be the solution of

$\left\\{\begin{array}[]{l}\triangle u_{f}=0,\\\\[5.0pt]
u_{f}|_{\partial\Omega}=f.\end{array}\right.$ (106)

Suppose there exists a point $x_{0}\in\Gamma$ such that

$\partial_{\nu}u_{f}(x_{0})=0,\quad\forall f.$ (107)

Let $G_{\Omega}(x,y)$ be the (Dirichlet) Green’s function associated to the
domain $\Omega$. We have that $G_{\Omega}(x,\cdot)\in
C^{\infty}(\overline{\Omega}\setminus\\{x\\})$ and

$u_{f}(x)=-\int_{\partial\Omega}f(y)\nu(y)\cdot\nabla_{y}G_{\Omega}(x,y)\,\text{d}S(y).$
(108)

We then get

$\partial_{\nu}u_{f}(x_{0})=-\int_{\partial\Omega}f(y)\nu(x_{0})\cdot\nabla_{x}\left(\nu(y)\cdot\nabla_{y}G_{\Omega}(x_{0},y)\right)\,\text{d}S(y)$
(109)

By (107) we conclude that

$\nu(y)\cdot\nabla_{y}\left(\nu(x_{0})\cdot\nabla_{x}G_{\Omega}(x_{0},y)\right)=0,\quad\forall
y\in\Sigma.$ (110)

Note that we also have

$\left\\{\begin{array}[]{l}\triangle_{y}\left(\nu(x_{0})\cdot\nabla_{x}G_{\Omega}(x_{0},y)\right)=0,\\\\[5.0pt]
\nu(x_{0})\cdot\nabla_{x}G_{\Omega}(x_{0},y)=0,\quad\forall
y\in\Sigma.\end{array}\right.$ (111)

By unique continuation

$\nu(x_{0})\cdot\nabla_{x}G_{\Omega}(x_{0},y)=0,\quad\forall y\in\Omega.$
(112)

This is a contradiction (e.g. since
$\nu(x_{0})\cdot\nabla_{x}G_{\Omega}(x_{0},y)\to\infty$ and $y\to x_{0}$).

We have shown that for any $x\in\Gamma$ there exists $f_{x}\in
C^{\infty}(\partial\Omega)$ such that $f_{x}\geq 0$, $f_{x}|_{\Gamma}=0$, and
$\partial_{\nu}u_{f_{x}}(x)<0$. As $\overline{\Gamma}$ is compact, we can find
$f_{1},\ldots,f_{N}\in C^{\infty}(\partial\Omega)$ such that $f_{j}\geq 0$,
$f_{j}|_{\Gamma}=0$, for $j=1,\ldots,N$, and

$\partial_{\nu}u_{f}|_{\overline{\Gamma}}<0,\quad f=\sum_{j=1}^{N}f_{j}.$
(113)

∎

If we choose $V(x)=U_{0}(x)+sU(x)$ in (105), where $U\geq 0$, $\triangle U=0$
and $U|_{\Gamma}=0$, then

$0=\left.\frac{\,\text{d}}{\,\text{d}s}\right|_{s=0}\int_{\Omega}(\epsilon^{(i)}(x)-\epsilon^{(ii)}(x))(U_{0}(x)+sU(x))^{\frac{1}{m}}W(x)\,\text{d}x\\\\[5.0pt]
=\int_{\Omega}(\epsilon^{(i)}(x)-\epsilon^{(ii)}(x))U_{0}^{\frac{1}{m}-1}(x)U(x)W(x)\,\text{d}x.$
(114)

Note that it is possible to differentiate in $s$ since
$U_{0}^{\frac{1}{m}-1}\in L^{1}(\Omega)$. We can move from non-negative $U$,
$W$ to complex valued ones in the obvious way using linearity. With the
notation

$F(x)=(\epsilon^{(i)}(x)-\epsilon^{(ii)}(x))U_{0}^{\frac{1}{m}-1}(x)$ (115)

we then have

$\int_{\Omega}F(x)U(x)W(x)\,\text{d}x=0$ (116)

for all functions $U$, $W$ harmonic in $\Omega$ and such that
$U|_{\Gamma}=W|_{\Gamma}=0$. A similar problem, with $F\in
L^{\infty}(\Omega)$, was considered in [17]. We will adapt their argument to
our case where $F\in L^{1}(\Omega)$.

