# Inverse problems for semilinear elliptic PDE with a general nonlinearity
$a(x,u)$

David Johansson  Department of Mathematics and Statistics, University of
Jyväskylä<EMAIL_ADDRESS>,  Janne Nurminen  Department of
Mathematics and Statistics, University of Jyväskylä<EMAIL_ADDRESS>and  Mikko Salo  Department of Mathematics and Statistics, University of
Jyäskylä<EMAIL_ADDRESS>
###### Abstract.

This article studies the inverse problem of recovering a nonlinearity in an
elliptic equation $\Delta u+a(x,u)=0$ from boundary measurements of solutions.
Previous results based on first order linearization achieve this under a sign
condition on $\partial_{u}a(x,u)$, and results based on higher order
linearization recover the Taylor series of $a(x,u)$ with respect to $u$. We
improve these results and show that a general nonlinearity, and not just its
Taylor series, is uniquely determined up to gauge near a fixed solution. Our
method is based on constructing a good solution map that locally parametrizes
solutions of the nonlinear equation by solutions of the linearized equation.

##  1\. Introduction

### Motivation

Let $\Omega\subseteq\mathbb{R}^{n}$, $n\geq 2$, be a bounded domain whose
boundary is assumed to be $C^{\infty}$ for simplicity, and let $a\in
C^{k}(\mathbb{R},C^{1,\alpha}(\overline{\Omega}))$ where $k\geq 3$ and
$0<\alpha<1$. We write $a=a(x,z)$ for $x\in\overline{\Omega}$ and
$z\in\mathbb{R}$, and consider equations of the form

(1.1) $\Delta u(x)+a(x,u(x))=0\text{ in }\Omega.$

In this article we study the inverse problem of identifying the function
$a(x,z)$ from certain boundary measurements of solutions of (1.1). For
example, the boundary measurements could be encoded by a Dirichlet-to-Neumann
(DN) map if the equation is well-posed, or more generally one could use the
(full) Cauchy data set

$C_{a}:=\\{(u|_{\partial\Omega},\partial_{\nu}u|_{\partial\Omega})\,:\,u\in
C^{2,\alpha}(\overline{\Omega})\text{ solves $\Delta u+a(x,u)=0$}\\}.$

That is, we wish to answer the question:

Does $C_{a}$ determine $a(x,z)$?

For linear equations $\Delta u+q(x)u=0$, the question above is a version of
Calderón’s inverse problem and there is large literature (see e.g. the survey
[Uhl14]). There are also many results for nonlinear equations. The first
generation of such results was based on _first order linearization_ , i.e. on
studying the (first) Fréchet derivative of the nonlinear DN map and using
existing results for linear equations. This method was introduced in [Isa93],
and further results for determining a nonlinearity $a(x,u)$ as in (1.1) were
given in IS, IN, ImanuvilovYamamoto2013. These results typically require
assumptions such as

(1.2) $\displaystyle a(x,0)=0,$ (1.3) $\displaystyle\partial_{u}a(x,u)\leq 0,$

which ensure well-posedness and a maximum principle. The assumption (1.3) was
weakened in [IN95], and [Sun10] gave a result without assuming (1.2). The
results show that one can recover $a(x,u)$ in (some subset of) the _reachable
set_

$E_{a}:=\\{(x,z)\,:\,x\in\overline{\Omega},\ z=u(x)\text{ for some solution
$u$ of $\Delta u+a(x,u)=0$}\\}.$

There are many related works for quasilinear and conductivity type equations.
References may be found in the survey articles Sun2005, Uhlmann2009.

The works FO, LLLS introduced a higher order linearization method in inverse
problems for nonlinear elliptic equations, motivated by the earlier work
[KLU18] for hyperbolic equations. This method applies to inverse problems for
equations like (1.1) without any positivity assumptions as in (1.3). Moreover,
unlike in the first order linearization method that reduced matters to known
results for linear equations, in higher order linearization the nonlinearity
is used as a tool that helps in solving inverse problems. In this way, one can
obtain results in partial data problems LLLS2, KU, ST or anisotropic problems
FO, LLLS, CFO23, FKO23 that are stronger than the known results for
corresponding linear equations.

However, the higher order linearization results for (1.1) start with the
assumptions that

(1.4) $\displaystyle a(x,0)=0,$ (1.5) $0$ is not a Dirichlet eigenvalue for
$\Delta+\partial_{u}a(x,0)$ in $\Omega$.

The first assumption ensures that $u\equiv 0$ is a solution of (1.1). The
second assumption ensures that the linearized equation is well-posed for small
Dirichlet data, and hence there is a nonlinear DN map $\Lambda_{a}$ for (1.1)
defined for small Dirichlet data. The additional assumption
$\partial_{u}a(x,0)=0$ also appears in many results. The works FO, LLLS then
show that $\Lambda_{a}$ (defined for small Dirichlet data) determines
$\partial_{u}^{j}a(\,\cdot\,,0)$ for many $j\geq 0$. If one additionally
assumes that $a(x,z)$ is real-analytic in $z$, then this is sufficient for
determining $a(x,z)$ completely.

### Results

Our aim is to consider inverse problems for (1.1) for general functions $a\in
C^{k}(\mathbb{R},C^{1,\alpha}(\overline{\Omega}))$. In particular, we wish to
remove the assumptions (1.2)–(1.3) in the first order linearization method and
(1.4)–(1.5) in the higher order linearization method. This requires certain
changes in the problem setup. First of all, the results in FO, LLLS are based
on looking at solutions of (1.1) close to $u\equiv 0$ and on well-posedness
for small data. Moreover, the linearized equation might not be well-posed in
general, but by Fredholm theory it is still well-posed for most Dirichlet data
(i.e. data that are $L^{2}$-orthogonal to a finite dimensional space). It
follows that there may not be a Dirichlet-to-Neumann map to work with.

For these reasons, in the general case we consider an arbitrary but fixed
solution $w\in C^{2,\alpha}(\overline{\Omega})$ of $\Delta w+a(x,w)=0$ and for
$\delta>0$ we define the local Cauchy data set

$C_{a}^{w,\delta}:=\\{(u|_{\partial\Omega},\partial_{\nu}u|_{\partial\Omega})\,:\,u\in
C^{2,\alpha}(\overline{\Omega})\text{ solves $\Delta u+a(x,u)=0$}\,\text{ and
}\,\lVert w-u\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq\delta\\}.$

If $w\equiv 0$, this would be analogous to small Dirichlet data. By the
Fredholm theory fact mentioned above one expects that there are many solutions
close to $w$, and we will prove a precise version of such a result.

Our first main theorem shows that if two nonlinearities $a_{1}$ and $a_{2}$
admit a common solution $w$ and if their local boundary measurements satisfy
the inclusion $C_{a_{1}}^{w,\delta}\subseteq C_{a_{2}}^{0,C}$, then
$a_{1}=a_{2}$ near the common solution $w$.

######  Theorem 1.1.

Let $a_{1},a_{2}\in C^{3}(\mathbb{R},C^{1,\alpha}(\overline{\Omega}))$, and
let $w\in C^{2,\alpha}(\overline{\Omega})$ solve $\Delta w+a_{1}(x,w)=0$ and
$\Delta w+a_{2}(x,w)=0$ in $\Omega$. If for some $\delta,C>0$ one has

$C_{a_{1}}^{w,\delta}\subseteq C_{a_{2}}^{0,C},$

then there is $\varepsilon>0$ such that

$a_{1}(x,w(x)+\lambda)=a_{2}(x,w(x)+\lambda),\qquad x\in\overline{\Omega},\ \
\lvert\lambda\rvert\leq\varepsilon.$

This theorem is based on using first order linearization and it is valid for
general nonlinearities. In particular, the sign condition (1.3) in the earlier
results mentioned above is not needed. We note that the higher order
linearization method does not require the sign condition either, but the
result in FO, LLLS for the case $w\equiv 0$ was
$\partial_{u}^{j}a_{1}(\,\cdot\,,0)=\partial_{u}^{j}a_{2}(\,\cdot\,,0)$ for
many $j\geq 0$ which is clearly weaker than the conclusion in Theorem 1.1.
Moreover, if the linearizations of $\Delta u+a_{j}(x,u)=0$ (linearized at the
solution $w$) happen to be well-posed, then by the arguments in Section 2
there are Dirichlet-to-Neumann maps $\Lambda_{a_{j}}$ defined for Dirichlet
data close to $w|_{\partial\Omega}$, and then by Theorem 1.1 one obtains
$a_{1}=a_{2}$ near points $(x,w(x))$ whenever
$\Lambda_{a_{1}}=\Lambda_{a_{2}}$.

######  Remark 1.2.

It is in order to explain the assumption $C_{a_{1}}^{w,\delta}\subseteq
C_{a_{2}}^{0,C}$ in the theorem. A more typical way of stating a uniqueness
result would be to say that $C_{a_{1}}=C_{a_{2}}$ (i.e. the full Cauchy data
sets of $a_{1}$ and $a_{2}$ agree) implies $a_{1}=a_{2}$ somewhere. However,
$C_{a_{1}}=C_{a_{2}}$ implies our assumption $C_{a_{1}}^{w,\delta}\subseteq
C_{a_{2}}^{0,C}$ in many cases, e.g. when the equation $\Delta u+a_{2}(x,u)=0$
is well-posed for Dirichlet data near $w|_{\partial\Omega}$ or when
$\lVert\partial_{u}a_{2}\rVert_{L^{\infty}(\Omega\times\mathbb{R})}<\infty$
(the latter fact follows from the Cauchy data estimates in Section 3).

More precisely, the assumption $C_{a_{1}}^{w,\delta}\subseteq C_{a_{2}}^{0,C}$
means that if $u_{1}$ solves $\Delta u_{1}+a_{1}(x,u_{1})=0$ with $\lVert
u_{1}-w\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq\delta$, then there is
$u_{2}$ solving $\Delta u_{2}+a_{2}(x,u_{2})=0$, having the same Cauchy data
as $u_{1}$, and satisfying $\lVert
u_{2}\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq C$. The existence of such a
constant $C$ is required in the proof to make sure that if $u_{1}$ is very
close to $w$, then $u_{2}$ will also be close to $w$ and we can use a
uniqueness result to guarantee that $u_{2}$ can be differentiated with respect
to some parameters if the same is true for $u_{1}$.

If the nonlinearities $a_{1}$ and $a_{2}$ do not admit a common solution, then
this inverse problem has a gauge invariance as observed in [Sun10]. If $a\in
C^{k}(\mathbb{R},C^{\alpha}(\overline{\Omega}))$ is a nonlinearity and
$\varphi\in C^{2,\alpha}(\overline{\Omega})$ is any function satisfying
$\varphi|_{\partial\Omega}=\partial_{\nu}\varphi|_{\partial\Omega}=0$, we
define

(1.6) $T_{\varphi}a(x,u):=\Delta\varphi(x)+a(x,u+\varphi(x)).$

Then $u$ solves $\Delta u+a(x,u(x))=0$ if and only if $v=u-\varphi$ solves
$\Delta v+T_{\varphi}a(x,v(x))=0$. It follows that the solutions of these two
equations have the same Cauchy data. Hence, if the Cauchy data sets for
$a_{1}$ and $a_{2}$ agree, one can only expect that
$a_{2}(x,u)=T_{\varphi}a_{1}(x,u)$ for $(x,u)$ in the reachable set. There are
a number of related works based on the first linearization, see e.g. IS, IN,
Sun. Recent works that involve a similar gauge invariance are given in LL23,
KLL23. We also mention the examples in IS, FKO23 showing that in general the
reachable set is not all of $\overline{\Omega}\times\mathbb{R}$.

Our next result shows that if one knows the Cauchy data for a nonlinearity $a$
and for solutions close to a given solution $w$, then one can recover $a$ near
points $(x,w(x))$ precisely up to the gauge mentioned above.

######  Theorem 1.3.

Let $a_{1},a_{2}\in C^{3}(\mathbb{R},C^{1,\alpha}(\overline{\Omega}))$, and
let $w_{1}\in C^{2,\alpha}(\overline{\Omega})$ solve $\Delta
w_{1}+a_{1}(x,w_{1})=0$ in $\Omega$. If

$C_{a_{1}}^{w_{1},\delta}\subseteq C_{a_{2}}^{0,C}$

for some $\delta,C>0$, then there is $\varepsilon>0$ such that

$\displaystyle a_{1}(x,w_{1}(x)+\lambda)=T_{\varphi}a_{2}(x,w_{1}(x)+\lambda)$

whenever $x\in\overline{\Omega}$ and $\lvert\lambda\rvert\leq\varepsilon$.
Here $\varphi\coloneqq w_{2}-w_{1}$ where $w_{2}\in
C^{2,\alpha}(\overline{\Omega})$ is the unique solution of $\Delta
w_{2}+a_{2}(x,w_{2})=0$ in $\Omega$ with
$w_{1}|_{\partial\Omega}=w_{2}|_{\partial\Omega}$ and
$\partial_{\nu}w_{1}|_{\partial\Omega}=\partial_{\nu}w_{2}|_{\partial\Omega}$.

Again, Theorem 1.3 is valid for general nonlinearities. Note that Theorem 1.1
is a corollary of Theorem 1.3 since $w_{2}=w_{1}$ in that case.

Both Theorem 1.1 and 1.3 are based on first order linearization and they rely
on the solution of an inverse problem for the linearized equation. In
contrast, many of the results based on higher order linearization do not rely
directly on the inverse problem for the linearized equation. In fact in these
results the equation often has a form where the unknown quantities only appear
in higher linearizations and not in the first linearization. For such
equations, nonlinearity often helps and one can obtain improved results in the
presence of nonlinearity.

In the case of the higher order linearization method, we can remove the
assumptions (1.4)–(1.5) that were present in most of the earlier results. The
following result is an example of what one can prove.

######  Theorem 1.4.

Let $a_{1},a_{2}\in C^{k+1}(\mathbb{R},C^{1,\alpha}(\overline{\Omega}))$ with
$k\geq 2$, let $w_{1}\in C^{2,\alpha}(\overline{\Omega})$ solve $\Delta
w_{1}+a_{1}(x,w_{1})=0$ in $\Omega$, and suppose that

$C_{a_{1}}^{w_{1},\delta}\subseteq C_{a_{2}}^{0,C}$

for some $\delta,C>0$. Let $w_{2}\in C^{2,\alpha}(\overline{\Omega})$ be the
unique solution of $\Delta w_{2}+a_{2}(x,w_{2})=0$ in $\Omega$ with
$w_{1}|_{\partial\Omega}=w_{2}|_{\partial\Omega}$ and
$\partial_{\nu}w_{1}|_{\partial\Omega}=\partial_{\nu}w_{2}|_{\partial\Omega}$.
Assume further that

(1.7) $\partial_{u}^{l}a_{1}(x,w_{1})=\partial_{u}^{l}a_{2}(x,w_{2}),\qquad
1\leq l\leq k-1.$

Then

$\displaystyle\int_{\Omega}(\partial_{u}^{k}a_{1}(x,w_{1})-\partial_{u}^{k}a_{2}(x,w_{2}))v_{1}\ldots
v_{k+1}\,dx=0$

for any $v_{j}$ solving the linear equation $\Delta
v_{j}+\partial_{u}a_{1}(x,w_{1})v_{j}=0$ in $\Omega$.

In other words, if $C_{a_{1}}^{w_{1},\delta}\subseteq C_{a_{2}}^{0,C}$ and
(1.7) holds, then
$\partial_{u}^{k}a_{1}(x,w_{1})-\partial_{u}^{k}a_{2}(x,w_{2})$ is
$L^{2}$-orthogonal to products of $k+1$ solutions of the same linear equation.
This is a typical conclusion in the higher order linearization method. Under
our current assumptions, Theorem 1.3, which is based on solving the inverse
problem for the linearized equation, already implies that
$\partial_{u}^{k}a_{1}(x,w_{1})=\partial_{u}^{k}a_{2}(x,w_{2})$. The point is
that as long as (1.7) holds (this is true e.g. for polynomial nonlinearities
$a_{j}(x,u)=q_{j}(x)u^{k}$ and $w_{j}\equiv 0$), Theorem 1.4 does not rely on
solving the inverse problem for the linearized equation and one can prove this
without assuming (1.4)–(1.5). Theorem 1.4 remains valid for more general
equations such as $\Delta_{g}u+a(x,u)=0$ with a smooth Riemannian metric $g$,
for which the linearized case is not fully understood. For these more general
equations one might be able to obtain improved results in the nonlinear case
as was done in FO, LLLS and subsequent works.

### Methods

Let us next describe the methods for proving the above results. The first
objective is to show that near a solution $w$ of $\Delta w+a(x,w)=0$, there
are many solutions $u_{v}=w+v+O(\lVert v\rVert^{2})$ of the same equation that
are parametrized by small solutions $v$ of the linearized equation

$\Delta v+\partial_{u}a(x,w)v=0.$

It will also be important that $u_{v}$ depends smoothly on $v$. We will prove
this by a standard argument using the implicit function theorem. This is
slightly delicate since the linearized equation may not be well-posed. In
order to have a solution $u_{v}$ depending smoothly on $v$ that is unique in a
suitable sense, one needs to use a solution operator for the linearized
equation that takes into account the finite dimensional obstructions to
solvability coming from Fredholm theory.

One can recast the previous result in a different language. If
$q(x)=\partial_{u}a(x,w(x))$ and

$\displaystyle V_{q}$ $\displaystyle=\\{v\in
C^{2,\alpha}(\overline{\Omega})\,:\,\Delta v+qv=0\text{ in $\Omega$}\\},$
$\displaystyle W_{a}$ $\displaystyle=\\{u\in
C^{2,\alpha}(\overline{\Omega})\,:\,\Delta u+a(x,u)=0\text{ in $\Omega$}\\},$

then $V_{q}$ is the solution space of $\Delta+q$ and $W_{a}$ is a Banach
manifold in $C^{2,\alpha}(\overline{\Omega})$ consisting of solutions of the
nonlinear equation. Then $V_{q}$ is the tangent space of $W_{a}$ at $w$, and
our result shows that

$S_{a,w}:v\mapsto u_{v}$

is a bijective smooth map from a neighborhood of $0$ in $V_{q}$ onto a
neighborhood of $w$ in $W_{a}$ with $DS_{a,w}(0)=\mathrm{Id}$. Similar ideas
appear e.g. in Eells, Palais, Sunada.