It is observed in [17, Section 3] that, without loss of generality, we may
assume that

$\Omega\subset\\{x\in\mathbb{R}^{n}:|x+e_{1}|<1\\},\quad\Gamma=\\{x\in\partial\Omega:x_{1}\leq-2c\\},$
(117)

where $c>0$ is a positive constant, and that $0\in\partial\Omega$. This is
proven by checking that the structure of the integral identity (116) remains
unchanged under an appropriately chosen conformal transformation. The same
holds here, and under the above mentioned transformation we still retain the
properties that $F\in L^{1}(\Omega)\cap C^{\infty}(\Omega)$ and that $F(x)$
only becomes unbounded as $x\to\Gamma$. For any vector $z\in\mathbb{C}^{n}$ we
will use the notation

$z=(z_{1},z^{\prime}),\quad z^{\prime}\in\mathbb{C}^{n-1}.$ (118)

###  4.1 A local result

Let $\zeta\in\mathbb{C}^{n}$ be such that $\zeta^{2}=0$ and let $\chi\in
C_{0}^{\infty}(\mathbb{R}^{n})$ be such that
$\mbox{supp\;}(\chi)\subset\\{x\in\mathbb{R}^{n}:x_{1}<-c\\}$ and
$\chi|_{\Gamma}=1$. We can construct harmonic functions

$U(x,\zeta,h)=e^{-\frac{i}{h}x\cdot\zeta}+R(x,\zeta,h),$ (119)

where $R$ solves

$\left\\{\begin{array}[]{l}\triangle R=0,\\\\[5.0pt]
R|_{\partial\Omega}=-(e^{-\frac{i}{h}x\cdot\zeta}\chi)|_{\partial\Omega},\end{array}\right.$
(120)

and $h$ is a positive parameter. In [30, Section 2] it is shown that there
exists $C>0$ so that the remainder terms satisfy the estimate

$||R(\cdot,\zeta,h)||_{C(\overline{\Omega})}\leq
C\left(1+\frac{|\zeta|^{k}}{h^{k}}\right)e^{-\frac{c}{h}\mathfrak{Im}\;\zeta_{1}}e^{\frac{1}{h}|\mathfrak{Im}\;\zeta^{\prime}|},$
(121)

whenever $\mathfrak{Im}\;\zeta_{1}\geq 0$.

Let $\zeta,\eta\in C^{n}$ both be such that $\zeta^{2}=\eta^{2}=0$,
$\mathfrak{Im}\;\zeta_{1},\mathfrak{Im}\;\eta_{1}\geq 0$. Choosing

$U(x)=U(x,\zeta,h),\quad W(x)=U(x,\eta,h)$ (122)

in (116) and using (121) to control the terms involving remainder terms we get
that

$\left|\int_{\Omega}e^{-\frac{i}{h}x\cdot(\zeta+\eta)}F(x)\,\text{d}x\right|\\\\[5.0pt]
\leq
C||F||_{L^{1}(\Omega)}\left(1+\frac{|\zeta|^{k}}{h^{k}}\right)\left(1+\frac{|\eta|^{k}}{h^{k}}\right)e^{-\frac{c}{h}\min(\mathfrak{Im}\;\zeta_{1},\mathfrak{Im}\;\eta_{1})}e^{\frac{2}{h}(|\mathfrak{Im}\;\zeta^{\prime}|+|\mathfrak{Im}\;\eta^{\prime}|)}.$
(123)

Let $\sigma=ie_{1}+e_{2}\in\mathbb{C}^{n}$ and suppose $a\in(0,\infty)$. If
$z\in\mathbb{C}^{n}$ is such that $|z-2iae_{1}|<2\varepsilon$ then (see [17,
Section 2]) there exist $\zeta,\eta\in C^{n}$, with $\zeta^{2}=\eta^{2}=0$,
such that

$z=\zeta+\eta,\quad|\zeta-a\sigma|<Ca\varepsilon,\quad|\eta+a\overline{\sigma}|<Ca\varepsilon,$
(124)

if $\varepsilon>0$ is sufficiently small. It follows that for such a $z$ we
have

$\left|\int_{\Omega}e^{-\frac{i}{h}x\cdot z}F(x)\,\text{d}x\right|\leq
C\left(\frac{a}{h}\right)^{2k}||f||_{L^{1}(\Omega)}e^{-\frac{ca}{2h}}e^{\frac{2C\varepsilon
a}{h}}.$ (125)