Next, starting from the assumption $C_{a_{1}}^{w,\delta}\subseteq
C_{a_{2}}^{0,C}$, we construct a solution $u_{1,v}$ as above for $a_{1}$, and
use the inclusion of Cauchy data sets to conclude that there is a solution
$u_{2,v}$ for $a_{2}$ having the same Cauchy data as $u_{1,v}$. We know that
$u_{1,v}$ depends smoothly on $v$, but this is not known for $u_{2,v}$. In
order to show that also $u_{2,v}$ depends smoothly on $v$, we prove
quantitative estimates showing that solutions of both the linearized and
nonlinear equations depend continuously on their Cauchy data. For one of these
results we invoke a standard Carleman estimate. Since $u_{2,v}$ depends
continuously on its Cauchy data, and since the Cauchy data of $u_{2,v}$ is the
same as for $u_{1,v}$ and the latter depends smoothly on $v$, we are able to
show that also $u_{2,v}$ depends smoothly on $v$. This also uses certain
functional analytic arguments following [OSSU20].

The first order linearization result, Theorem 1.1, is proved as follows. If
$C_{a_{1}}^{w,\delta}\subseteq C_{a_{2}}^{0,C}$ and if $u_{1,v_{t}}$ and
$u_{2,v_{t}}$ are as described above with $v_{t}=v+th$, we differentiate the
equations $\Delta u_{j,v_{t}}+a_{j}(x,u_{j,v_{t}})=0$ with respect to $t$,
subtract the resulting equations, and integrate against a solution of
$\Delta\tilde{v}_{2}+\partial_{u}a_{2}(x,u_{2,v})\tilde{v}_{2}=0$ to obtain

$\int_{\Omega}(\partial_{u}a_{1}(x,u_{1,v})-\partial_{u}a_{2}(x,u_{2,v}))\tilde{v}_{1}\tilde{v}_{2}\,dx=0$

where $\tilde{v}_{1}=DS_{a_{1},w}(v)h$. Then we use the bijectivity of
$S_{a,w}$ above to conclude that any solution $\tilde{v}_{1}$ of
$\Delta\tilde{v}_{1}+\partial_{u}a_{1}(x,u_{1,v})\tilde{v}_{1}=0$ can be
written as $DS_{a_{1},w}(v)h$ for some $h$. Density of products of solutions
as in the standard Calderón problem SylvesterUhlmann, Bukhgeim implies that

$\partial_{u}a_{1}(x,u_{1,v})=\partial_{u}a_{2}(x,u_{2,v})\text{ for any small
$v\in V_{q}$.}$

Next we show that $\varphi_{v}\coloneqq u_{2,v}-u_{1,v}$ is independent of
$v$, by observing that the derivative of $\varphi_{v}$ with respect to $v$ is
identically $0$, because it solves a linear elliptic equation and has zero
Cauchy data. Since $\varphi_{0}=w-w=0$, we obtain

$\partial_{u}a_{1}(x,u_{1,v})=\partial_{u}a_{2}(x,u_{1,v})\text{ for any small
$v\in V_{q}$.}$

(In the setting of Theorem 1.3 one has $u_{2,v}=u_{1,v}+\varphi$ instead.) It
then remains to show that for any fixed $x_{0}$, the values $u_{1,v}(x_{0})$
generate an interval $[w(x_{0})-\varepsilon,w(x_{0})+\varepsilon]$ by varying
$v$. This follows since $u_{1,v}=w+v+O(\lVert v\rVert^{2})$ and since one can
generate linear solutions $v$ with $v(x_{0})\neq 0$ by using Runge
approximation. This concludes the outline of proof of Theorem 1.1. The proof
of Theorem 1.3 is analogous, except that $\varphi$ will be a nonzero function
that is independent of $v$.

Finally, we use the higher order linearization method and differentiate $k$
times the equations $\Delta u_{j,v}+a_{j}(x,u_{j,v})=0$ with respect to $v$.
Subtracting the resulting equations, using the assumption (1.7) and
integrating against a solution $v_{k+1}$, we arrive at Theorem 1.4.

The article is organized as follows. In Section 2 we construct a solution map
for equation (1.1). Section 3 is dedicated to quantitative uniqueness results
for (1.1) and its linearization. The linearization methods require two smooth
solution maps and the second one is constructed in Section 4. In Section 5 we
use first order linearization to prove Theorems 1.1 and 1.3. Finally, in
Section 6 we give the proof of Theorem 1.4. At the end we have an Appendix
where we give a Runge approximation result for the first linearization of
(1.1).

### Acknowledgements

The authors are partly supported by the Academy of Finland (Centre of
Excellence in Inverse Modelling and Imaging, grant 353091). M.S. was also
supported by the European Research Council under Horizon 2020 (ERC CoG
770924).

##  2\. A smooth solution operator

The linearization methods used in this work are based on constructing
solutions $u=u_{v}$ of

(2.1) $\Delta u+a(x,u)=0\text{ in }\Omega,$

so that $u_{v}$ is close to a fixed solution $w$ of (2.1) and depends smoothly
on a small solution $v$ of the linearized equation

(2.2) $\Delta v+\partial_{u}a(x,w)v=0\text{ in }\Omega.$

In order to parametrize solutions of (2.1) on solutions of (2.2), we need to
be able to single out suitable solutions of the linearized equation which may
not be well-posed. This is done in Lemma 2.1 by using the Fredholm
alternative. We then construct solutions $u_{v}$ of (2.1) by solving a
nonlinear fixed point equation. This fixed point equation is solved in Lemma
2.3. The construction of the smooth solution map $v\to u_{v}$ for the equation
(2.1) is completed in Lemma 2.4. Finally in Lemma 2.5 we show that the first
Fréchet derivative of the solution operator is an isomorphism between spaces
of solutions to the linearized equation.

Before we proceed to the results of this section, let us define some spaces
and mappings that are used throughout. Let $\Omega\subseteq\mathbb{R}^{n}$
with $n\geq 2$ be a bounded open set with $C^{\infty}$ boundary, and let $q\in
C^{1,\alpha}(\overline{\Omega})$ be real valued where $0<\alpha<1$. First we
have the kernel $N_{q}$ of the operator $\Delta+q:H_{0}^{1}(\Omega)\to
H^{-1}(\Omega)$ and the space of Neumann data $\partial_{\nu}N_{q}$ of the
functions in $N_{q}$, i.e.

$\displaystyle N_{q}$ $\displaystyle=\\{\psi\in
H^{1}_{0}(\Omega)\,:\,(\Delta+q)\psi=0\\},$ $\displaystyle\partial_{\nu}N_{q}$
$\displaystyle=\\{\partial_{\nu}\psi|_{\partial\Omega}\,:\,\psi\in N_{q}\\}.$

These spaces appear due to the use of the Fredholm alternative. These are
finite dimensional spaces, and since $q\in C^{1,\alpha}(\overline{\Omega})$,
elliptic regularity [Bro62, Theorem 2.2] ensures that $N_{q}\subseteq
C^{3,\alpha}(\overline{\Omega})$ and $\partial_{\nu}N_{q}\subseteq
C^{2,\alpha}(\partial\Omega)$. The last fact is the only reason why we assume
$q\in C^{1,\alpha}(\overline{\Omega})$ (otherwise $q\in
C^{\alpha}(\overline{\Omega})$ would have been sufficient). We let
$\\{\partial_{\nu}\psi_{1},\ldots,\partial_{\nu}\psi_{m}\\}$ be an orthonormal
basis of $\partial_{\nu}N_{q}$ with respect to the
$L^{2}(\partial\Omega)$-inner product.

We now show that even in the case when $0$ is a Dirichlet eigenvalue of
$\Delta+q$ in $\Omega$, the equation $(\Delta+q)u=F$ has a solution $u$ for
any $F$ and one can prescribe the Dirichlet data of $u$ in the
$L^{2}(\partial\Omega)$-orthocomplement of the finite dimensional space
$\partial_{\nu}N_{q}$. Below, the notation $\perp$ will always mean
$L^{2}$-orthogonality.

######  Lemma 2.1.

Let $q\in C^{1,\alpha}(\overline{\Omega})$. For any $F\in
C^{\alpha}(\overline{\Omega})$ and $f\in C^{2,\alpha}(\partial\Omega)$, there
is a unique function $\Phi=\Phi(F,f)\in\partial_{\nu}N_{q}$ such that the
problem

(2.3) $\begin{cases}\Delta u+qu=F\quad&\text{in }\Omega,\\\
u=f-\Phi\quad&\text{on }\partial\Omega,\end{cases}$

admits a solution $u\in C^{2,\alpha}(\overline{\Omega})$. The function $\Phi$
is given by

(2.4)
$\Phi(F,f)=\sum_{j=1}^{m}\left(\int_{\Omega}F\psi_{j}\,dx+\int_{\partial\Omega}f\partial_{\nu}\psi_{j}\,dS\right)\partial_{\nu}\psi_{j}.$

Moreover, there is unique solution $u_{F,f}=G_{q}(F,f)$ such that
$u_{F,f}\perp N_{q}$. The solution $u_{F,f}$ depends linearly on $F$ and $f$
and satisfies

(2.5) $\lVert u_{F,f}\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq C(\lVert
F\rVert_{C^{\alpha}(\overline{\Omega})}+\lVert
f\rVert_{C^{2,\alpha}(\partial\Omega)}),$

where $C$ is independent of $F$ and $f$.

###### Proof.

We first consider the case of solvability in $H^{2}(\Omega)$ with zero
Dirichlet data. If $X=H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$ equipped with the
$H^{2}(\Omega)$ norm, then by Fredholm theory [Eva10, Theorem 4 in Section
6.2.3] and elliptic regularity [Eva10, Theorem 4 in Section 6.3.2] the map

$T:X\to L^{2}(\Omega),\ Tv=(\Delta+q)v$

is Fredholm, i.e. it has finite dimensional kernel $N_{q}$ and its range
$\mathrm{Ran}(T)=\\{F\in L^{2}(\Omega)\,:\,F\perp N_{q}\\}$ has finite
codimension. It follows that the induced map

$T_{1}:X/N_{q}\to\mathrm{Ran}(T)$

is bounded and bijective, hence invertible by the open mapping theorem. The
space $X/N_{q}$ can be identified with $Y=\\{u\in X\,:\,u\perp N_{q}\\}$, and
$T$ becomes an isomorphism from $Y$ onto $\mathrm{Ran}(T)$. (To see this, let
$E:X\to L^{2}(\Omega)$ be the restriction to $X$ of the $L^{2}$-orthogonal
projection onto $N_{q}$. Then
$X=\operatorname{Ran}(E)\oplus\operatorname{Ker}(E)=N_{q}\oplus Y$ [Con90,
Theorem 13.2 b)], and the map $Y\to X/N_{q}$, $u\mapsto[u]$ identifies $Y$
with $X/N_{q}$.) It follows that for any $F\in\mathrm{Ran}(T)$ there is a
unique $v_{F}\in X$ with $v_{F}\perp N_{q}$ such that $T(v_{F})=F$. In other
words, for any $F\in L^{2}(\Omega)$ with $F\perp N_{q}$ there is a unique
$v_{F}\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$ with $v_{F}\perp N_{q}$ that
depends linearly on $F$ and solves

$(\Delta+q)v=F\text{ in $\Omega$},\qquad v|_{\partial\Omega}=0,$

and one has

(2.6) $\lVert v_{F}\rVert_{H^{2}(\Omega)}\leq C\lVert
F\rVert_{L^{2}(\Omega)}.$

We can obtain a similar statement in Hölder spaces. Let $F\in
C^{\alpha}(\overline{\Omega})$ with $F\perp N_{q}$ and let $v_{F}$ be as
above. By elliptic regularity, $v_{F}\in C^{2,\alpha}(\overline{\Omega})$ and

(2.7) $\lVert v_{F}\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq C\lVert
F\rVert_{C^{\alpha}(\overline{\Omega})}.$

(More precisely, from [Bro62, Theorem 2.2] and [GT01, Lemma 6.18 and Problem
6.2] we obtain $v_{F}\in C^{2,\alpha}(\overline{\Omega})$ and

(2.8) $\lVert v_{F}\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq C(\lVert
v_{F}\rVert_{C(\overline{\Omega})}+\lVert
F\rVert_{C^{\alpha}(\overline{\Omega})}).$

From Theorem $8.15$ and the remark around equation $8.38$ in [GT01] it follows
that $\lVert v_{F}\rVert_{C(\overline{\Omega})}\leq C\lVert
v_{F}\rVert_{L^{2}(\Omega)}$. Using this with (2.6) and (2.8) yields (2.7).)

We now consider (2.3). To study the uniqueness of $u$, we fix a bounded
extension operator

$E_{q}:C^{2,\alpha}(\partial\Omega)\to C^{2,\alpha}(\overline{\Omega})\text{
with $E_{q}h|_{\partial\Omega}=h$ and $E_{q}h\perp N_{q}$ for all $h\in
C^{2,\alpha}(\partial\Omega)$}.$

In fact it is enough to take $E_{q}=(\mathrm{Id}-P_{N_{q}})E$, where $E$ is
any bounded extension operator [Tri83, Theorem 3.3.3 and eq. 1) on p. 51] and
$P_{N_{q}}$ is the $L^{2}(\Omega)$-orthogonal projection to $N_{q}$. We see
that $u$ solves (2.3) iff $u=E_{q}(f-\Phi)+v$, where $v$ solves

(2.9) $\begin{cases}\Delta v+qv=\tilde{F}\quad&\text{in }\Omega,\\\
v=0\quad&\text{on }\partial\Omega,\end{cases}$

where we wrote $\tilde{F}=F-(\Delta+q)E_{q}(f-\Phi)$. We wish to find a
function $\Phi\in\partial_{\nu}N_{q}$ such that $\tilde{F}\perp
L^{2}(\Omega)$. If $\psi\in N_{q}$, integrating by parts gives

$\int_{\Omega}\tilde{F}\psi\,dx=\int_{\Omega}F\psi\,dx+\int_{\partial\Omega}(f-\Phi)\partial_{\nu}\psi\,dS.$

Thus $\tilde{F}\perp N_{q}$ iff $\Phi$ satisfies for all $\psi\in N_{q}$ the
condition

$\int_{\partial\Omega}\Phi\partial_{\nu}\psi\,dS=\int_{\Omega}F\psi\,dx+\int_{\partial\Omega}f\partial_{\nu}\psi\,dS.$

This holds for $\Phi$ iff $\Phi=\Phi(F,f)$ is given by (2.4). For this $\Phi$,
we let $v_{\tilde{F}}\perp N_{q}$ be the solution of (2.9) satisfying (2.6).
Then $u_{F,f}:=E_{q}(f-\Phi(F,f))+v_{\tilde{F}}\perp N_{q}$ satisfies the
required estimate (2.5). ∎

Next we prove an auxiliary lemma which is used in several places in the
remainder of the article.

######  Lemma 2.2.

Let $a\in C^{k}(\mathbb{R},C^{\alpha}(\overline{\Omega}))$ and $f\in
C(\overline{\Omega})$ and let $l\leq k$. Then

(2.10)
$\lVert\partial_{u}^{l}a(x,f(x))\rVert_{C^{\alpha}(\overline{\Omega})}\leq\lVert
a\rVert_{C^{k}([-M,M],C^{\alpha}(\overline{\Omega}))}$

where $M=\lVert f\rVert_{C(\overline{\Omega})}$.

###### Proof.

By manipulating the supremum in the definition of the norm we have

$\displaystyle\lVert\partial_{u}^{l}a(x,f)\rVert_{C^{\alpha}(\overline{\Omega})}=$
$\displaystyle\sup_{x\in\overline{\Omega}}\lvert\partial_{u}^{l}a(x,f(x))\rvert+\sup_{\begin{subarray}{c}x,y\in\overline{\Omega}\\\
x\neq
y\end{subarray}}\frac{\lvert\partial_{u}^{l}a(x,f(x))-\partial_{u}^{l}a(y,f(y))\rvert}{\lvert
x-y\rvert^{\alpha}}$ $\displaystyle\leq$
$\displaystyle\sup_{x\in\overline{\Omega}}\lVert\partial_{u}^{l}a(x)\rVert_{C([-M,M])}+\sup_{\begin{subarray}{c}x,y\in\overline{\Omega}\\\
x\neq
y\end{subarray}}\sup_{\eta,\theta\in[-M,M]}\frac{\lvert\partial_{u}^{l}a(x,\eta)-\partial_{u}^{l}a(y,\theta)\rvert}{\lvert
x-y\rvert^{\alpha}}$ $\displaystyle\leq$
$\displaystyle\sup_{x\in\overline{\Omega}}\lVert
a(x)\rVert_{C^{l}([-M,M])}+\sup_{\begin{subarray}{c}x,y\in\overline{\Omega}\\\
x\neq y\end{subarray}}\frac{\lVert a(x)-a(y)\rVert_{C^{l}([-M,M])}}{\lvert
x-y\rvert^{\alpha}}$ $\displaystyle=$ $\displaystyle\lVert
a\rVert_{C^{k}([-M,M],C^{\alpha}(\overline{\Omega}))}.\qed$

Next we study a fixed point equation related to the linearized equation.
Below, we let

(2.11) $B_{\delta}=\\{u\in C^{2,\alpha}(\overline{\Omega})\,:\,\lVert
u\rVert_{C^{2,\alpha}(\overline{\Omega})}<\delta\\}.$

######  Lemma 2.3.

Let $a\in C^{3}(\mathbb{R},C^{1,\alpha}(\overline{\Omega}))$ and $w\in
C^{2,\alpha}(\overline{\Omega})$ be a solution of $\Delta w+a(x,w)=0$ in
$\Omega.$ Let $q=\partial_{u}a(x,w)$ and let $G_{q}(F,f)$ be the solution
operator of

$\begin{cases}\Delta u+qu=F&\quad\text{in }\Omega\\\
u=f-\Phi(F,f)&\quad\text{on }\partial\Omega\end{cases}$

provided by Lemma 2.1. Define $R_{v}(r)=R(v+r)$ where
$R:C^{2,\alpha}(\overline{\Omega})\to C^{\alpha}(\overline{\Omega})$ is given
by

(2.12)
$R(h)(x)\coloneqq\int_{0}^{1}[\partial_{u}a(x,w(x)+th(x))-\partial_{u}a(x,w(x))]h(x)\,dt.$

For fixed $v\in C^{2,\alpha}(\overline{\Omega})$ define
$T_{v}:C^{2,\alpha}(\overline{\Omega})\to C^{2,\alpha}(\overline{\Omega})$ by
$T_{v}(r)=-G_{q}(R_{v}(r),0)$.