Let $\mathcal{T}$ denote the Segal-Bargmann transform

$\mathcal{T}F(z)=\int_{\Omega}e^{-\frac{1}{2h}(z-y)^{2}}F(y)\,\text{d}y,\quad
z\in\mathbb{C}^{n}.$ (126)

Note that

$(z-y)^{2}=(\mathfrak{Re}\;z-y)^{2}-(\mathfrak{Im}\;z)^{2}+2i(\mathfrak{Re}\;z-y)\cdot\mathfrak{Im}\;z,$
(127)

so

$|\mathcal{T}F(z)|\leq
e^{\frac{1}{2h}|\mathfrak{Im}\;z|^{2}}||F||_{L^{1}(\Omega)}.$ (128)

When $\mathfrak{Re}\;z_{1}\geq 0$, since
$\Omega\subset\\{x\in\mathbb{R}^{n}:x_{1}\leq 0\\}$, we have the sharper
estimate

$|\mathcal{T}F(z)|\leq
e^{\frac{1}{2h}(|\mathfrak{Im}\;z|^{2}-|\mathfrak{Re}\;z_{1}|^{2})}||F||_{L^{1}(\Omega)}.$
(129)

It is observed in [17, Section 4] that the Segal-Bargman transform can also be
written as

$\mathcal{T}F(z)=(2\pi
h)^{-\frac{n}{2}}\int_{\mathbb{R}^{n}\times\Omega}e^{-\frac{1}{2h}(z^{2}+t^{2})}e^{-\frac{i}{h}y\cdot(t+iz)}F(y)\,\text{d}t\,\text{d}y.$
(130)

Then when $\mathfrak{Re}\;z_{1}\geq 0$,

$|\mathcal{T}F(z)|\leq(2\pi
h)^{-\frac{n}{2}}\int_{\mathbb{R}^{n}}e^{\frac{1}{2h}(|\mathfrak{Im}\;z|^{2}-|\mathfrak{Re}\;z|^{2}-t^{2})}\left|\int_{\Omega}e^{-\frac{i}{h}y\cdot(t+iz)}F(y)\,\text{d}y\right|\,\text{d}t\\\\[5.0pt]
\leq(2\pi
h)^{-\frac{n}{2}}e^{\frac{1}{2h}(|\mathfrak{Im}\;z|^{2}-|\mathfrak{Re}\;z|^{2})}\left(\int_{|t|\leq\varepsilon
a}e^{-\frac{t^{2}}{2h}}\left|\int_{\Omega}e^{-\frac{i}{h}y\cdot(t+iz)}F(y)\,\text{d}y\right|\,\text{d}t\right.\\\\[5.0pt]
+\left.\int_{|t|\geq\varepsilon
a}e^{-\frac{t^{2}}{2h}}\left|\int_{\Omega}e^{-\frac{i}{h}y\cdot(t+iz)}F(y)\,\text{d}y\right|\right)\\\\[5.0pt]
\leq
e^{\frac{1}{2h}(|\mathfrak{Im}\;z|^{2}-|\mathfrak{Re}\;z|^{2})}\left(\sup_{|t|\leq
a\varepsilon}\left|\int_{\Omega}e^{-\frac{i}{h}y\cdot(t+iz)}F(y)\,\text{d}y\right|\right.\\\\[5.0pt]
+\left.\sqrt{2}e^{\frac{1}{h}|\mathfrak{Re}\;z^{\prime}|}e^{-\frac{\varepsilon^{2}a^{2}}{4h}}\int_{\Omega}|F(y)|\,\text{d}y\right).$
(131)

When $|z-2ae_{1}|<\varepsilon a$, by (125), the above gives

$|\mathcal{T}F(z)|\leq
C\left(\frac{a}{h}\right)^{2k}||f||_{L^{1}(\Omega)}e^{\frac{1}{2h}(|\mathfrak{Im}\;z|^{2}-|\mathfrak{Re}\;z|^{2})}\left(e^{-\frac{ca}{2h}}e^{\frac{2C\varepsilon
a}{h}}+e^{-\frac{\varepsilon^{2}a^{2}}{4h}}e^{\frac{\varepsilon
a}{h}}\right).$ (132)

We are free to choose the parameters $\varepsilon$ and $a$, and with
$\varepsilon$ sufficiently small and $a$ sufficiently large we have