Under the above assumptions, there exists $\delta>0$ such that
$T_{v}|_{B_{\delta}}:B_{\delta}\to B_{\delta}$ is a contraction. Furthermore,

(2.13) $\lVert T_{v}(h)\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq C\lVert
v+h\rVert_{C^{2,\alpha}(\overline{\Omega})}^{2},\qquad h\in B_{\delta}.$

Consequently there exists a unique $r\in B_{\delta}$ solving the fixed point
equation $r=T_{v}(r)$. The function $r$ is also the unique solution with
$r\perp N_{q}$ of

(2.14) $\begin{cases}\Delta r+\partial_{u}a(x,w)r=-R(v+r)&\quad\text{in
}\Omega\\\ r|_{\partial\Omega}\in\partial_{\nu}N_{q}&\quad\text{on
}\partial\Omega,\end{cases}$

and necessarily $r|_{\partial\Omega}=\Phi(R(v+r),0)$.

###### Proof.

We first show that $T_{v}$ maps $B_{\delta}$ into itself when $\delta$ is
small enough. Let $v,r\in B_{\delta}$ where initially $\delta\leq 1$. By the
mapping properties for $G$ in Lemma 2.1 and the fundamental theorem of
calculus, we have

(2.15) $\displaystyle\lVert T_{v}(r)\rVert_{C^{2,\alpha}(\overline{\Omega})}$
$\displaystyle\leq C\lVert R(v+r)\rVert_{C^{\alpha}(\overline{\Omega})}$
$\displaystyle=C\left\lVert\int_{0}^{1}[\partial_{u}a(x,w+t(v+r))-\partial_{u}a(x,w)](v+r)\,dt\right\rVert_{C^{\alpha}(\overline{\Omega})}$
$\displaystyle=C\left\lVert\int_{0}^{1}\int_{0}^{1}\partial_{u}^{2}a(x,w+st(v+r))t(v+r)^{2}\,ds\,dt\right\rVert_{C^{\alpha}(\overline{\Omega})}$

From Lemma 2.2 we get for $s,t\in[0,1]$ that

(2.16)
$\lVert\partial_{u}^{2}a(x,w+st(v+r))\rVert_{C^{\alpha}(\overline{\Omega})}\leq
C_{w,a}.$

Since $v\in B_{\delta}$ we have, by using (2.16) in (2.15), that

$\lVert T_{v}(r)\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq C\lVert
v+r\rVert_{C^{\alpha}(\overline{\Omega})}^{2}\leq C\lVert
v+r\rVert_{C^{2,\alpha}(\overline{\Omega})}^{2}\leq C\delta^{2}.$

The second inequality above proves (2.13). For $\delta$ small enough we get

$\lVert T_{v}(r)\rVert_{C^{k,\alpha}(\overline{\Omega})}\leq\delta$

and conclude that $T_{v}$ indeed maps $B_{\delta}$ into itself.

Next we show the contraction property of $T_{v}.$ Let $r_{1},r_{2}\in
B_{\delta}$. Then, as in (2.15), we have

$\lVert T_{v}(r_{1})-T_{v}(r_{2})\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq
C\lVert R_{v}(r_{1})-R_{v}(r_{2})\rVert_{C^{\alpha}(\overline{\Omega})}.$

Denote $u_{i}=v+r_{i}$, $i=1,2$. Then

$\displaystyle R_{v}(r_{1})-R_{v}(r_{2})=$
$\displaystyle\int_{0}^{1}\left(\partial_{u}a(x,w+tu_{1})-\partial_{u}a(x,w)\right)u_{1}-\left(\partial_{u}a(x,w+tu_{2})-\partial_{u}a(x,w)\right)u_{2}\,dt$
$\displaystyle=$
$\displaystyle\int_{0}^{1}(\partial_{u}a(x,w+tu_{1})-\partial_{u}a(x,w)+\partial_{u}a(x,w+tu_{2})-\partial_{u}a(x,w))(u_{1}-u_{2})$
$\displaystyle-(\partial_{u}a(x,w+tu_{2})-\partial_{u}a(x,w))u_{1}+(\partial_{u}a(x,w+tu_{1})-\partial_{u}a(x,w))u_{2}\,dt$
$\displaystyle=$
$\displaystyle\int_{0}^{1}(u_{1}-u_{2})\left(tu_{1}\int_{0}^{1}\partial_{u}^{2}a(x,w+stu_{1})\,ds+tu_{2}\int_{0}^{1}\partial_{u}^{2}a(x,w+stu_{2})\,ds\right)$
$\displaystyle-
tu_{1}u_{2}\int_{0}^{1}\partial_{u}^{2}a(x,w+stu_{2})\,ds+tu_{1}u_{2}\int_{0}^{1}\partial_{u}^{2}a(x,w+stu_{1})\,ds\,dt$
$\displaystyle=$
$\displaystyle\int_{0}^{1}(u_{1}-u_{2})\Bigg{(}tu_{1}\int_{0}^{1}\partial_{u}^{2}a(x,w+stu_{1})\,ds+tu_{2}\int_{0}^{1}\partial_{u}^{2}a(x,w+stu_{2})\,ds$
$\displaystyle+t^{2}u_{1}u_{2}\int_{0}^{1}\int_{0}^{1}s\partial_{u}^{3}a(x,w+ystu_{1}+(1-y)stu_{2})\,dy\,ds\Bigg{)}\,dt.$

Let us estimate the norm of the last expression term by term. Since
$v,r_{i}\in B_{\delta}$ then $u_{i}\in B_{2\delta}$ for $i\in\\{1,2\\}$. Using
this and (2.16) we have that

$\displaystyle\left\lVert
tu_{i}\int_{0}^{1}\partial_{u}^{2}a(x,w+stu_{i})\,ds\right\rVert_{C^{\alpha}(\overline{\Omega})}$
$\displaystyle\leq t\lVert
u_{i}\rVert_{C^{\alpha}(\overline{\Omega})}\int_{0}^{1}\lVert\partial_{u}^{2}a(x,w+stu_{i})\rVert_{C^{\alpha}(\overline{\Omega})}\,ds$
$\displaystyle\leq t\delta C.$

Just as in (2.16) we get
$\lVert\partial_{u}^{3}a(x,w+h)\rVert_{C^{\alpha}(\overline{\Omega})}\leq
C_{w,a}$. Using this we estimate

$\displaystyle\left\lVert
t^{2}u_{1}u_{2}\int_{0}^{1}\int_{0}^{1}s\partial_{u}^{3}a(x,w+ystu_{1}+(1-y)stu_{2})\,dy\,ds\right\rVert_{C^{\alpha}(\overline{\Omega})}$
$\displaystyle\leq t^{2}\lVert
u_{1}u_{2}\rVert_{C^{\alpha}(\overline{\Omega})}\int_{0}^{1}\int_{0}^{1}s\lVert\partial_{u}^{3}a(x,w+ystu_{1}+(1-y)stu_{2})\rVert_{C^{\alpha}(\overline{\Omega})}\,dy\,ds$
$\displaystyle\leq t^{2}\delta^{2}C$

Finally, for small enough $\delta>0$ we have

$\displaystyle\lVert T_{v}(r_{1})-T_{v}(r_{2})\rVert_{C^{2,\alpha}(\Omega)}$
$\displaystyle\leq C_{a}\int_{0}^{1}\lVert
u_{1}-u_{2}\rVert_{C^{\alpha}(\overline{\Omega})}(2t\delta
C+t^{2}\delta^{2}C)\,dt$ $\displaystyle\leq\frac{1}{2}\lVert
u_{1}-u_{2}\rVert_{C^{2,\alpha}(\overline{\Omega})}$
$\displaystyle=\frac{1}{2}\lVert
r_{1}-r_{2}\rVert_{C^{2,\alpha}(\overline{\Omega})}.$

Thus $T_{v}$ is a contraction and the Banach fixed point theorem ensures
existence and uniqueness of solution to the equation $r=T_{v}(r)$ in
$B_{\delta}$. The definition of $T_{v}$ ensures that $r$ also solves (2.14). ∎

We now construct the smooth solution map $S_{a,w}$, which maps small solutions
$v$ of the linearized equation $\Delta v+\partial_{u}a(x,w)v=0$ to solutions
$u$ of the nonlinear equation $\Delta u+a(x,u)=0$ that are close to some fixed
solution $w$. Below, if $F:U\to Y$ is a $C^{1}$ map where $X$ and $Y$ are
Banach spaces and $U\subseteq X$ is open, we will denote its Fréchet
derivative by

$DF(x)=F^{\prime}(x).$

Recall that $B_{\delta}$ is given by (2.11).

######  Lemma 2.4.

Let $a\in C^{k}(\mathbb{R},C^{1,\alpha}(\overline{\Omega}))$, $k\geq 3$. Let
$w\in C^{2,\alpha}(\overline{\Omega})$ be a solution of $\Delta w+a(x,w)=0$.
Let $q(x)=\partial_{u}a(x,w(x))$. Then there exist $\delta,C>0$ and a
$C^{k-1}$ map $Q=Q_{a,w}:B_{\delta}\to B_{\delta}$ satisfying

$\displaystyle Q(B_{\delta})$ $\displaystyle\subseteq N_{q}^{\perp},$
$\displaystyle Q(B_{\delta})|_{\partial\Omega}$
$\displaystyle\subseteq\partial_{\nu}N_{q},$ $\displaystyle Q(0)$
$\displaystyle=DQ(0)=0$

and

(2.17) $\lVert Q(v)\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq C\lVert
v\rVert_{C^{2,\alpha}(\overline{\Omega})}^{2},$

such that $S_{a}\colon B_{\delta}\to C^{2,\alpha}(\overline{\Omega})$ defined
by $u=S_{a,w}(v)=w+v+Q(v)$ is a $C^{k-1}$ map satisfying

(2.18) $\Delta u+a(x,u)=\Delta v+qv\quad\text{in }\Omega$

with $S_{a,w}^{\prime}(0)v=v$. In particular, if $v$ solves $\Delta v+qv=0$,
then $u=S_{a,w}(v)$ solves $\Delta u+a(x,u)=0$.

Conversely, if $\delta$ is small enough, then given any solution $u\in
C^{2,\alpha}(\overline{\Omega})$ of $\Delta u+a(x,u)=0$ with $\lVert
u-w\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq\delta$ there exists a unique
solution $v\in C^{2,\alpha}(\overline{\Omega})$ of $\Delta v+qv=0$ such that
$u=S_{a,w}(v)$. The function $v$ is explicitly given by

(2.19) $v=P_{N_{q}}(u-w)+G_{q}(0,(u-w)|_{\partial\Omega}),$

and one has $\lVert v\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq C\lVert
u-w\rVert_{C^{2,\alpha}(\overline{\Omega})}$.

###### Proof.

We first construct the map $Q$. Let $v\in B_{\delta}$. We look for a solution
$u$ of (2.18) having the form $u=w+v+r$ and formulate a fixed point equation
for $r$. Taylor expansion gives

$\displaystyle\Delta u+a(x,u)$ $\displaystyle=\Delta(w+v+r)+a(x,w+v+r)$
$\displaystyle=\Delta w+\Delta(v+r)+a(x,w)+\partial_{u}a(x,w)(v+r)+R_{v}(r)$

where
$R_{v}(r)(x)=\int_{0}^{1}[\partial_{u}a(x,w(x)+t[v(x)+r(x)])-\partial_{u}a(x,w(x))][v(x)+r(x)]\,dt$.
Since $w$ is a solution of $\Delta w+a(x,w)=0$, we see that $u$ solves $\Delta
u+a(x,u)=\Delta v+qv$ if $r$ satisfies

$\Delta r+\partial_{u}a(x,w)r+R_{v}(r)=0.$

For each $v\in B_{\delta},$ Lemma 2.3 ensures existence and uniqueness of a
solution $r=r_{v}$ in $B_{\delta}$ with $r\perp N_{q}$ and
$r|_{\partial\Omega}\in\partial_{\nu}N_{q}$. Hence the mapping $v\mapsto
r_{v}$ is well-defined for $v\in B_{\delta}$. Next we use the implicit
function theorem to show that this mapping is $C^{k-1}$.

Let $F:C^{2,\alpha}(\overline{\Omega})\times
C^{2,\alpha}(\overline{\Omega})\to C^{2,\alpha}(\overline{\Omega})$ be defined
by

$F(v,r)=r-T_{v}(r)=r+G_{q}(R_{v}(r),0).$

From the definition of $R_{v}$ and $G_{q}$ it follows that $F(0,0)=0$. Next,
$R_{v}$ is $C^{k-1}$ since $\partial_{u}a\in
C^{k-1,\alpha}(\mathbb{R};C^{1,\alpha}(\overline{\Omega}))$. Consequently, $F$
is $C^{k-1}$ since $G_{q}$ is linear. Moreover

$\displaystyle D_{r}F|_{(0,0)}(h)$
$\displaystyle=h+G_{q}\left(\int_{0}^{1}\partial_{u}^{2}a(x,w+t(v+r))(v+r)+\partial_{u}a(x,w+t(v+r))-\partial_{u}a(x,w)\,dt|_{(v,r)=(0,0)},0\right)$
$\displaystyle=h$

and this is a linear homeomorphism from $C^{2,\alpha}(\overline{\Omega})$ to
itself. Thus the implicit function theorem [HG27, Theorem 4] ensures the
existence of open balls $B_{\delta_{1}},B_{\delta_{2}}$ and a $C^{k-1}$ map
$Q\colon B_{\delta_{1}}\to B_{\delta_{2}}$ such that

$F(v,Q(v))=0.$

Since $r_{v}$ found by Lemma 2.3 above is the unique solution of
$F(v,\,\cdot\,)=0$ in $B_{\delta}$ for $v\in B_{\delta}$, we conclude that for
$\delta<\min\\{\delta_{1},\delta_{2}\\}$, $r_{v}$ belongs to $B_{\delta_{2}}$
and hence $Q(v)=r_{v}$. We have now shown that for each $v\in B_{\delta}$
there is a unique $r_{v}\in B_{\delta}$ with $r\perp N_{q}$ and
$r|_{\partial\Omega}\in\partial_{\nu}N_{q}$ such that $u_{v}=w+v+r_{v}$ is a
solution of the equation $\Delta u_{v}+a(x,u_{v})=\Delta v+qv$. Moreover, the
map $v\mapsto u_{v}=S_{a,w}(v)$ is $C^{k-1}$.

Next we show that $Q$ satisfies the other properties in the statement. The
estimate (2.13) implies

$\lVert r\rVert_{C^{2,\alpha}(\overline{\Omega})}=\lVert
T_{v}(r)\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq C\lVert
v\rVert_{C^{2,\alpha}(\overline{\Omega})}^{2}+C\lVert
r\rVert_{C^{2,\alpha}(\overline{\Omega})}^{2}$

and from this we get that

$\lVert v\rVert_{C^{2,\alpha}(\overline{\Omega})}^{2}\geq\lVert
r\rVert_{C^{2,\alpha}(\overline{\Omega})}-C\lVert
r\rVert_{C^{2,\alpha}(\overline{\Omega})}^{2}\geq\lVert
r\rVert_{C^{2,\alpha}(\overline{\Omega})}(1-C\delta).$

For $\delta$ small enough, and since $Q(v)=r$, we have

$\lVert Q(v)\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq C\lVert
v\rVert_{C^{2,\alpha}(\overline{\Omega})}^{2}.$

This proves (2.17) and shows that $Q(0)=0$. Using (2.17) together with
$Q(0)=0$ implies that $DQ(0)=0$. Since $Q(v)=r_{v}$, we have $Q(v)\in
N_{q}^{\perp}$ and $Q(v)|_{\partial\Omega}\in\partial_{\nu}N_{q}$.