$|\mathcal{T}F(z)|\leq
Ch^{-2k}||F||_{L^{1}(\Omega)}e^{\frac{1}{2h}(|\mathfrak{Im}\;z|^{2}-|\mathfrak{Re}\;z|^{2}-\frac{ca}{2})}.$
(133)

Let

$\Phi(z_{1})=\left\\{\begin{array}[]{ll}|\mathfrak{Im}\;z_{1}|^{2},&\text{for
}\mathfrak{Re}\;z_{1}\leq 0\\\
|\mathfrak{Im}\;z_{1}|^{2}-|\mathfrak{Re}\;z_{1}|^{2},&\text{for
}\mathfrak{Re}\;z_{1}\geq 0.\end{array}\right.$ (134)

For $z_{1}\in\mathbb{C}$, $x^{\prime}\in\mathbb{R}^{n-1}$, the above estimates
can be combined into

$e^{-\frac{1}{2h}\Phi(z_{1})}|\mathcal{T}F(z_{1},x^{\prime})|\\\\[5.0pt] \leq
Ch^{-2k}||F||_{L^{1}(\Omega)}\left\\{\begin{array}[]{ll}1,&\text{for
}z_{1}\in\mathbb{C},\\\\[5.0pt] e^{-\frac{ca}{4h}},&\text{for
}|z_{1}-2a|\leq\frac{\varepsilon a}{2},|x^{\prime}|<\frac{\varepsilon
a}{2}.\end{array}\right.$ (135)

This is analogous to the estimate [17, equation (4.7)]. As in the original
proof we can now apply [17, Lemma 4.1] to conclude that there exist
$c^{\prime},\delta>0$ such that

$|\mathcal{T}F(x|\leq
Ch^{-2k}||F||_{L^{1}(\Omega)}e^{-\frac{c^{\prime}}{2h}},$ (136)

whenever $x\in\Omega\cap\\{x\in\mathbb{R}^{n}:|x_{1}|\leq\delta\\}$. For such
an $x$ let $\varphi\in C_{0}^{\infty}(\Omega)$ be such that $\varphi(y)=1$ for
$y$ in a small neighborhood of $x$. Then we have that as $h\to 0$

$(2\pi
h)^{-\frac{n}{2}}\int_{\Omega}e^{-\frac{1}{2h}(x-y)^{2}}\varphi(y)F(y)\,\text{d}y\to\varphi(x)F(x)=F(x),$
(137)

and by the dominated convergence theorem

$(2\pi
h)^{-\frac{n}{2}}\int_{\Omega}e^{-\frac{1}{2h}(x-y)^{2}}(1-\varphi(y))F(y)\,\text{d}y\to
0.$ (138)

It follows that $F(x)=0$ for any
$x\in\Omega\cap\\{x\in\mathbb{R}^{n}:|x_{1}|\leq\delta\\}$.

###  4.2 The global result

Suppose $\Omega_{1}\subset\Omega_{2}\subset\mathbb{R}^{n}$ are two bounded
smooth domains such that $\Omega_{2}\setminus\overline{\Omega_{1}}$ is not
empty. We further assume that $\partial\Omega_{1}\cap\partial\Omega_{2}$ is an
open subset of $\partial\Omega_{1}$ and it has $C^{\infty}$ boundary. The
following lemma follows easily, via the Sobolev embedding theorem, from a
result in [30].

######  Lemma 2 (see [30, Lemma 2.6]).

The space

$\left\\{\int_{\Omega_{2}}G_{\Omega_{2}}(\cdot,y)a(y)\,\text{d}y:a\in
C^{\infty}(\overline{\Omega_{2}}),\mbox{supp\;}a\subset\Omega_{2}\setminus\overline{\Omega_{1}}\right\\}$
(139)

is dense in

$\left\\{U\in C^{\infty}(\overline{\Omega_{1}}):\triangle
U=0,U|_{\partial\Omega_{1}\cap\partial\Omega_{2}}=0\right\\},$ (140)

in the $L^{\infty}(\Omega_{1})$ topology.