We now prove the converse statement. Suppose that $u\in
C^{2,\alpha}(\overline{\Omega})$ solves $\Delta u+a(x,u)=0$ in $\Omega$ and
$\lVert u-w\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq\delta$. We write
$\tilde{u}=u-w$ and want to construct $v$ solving $\Delta v+qv=0$ such that
$\tilde{u}=v+Q(v)$. Denote by $P_{N_{q}}$ and $P_{\partial_{\nu}N_{q}}$ the
$L^{2}$-orthogonal projections to the finite dimensional spaces $N_{q}$ and
$\partial_{\nu}N_{q}$, respectively. Motivated by the conditions
$Q(B_{\delta})\subseteq N_{q}^{\perp}$ and
$Q(B_{\delta})|_{\partial\Omega}\subseteq\partial_{\nu}N_{q}$ we define

$\psi=P_{N_{q}}\tilde{u},$

and let $\varphi$ to be the unique solution given by Lemma 2.1 of the problem

$\Delta\varphi+q\varphi=0\text{ in $\Omega$},\qquad\varphi\perp
N_{q},\qquad\varphi|_{\partial\Omega}=(\mathrm{Id}-P_{\partial_{\nu}N_{q}})(\tilde{u}|_{\partial\Omega}).$

Let $v=\varphi+\psi$, which means that $v$ is given by (2.19). It follows that
$\Delta v+qv=0$ and
$v|_{\partial\Omega}=(\mathrm{Id}-P_{\partial_{\nu}N_{q}})(\tilde{u}|_{\partial\Omega})$.
We also have

$\lVert v\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq
C\lVert\tilde{u}\rVert_{C^{2,\alpha}(\overline{\Omega})}=C\lVert
u-w\rVert_{C^{2,\alpha}(\overline{\Omega})}.$

It remains to show that $r=\tilde{u}-v=u-w-v$ satisfies $r=Q(v)$. By the above
conditions we have $r\perp N_{q}$ and
$r|_{\partial\Omega}\in\partial_{\nu}N_{q}$, and $r$ satisfies

$\displaystyle(\Delta+q)r$
$\displaystyle=(\Delta+q)(u-w)=-(a(x,u)-a(x,w)-q(u-w))$
$\displaystyle=-(a(x,w+v+r)-a(x,w)-q(v+r))=-R_{v}(r).$

The first part of the proof implies that $r=Q(v)$ if $\delta$ is chosen small
enough. This proves that $u=S_{a,w}(v)$. To show that $v$ is unique suppose
that $u=S_{a,w}(\tilde{v})$ for another solution $\tilde{v}$. Then the
definition of $S_{a,w}$ gives

$v-\tilde{v}=Q(\tilde{v})-Q(v).$

Thus $v-\tilde{v}\perp N_{q}$ and
$v-\tilde{v}|_{\partial\Omega}\in\partial_{\nu}N_{q}$. Since
$(\Delta+q)(v-\tilde{v})=0$, Lemma 2.1 implies $v=\tilde{v}$ showing that $v$
is unique. ∎

######  Lemma 2.5.

In the setting of Lemma 2.4, if $v$ is small and solves $\Delta v+qv=0$ for
$q=\partial_{u}a(x,w)$, define

$\displaystyle q_{v}$ $\displaystyle=\partial_{u}a(x,S_{a,w}(v)),$
$\displaystyle V_{\tilde{q}}$ $\displaystyle=\\{h\in
C^{2,\alpha}(\overline{\Omega})\,:\,\Delta h+\tilde{q}h=0\\}$

If $v\in V_{q}$ is small, the map $DS_{a,w}(v)$ is an isomorphism from $V_{q}$
onto $V_{q_{v}}$.

###### Proof.

Let $v$ be a small solution of $\Delta v+qv=0$ and $v_{t}=v+th$ where $h\in
V_{q}$. Then $u_{t}=S_{a,w}(v_{t})$ solves

$\Delta u_{t}+a(x,u_{t})=0.$

Since $u_{t}$ is $C^{1}$ in $t$, the function
$\dot{u}_{0}=\partial_{t}u_{t}|_{t=0}=DS_{a,w}(v)h$ satisfies

$\Delta\dot{u}_{0}+\partial_{u}a(x,S_{a,w}(v))\dot{u}_{0}=0.$

Thus $DS_{a,w}(v)$ maps $V_{q}$ into $V_{q_{v}}$.

Now suppose that $v\in V_{q}$ is small and $\tilde{h}\in V_{q_{v}}$. For $t$
small define $u_{t}=S_{a,S_{a,w}(v)}(t\tilde{h})$. By the converse part of
Lemma 2.4, if $v$ and $t$ are small enough one has $u_{t}=S_{a,w}(v_{t})$ for
a unique small solution $v_{t}\in V_{q}$, and $v_{t}$ is given by

$v_{t}=P_{N_{q}}(u_{t}-w)+G_{q}(0,(u_{t}-w)|_{\partial\Omega}).$

In particular, $v_{t}$ is $C^{1}$ in $t$, and since
$S_{a,w}(v_{0})=u_{0}=S_{a,S_{a,w}(v)}(0)=S_{a,w}(v)$ uniqueness gives
$v_{0}=v$. Differentiating the identities $u_{t}=S_{a,w}(v_{t})$ and
$u_{t}=S_{a,S_{a,w}(v)}(t\tilde{h})$ and using $DS(0)=\mathrm{Id}$ gives

$DS_{a,w}(v)\dot{v}_{0}=\dot{u}_{0}=DS_{a,S_{a,w}(v)}(0)\tilde{h}=\tilde{h}.$

This shows that $DS_{a,w}(v):V_{q}\to V_{q_{v}}$ is surjective.

Finally, suppose that $h\in V_{q}$ satisfies $DS_{a,w}(v)h=0$. Since
$S_{a,w}(v)=w+v+Q_{a,w}(v)$, we have

$h+DQ_{a,w}(v)h=0.$

But $DQ_{a,w}(0)=0$, which implies that $\lVert DQ_{a,w}(v)\rVert\leq 1/2$
when $v$ is sufficiently small. Here we used that $Q$ is a $C^{k-1}$ map where
$k\geq 3$. This implies that $\lVert h\rVert\leq\frac{1}{2}\lVert h\rVert$,
showing that $h=0$. Thus $DS_{a,w}(v):V_{q}\to V_{q_{v}}$ is bijective and
bounded, and by the open mapping theorem it is an isomorphism. ∎

##  3\. Estimates for solutions in terms of their Cauchy data

In this section we prove estimates for functions in terms of their Cauchy data
and in particular for solutions of the nonlinear equation

(3.1) $\Delta u+a(x,u)=0\text{ in }\Omega.$

The estimate for (3.1) is used in section 4 when constructing the second
solution map required for the linearization methods.

First we obtain an auxiliary regularity estimate that is then used to prove
the quantitative results.

######  Lemma 3.1.

Let $\Omega\subseteq\mathbb{R}^{n}$ be a bounded open set with $C^{\infty}$
boundary and let $q\in C^{\alpha}(\overline{\Omega})$. There is $C>0$ such
that for any $u\in C^{2,\alpha}(\overline{\Omega})$ we have

$\lVert u\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq C(\lVert
u\rVert_{C^{2,\alpha}(\partial\Omega)}+\lVert(\Delta+q)u\rVert_{C^{\alpha}(\overline{\Omega})}+\lVert
u\rVert_{H^{1}(\Omega)}).$

###### Proof.

Consider the Banach space $X=C^{2,\alpha}(\partial\Omega)\times
C^{\alpha}(\overline{\Omega})\times H^{1}(\Omega)$ with norm

$\lVert(f,F,v)\rVert_{X}=\lVert f\rVert_{C^{2,\alpha}(\partial\Omega)}+\lVert
F\rVert_{C^{\alpha}(\overline{\Omega})}+\lVert v\rVert_{H^{1}(\Omega)}.$

We define the map

$T:C^{2,\alpha}(\overline{\Omega})\to X,\ \
T(u)=(u|_{\partial\Omega},(\Delta+q)u,j(u)),$

where $j$ is the inclusion $C^{2,\alpha}(\overline{\Omega})\to H^{1}(\Omega)$.
Then $T$ is bounded, linear and injective. We claim that $T$ has closed range.
To see this, suppose that $u_{j}\in C^{2,\alpha}(\overline{\Omega})$ and
$T(u_{j})\to(f,F,v)$ in $X$. Then $u_{j}\to v$ in $H^{1}(\Omega)$,
$u_{j}|_{\partial\Omega}\to f$ in $C^{2,\alpha}(\partial\Omega)$ and
$(\Delta+q)u_{j}\to F$ in $C^{\alpha}(\overline{\Omega})$. On the other hand
$(\Delta+q)u_{j}\to(\Delta+q)v$ in $H^{-1}(\Omega)$ and
$u_{j}|_{\partial\Omega}\to v|_{\partial\Omega}$ in $H^{1/2}(\partial\Omega)$,
and by uniqueness of limits one has $(\Delta+q)v=F$ and
$v|_{\partial\Omega}=f$. By elliptic regularity, the weak solution $v$
satisfies $v\in C^{2,\alpha}(\overline{\Omega})$. Thus $(f,F,v)=T(v)$ and
$\mathrm{Ran}(T)$ is closed.

We have proved that $T:C^{2,\alpha}(\overline{\Omega})\to\mathrm{Ran}(T)$ is a
bounded linear bijection between Banach spaces. By the open mapping theorem it
has a bounded inverse $S:\mathrm{Ran}(T)\to C^{2,\alpha}(\overline{\Omega})$,
and thus for any $u\in C^{2,\alpha}(\overline{\Omega})$ one has

$\lVert u\rVert_{C^{2,\alpha}(\overline{\Omega})}=\lVert
STu\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq C\lVert Tu\rVert_{X}.$

This proves the claim. ∎

Next we show a quantitative uniqueness result that follows by combining Lemma
3.1 with the unique continuation principle. This is the used in Section 4
related to the first linearization of (3.1).