Let $x_{1}\in\Omega$ and $x_{0}\in\Sigma$. Let
$\theta:[0,1]\to\overline{\Omega}$ be a $C^{1}$ curve connecting
$x_{0}=\theta(0)$ to $x_{1}=\theta(1)$, such that $\theta((0,1])\subset\Omega$
and $\theta^{\prime}(0)$ is the interior normal to $\partial\Omega$ at
$x_{0}$. Let

$\Theta_{\varepsilon}(t)=\\{x\in\overline{\Omega}:dist(x,\theta([0,t]))\leq\varepsilon\\},\quad
t\in[0,1],$ (141)

and let

$I=\left\\{t\in[0,1]:F|_{\Theta_{\varepsilon}(t)\cap\Omega}=0\right\\}.$ (142)

Note that $I$ is a closed subset of $[0,1]$. If $\varepsilon>0$ is small
enough, the previous subsection implies that $I$ is not empty.

Suppose $t\in I$. Clearly then $[0,t]\subset I$. For sufficiently small
$\varepsilon$ we have that
$\partial\Theta_{\varepsilon}(t)\cap\Omega\subset\Sigma$. We can find a smooth
domain $\Omega_{1}$ such that

$\Omega\setminus\Theta_{\varepsilon}(t)\subset\Omega_{1},\quad\Gamma\subset\partial\Omega\cap\partial\Omega_{1},$
(143)

and a second smooth domain $\Omega_{2}$ such that

$\Omega\subset\Omega_{2},\quad\partial\Omega\cap\partial\Omega_{1}\subset\partial\Omega\cap\partial\Omega_{2}.$
(144)

We extend $F$ to be zero on $\Omega_{1}\setminus\overline{\Omega}$,
$\Omega_{2}\setminus\overline{\Omega}$.

Let
$K:(\Omega_{2}\setminus\overline{\Omega_{1}})\times(\Omega_{2}\setminus\overline{\Omega_{1}})\to\mathbb{R}$
be

$K(x,s)=\int_{\Omega_{1}}F(y)G_{\Omega_{2}}(x,y)G_{\Omega_{2}}(s,y)\,\text{d}y\\\\[5.0pt]
=\int_{\Omega}F(y)G_{\Omega_{2}}(x,y)G_{\Omega_{2}}(s,y)\,\text{d}y.$ (145)

$K$ is harmonic in both $x$ and $s$. If
$x,s\in\Omega_{2}\setminus\overline{\Omega}$, by (116), we have that
$K(x,s)=0$. By unique continuation it follows that $K(x,s)=0$ for all
$x,s\in\Omega_{2}\setminus\overline{\Omega_{1}}$. If $a,b\in
C^{\infty}(\overline{\Omega_{2}})$,
$\mbox{supp\;}a,\mbox{supp\;}b\subset\Omega_{2}\setminus\overline{\Omega_{1}}$,
then

$\int K(x,s)a(x)b(s)\,\text{d}x\,\text{d}s=0,$ (146)

and by the above approximation lemma we have that

$\int_{\Omega_{1}}F(y)U(y)W(y)\,\text{d}y=0,$ (147)

for all $U,W\in C^{\infty}(\overline{\Omega_{1}})$ which are harmonic in
$\Omega_{1}$ and such that
$U|_{\partial\Omega_{1}\cap\partial\Omega_{2}}=W|_{\partial\Omega_{1}\cap\partial\Omega_{2}}=0$.

Applying the result of the previous subsection to the domain $\Omega_{1}$, we
can conclude the there is a $t^{\prime}>t$ such that $t^{\prime}\in I$. This
is enough to conclude that $I$ is open in $[0,1]$, and hence $I=[0,1]$. This
gives that $F(x_{1})=0$, which completes the proof of Theorem 3.

#### Acknowledgments:

C.C. was supported by NSF of China under grants 11931011 and 11971333. The
research of T.G. is supported by the Collaborative Research Center, membership
no. 1283, Universität Bielefeld. GU was partly supported by NSF, a Walker
Family Professorship at UW, and a Si Yuan Professorship at IAS, HKUST. He was
also supported by a Simons Fellowship. Part of this work was done while he was
visiting IPAM in Fall 2021.

## References

  * [1] M. Bertsch.  A class of degenerate diffusion equations with a singular nonlinear term.  Nonlinear Analysis: Theory, Methods & Applications, 7(1):117–127, 1983.
  * [2] T. Brander.  Calderón problem for the p-Laplacian: First order derivative of conductivity on the boundary.  Proceedings of the American Mathematical Society, 144(1):177–189, 2016.
  * [3] T. Brander, B. Harrach, M. Kar, and M. Salo.  Monotonicity and enclosure methods for the p-Laplace equation.  SIAM Journal on Applied Mathematics, 78(2):742–758, 2018.
  * [4] T. Brander, J. Ilmavirta, and M. Kar.  Superconductive and insulating inclusions for linear and non-linear conductivity equations.  Inverse Problems & Imaging, 12(1):91–123, 2018.
  * [5] T. Brander, M. Kar, and M. Salo.  Enclosure method for the p-Laplace equation.  Inverse Problems, 31(4):045001, 2015.
  * [6] A. L. Bukhgeim.  Recovering a potential from cauchy data in the two-dimensional case.  Journal of Inverse and Ill-Posed Problems, 16(1):19–33, 2008.
  * [7] A. P. Calderón.  On an inverse boundary value problem.  In Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), pages 65–73. Soc. Brasil. Mat., Rio de Janeiro, 1980.
  * [8] P. Caro and Y. Kian.  Determination of convection terms and quasi-linearities appearing in diffusion equations.  arXiv preprint arXiv:1812.08495, 2018.
  * [9] C. I. Cârstea.  On an inverse boundary value problem for a nonlinear time harmonic Maxwell system.  Journal of Inverse and Ill-posed Problems, 1(ahead-of-print), 2020\.
  * [10] C. I. Cârstea and A. Feizmohammadi.  A density property for tensor products of gradients of harmonic functions and applications.  arXiv preprint arXiv:2009.11217, 2020.
  * [11] C. I. Cârstea and A. Feizmohammadi.  An inverse boundary value problem for certain anisotropic quasilinear elliptic equations.  Journal of Differential Equations, 284:318–349, 2021.
  * [12] C. I. Cârstea, A. Feizmohammadi, Y. Kian, K. Krupchyk, and G. Uhlmann.  The Calder$\backslash$’$\\{$o$\\}$ n inverse problem for isotropic quasilinear conductivities.  Advances in Mathematics, 391:107956, 2021.
  * [13] C. I. Cârstea, T. Ghosh, and G. Nakamura.  An inverse boundary value problem for the inhomogeneous porous medium equation.  arXiv preprint arXiv:2105.01368, 2021.
  * [14] C. I. Cârstea and M. Kar.  Recovery of coefficients for a weighted p-Laplacian perturbed by a linear second order term.  arXiv preprint arXiv:2001.01436, 2020.
  * [15] C. I. Cârstea, G. Nakamura, and M. Vashisth.  Reconstruction for the coefficients of a quasilinear elliptic partial differential equation.  Applied Mathematics Letters, 2019.
  * [16] M. Choulli and Y. Kian.  Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term.  Journal de Mathématiques Pures et Appliquées, 114:235–261, 2018.
  * [17] D. Dos Santos Ferreira, C. E Kenig, J. Sjöstrand, and G. Uhlmann.  On the linearized local calder$\backslash$’on problem.  Mathematical research letters, 16(5):955–970, 2009.
  * [18] H. Egger, J.-F. Pietschmann, and M. Schlottbom.  Simultaneous identification of diffusion and absorption coefficients in a quasilinear elliptic problem.  Inverse Problems, 30(3):035009, 2014.
  * [19] A. Feizmohammadi and L. Oksanen.  An inverse problem for a semi-linear elliptic equation in Riemannian geometries.  Journal of Differential Equations, 269(6):4683–4719, 2020.
  * [20] C.-Y. Guo, M. Kar, and M. Salo.  Inverse problems for p-Laplace type equations under monotonicity assumptions.  Rend. Istit. Mat. Univ. Trieste, 48:79–99, 2016.
  * [21] D. Hervas and Z. Sun.  An inverse boundary value problem for quasilinear elliptic equations.  Communications in Partial Differential Equations, 27(11-12):2449–2490, 2002.
  * [22] O. Imanuvilov, G. Uhlmann, and M. Yamamoto.  The calderón problem with partial data in two dimensions.  Journal of the American Mathematical Society, 23(3):655–691, 2010\.
  * [23] V. Isakov.  On uniqueness in inverse problems for semilinear parabolic equations.  Archive for Rational Mechanics and Analysis, 124(1):1–12, 1993\.
  * [24] V. Isakov.  Uniqueness of recovery of some quasilinear partial differential equations.  Communications in Partial Differential Equations, 26(11-12):1947–1973, 2001.
  * [25] V. Isakov and A. I. Nachman.  Global uniqueness for a two-dimensional semilinear elliptic inverse problem.  Transactions of the American Mathematical Society, 347(9):3375–3390, 1995.
  * [26] V. Isakov and J. Sylvester.  Global uniqueness for a semilinear elliptic inverse problem.  Communications on Pure and Applied Mathematics, 47(10):1403–1410, 1994.
  * [27] H. Kang and G. Nakamura.  Identification of nonlinearity in a conductivity equation via the Dirichlet-to-Neumann map.  Inverse Problems, 18(4):1079, 2002.
  * [28] M. Kar and J.-N. Wang.  Size estimates for the weighted p-Laplace equation with one measurement.  Discrete & Continuous Dynamical Systems-B, 22(11):0, 2017.
  * [29] C. E. Kenig, J. Sjöstrand, and G. Uhlmann.  The Calderón problem with partial data.  Annals of Mathematics, 165:567–591, 2007.
  * [30] Y. Kian, K. Krupchyk, and G. Uhlmann.  Partial data inverse problems for quasilinear conductivity equations.  arXiv preprint arXiv:2010.11409, 2020.
  * [31] Y. Kian and G. Uhlmann.  Recovery of nonlinear terms for reaction diffusion equations from boundary measurements.  arXiv preprint arXiv:2011.06039, 2020.
  * [32] K. Krupchyk and G. Uhlmann.  Partial data inverse problems for semilinear elliptic equations with gradient nonlinearities.  arXiv preprint arXiv:1909.08122, 2019.
  * [33] K. Krupchyk and G. Uhlmann.  A remark on partial data inverse problems for semilinear elliptic equations.  Proceedings of the American Mathematical Society, 148(2):681–685, 2020.
  * [34] O. A. Ladyzhenskaia, V. A. Solonnikov, and N. N. Ural’tseva.  Linear and quasi-linear equations of parabolic type, volume 23.  American Mathematical Soc., 1988.
  * [35] M. Lassas, T. Liimatainen, Y.-H. Lin, and M. Salo.  Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations.  arXiv preprint arXiv:1905.02764, 1905.
  * [36] M. Lassas, T. Liimatainen, Y.-H. Lin, and M. Salo.  Inverse problems for elliptic equations with power type nonlinearities.  arXiv preprint arXiv:1903.12562, 2019.
  * [37] G. M. Lieberman.  Second Order Parabolic Differential Equations.  World Scientific, 1996.
  * [38] C. Munoz and G. Uhlmann.  The Calderón problem for quasilinear elliptic equations.  Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 2020.
  * [39] A. I. Nachman.  Global uniqueness for a two-dimensional inverse boundary value problem.  Annals of Mathematics, pages 71–96, 1996.
  * [40] T. Namba.  Density-dependent dispersal and spatial distribution of a population.  Journal of Theoretical Biology, 86(2):351–363, 1980.
  * [41] P. Rosenau and S. Kamin.  Thermal waves in an absorbing and convecting medium.  Physica D: Nonlinear Phenomena, 8(1-2):273–283, 1983.
  * [42] A. E. Scheidegger.  The physics of flow through porous media.  University of Toronto Press, 1974.
  * [43] R. Shankar.  Recovering a quasilinear conductivity from boundary measurements.  arXiv preprint arXiv:1910.07890, 2019.
  * [44] Z. Sun.  On a quasilinear inverse boundary value problem.  Mathematische Zeitschrift, 221(1):293–305, 1996.
  * [45] Z. Sun.  Anisotropic inverse problems for quasilinear elliptic equations.  In Journal of Physics: Conference Series, volume 12, page 015. IOP Publishing, 2005.
  * [46] Z. Sun.  An inverse boundary-value problem for semilinear elliptic equations.  Electronic Journal of Differential Equations (EJDE)[electronic only], 2010:Paper–No, 2010.
  * [47] Z. Sun and G. Uhlmann.  Inverse problems in quasilinear anisotropic media.  American Journal of Mathematics, 119(4):771–797, 1997.
  * [48] J. Sylvester and G. Uhlmann.  A global uniqueness theorem for an inverse boundary value problem.  Annals of Mathematics, pages 153–169, 1987.
  * [49] J. L. Vázquez.  The porous medium equation: mathematical theory.  Oxford University Press on Demand, 2007.