######  Lemma 3.2.

Let $\Omega\subseteq\mathbb{R}^{n}$ be a bounded open set with $C^{\infty}$
boundary and let $q\in C^{\alpha}(\overline{\Omega})$. There is $C>0$ such
that for any $u\in C^{2,\alpha}(\overline{\Omega})$ we have

$\lVert u\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq C(\lVert
u\rVert_{C^{2,\alpha}(\partial\Omega)}+\lVert\partial_{\nu}u\rVert_{C^{1,\alpha}(\partial\Omega)}+\lVert\Delta
u+qu\rVert_{C^{\alpha}(\overline{\Omega})}).$

###### Proof.

We argue by contradiction and assume that for any $m$ there is $u_{m}$ such
that

(3.2) $\lVert u_{m}\rVert_{C^{2,\alpha}(\overline{\Omega})}>m(\lVert
u_{m}\rVert_{C^{2,\alpha}(\partial\Omega)}+\lVert\partial_{\nu}u_{m}\rVert_{C^{1,\alpha}(\partial\Omega)}+\lVert(\Delta+q)u_{m}\rVert_{C^{\alpha}(\overline{\Omega})}).$

On the other hand, Lemma 3.1 implies that

$\lVert u_{m}\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq C(\lVert
u_{m}\rVert_{C^{2,\alpha}(\partial\Omega)}+\lVert(\Delta+q)u_{m}\rVert_{C^{\alpha}(\overline{\Omega})}+\lVert
u_{m}\rVert_{H^{1}(\Omega)}).$

Normalize $u_{m}$ so that $\lVert u_{m}\rVert_{H^{1}(\Omega)}=1$. Then using
(3.2) yields

$\displaystyle\lVert u_{m}\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq
C(\frac{1}{m}\lVert u_{m}\rVert_{C^{2,\alpha}(\overline{\Omega})}+1)$

Then $\lVert u_{m}\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq C$ uniformly
when $m$ is sufficiently large.

By Theorem $1.34$ in [AF03] the embedding $C^{2,\alpha}(\overline{\Omega})\to
C^{2}(\overline{\Omega})$ is compact. Hence there is a subsequence, still
denoted $u_{m},$ that converges in $C^{2}(\overline{\Omega})$ to some $u\in
C^{2}(\overline{\Omega}).$ On the other hand, from (3.2) and the bound $\lVert
u_{m}\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq C$ we see that

$u_{m}|_{\partial\Omega}\to 0,\qquad\partial_{\nu}u_{m}|_{\partial\Omega}\to
0,\qquad(\Delta+q)u_{m}\to 0$

in the respective spaces. By uniqueness of limits we have
$u|_{\partial\Omega}=0$, $\partial_{\nu}u|_{\partial\Omega}=0$, and
$(\Delta+q)u=0$. Consequently, $u\equiv 0$ by unique continuation, which
contradicts $\lVert u\rVert_{H^{1}(\Omega)}=\lim\lVert
u_{m}\rVert_{H^{1}(\Omega)}=1$. ∎

Finally, we invoke a Carleman estimate to show that solutions of semilinear
equations of the form (3.1) are uniquely and stably determined by their Cauchy
data.

######  Lemma 3.3.

Let $a\in C^{2}(\mathbb{R},C^{\alpha}(\overline{\Omega}))$, and let $u_{0}\in
C^{2,\alpha}(\overline{\Omega})$ solve $\Delta u_{0}+a(x,u_{0})=0$ in
$\Omega$. If $u\in C^{2,\alpha}(\overline{\Omega})$ is any other solution of
$\Delta u+a(x,u)=0$ in $\Omega$ and $\lVert
u\rVert_{C^{2,\alpha}(\overline{\Omega})}$, $\lVert
u_{0}\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq M$, then

(3.3) $\lVert u-u_{0}\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq
C(M,a)(\lVert
u-u_{0}\rVert_{C^{2,\alpha}(\partial\Omega)}+\lVert\partial_{\nu}(u-u_{0})\rVert_{C^{1,\alpha}(\partial\Omega)}).$

###### Proof.

We use a standard Carleman estimate (see e.g. [Cho21, Theorem 4.1]): there are
$C,\tau_{0}>0$ and $\varphi\in C^{\infty}(\overline{\Omega})$ such that when
$\tau\geq\tau_{0}$, one has

$\lVert e^{\tau\varphi}v\rVert_{L^{2}(\Omega)}+\frac{1}{\tau}\lVert
e^{\tau\varphi}\nabla v\rVert_{L^{2}(\Omega)}\leq\frac{C}{\tau^{3/2}}\lVert
e^{\tau\varphi}\Delta v\rVert_{L^{2}(\Omega)}+C\lVert
e^{\tau\varphi}v\rVert_{L^{2}(\partial\Omega)}+\frac{C}{\tau}\lVert
e^{\tau\varphi}\nabla v\rVert_{L^{2}(\partial\Omega)}$

for any $v\in C^{2}(\overline{\Omega})$. We apply this with $v=u-u_{0}$ and
use the fact that

(3.4) $-\Delta
v=a(x,u)-a(x,u_{0})=\left[\int_{0}^{1}\partial_{u}a(x,(1-t)u_{0}+tu)\,dt\right]v.$

Since $|u|,|u_{0}|\leq M$, we get from Lemma 2.2 that

(3.5) $\lvert\Delta v(x)\rvert\leq C(M,a)\lvert v(x)\rvert.$

Thus, choosing $\tau=\tau(M,a)$ large but fixed, we get

$\frac{1}{2}\lVert e^{\tau\varphi}v\rVert_{L^{2}(\Omega)}+\frac{1}{\tau}\lVert
e^{\tau\varphi}\nabla v\rVert_{L^{2}(\Omega)}\leq C(\lVert
e^{\tau\varphi}v\rVert_{L^{2}(\partial\Omega)}+\lVert e^{\tau\varphi}\nabla
v\rVert_{L^{2}(\partial\Omega)}).$

Since $c(M,a)\leq e^{\tau\varphi}\leq C(M,a)$, we have

(3.6) $\lVert v\rVert_{H^{1}(\Omega)}\leq C(M,a)(\lVert
v\rVert_{H^{1}(\partial\Omega)}+\lVert\partial_{\nu}v\rVert_{L^{2}(\partial\Omega)}).$

We still need to estimate $\lVert v\rVert_{C^{2,\alpha}(\overline{\Omega})}$.
First, Lemma 3.1 gives

$\lVert v\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq C\left(\lVert
v\rVert_{C^{2,\alpha}(\partial\Omega)}+\lVert\Delta
v\rVert_{C^{\alpha}(\overline{\Omega})}+\lVert
v\rVert_{H^{1}(\Omega)}\right).$

From (3.4) we observe that

$\lVert\Delta v\rVert_{C^{\alpha}(\overline{\Omega})}\leq
C\left[\int_{0}^{1}\lVert\partial_{u}a(\,\cdot\,,(1-t)u_{0}(\,\cdot\,)+tu(\,\cdot\,))\rVert_{C^{\alpha}(\overline{\Omega})}\,dt\right]\lVert
v\rVert_{C^{\alpha}(\overline{\Omega})}.$

By using Lemma 2.2 to estimate the integral from above by a constant depending
on $a,u$ and $u_{0}$ we have

$\lVert\Delta v\rVert_{C^{\alpha}(\overline{\Omega})}\leq C\lVert
v\rVert_{C^{\alpha}(\overline{\Omega})}.$

Thus we get

(3.7) $\lVert v\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq C\left(\lVert
v\rVert_{C^{2,\alpha}(\partial\Omega)}+\lVert
v\rVert_{C^{\alpha}(\overline{\Omega})}+\lVert
v\rVert_{H^{1}(\Omega)}\right).$

Next, we have by the Sobolev embedding [AF03, Theorem 4.12 Part 2] that
$W^{1,s}\subseteq C^{\alpha}$ where $s=\frac{n}{1-\alpha}$. Using this and
[AF03, Theorem $5.2$ $(3)$] we obtain that

$\lVert v\rVert_{C^{\alpha}(\overline{\Omega})}\leq C\lVert
v\rVert_{W^{1,s}(\Omega)}\leq C\lVert v\rVert_{W^{2,s}(\Omega)}^{1/2}\lVert
v\rVert_{L^{s}(\Omega)}^{1/2}.$

Then we use interpolation of $L^{p}$-spaces (see for example [Eva10, Appendix
B]) to get

$\lVert v\rVert_{L^{s}(\Omega)}\leq C\lVert
v\rVert_{L^{2}(\Omega)}^{\lambda}\lVert v\rVert_{L^{r}(\Omega)}^{1-\lambda}$

for some $r>s$. Estimating the $L^{r}$\- and $W^{2,s}$-norms by the
$C^{2,\alpha}$-norm we have

$\lVert v\rVert_{C^{\alpha}(\overline{\Omega})}\leq C\lVert
v\rVert_{C^{2,\alpha}(\overline{\Omega})}^{(2-\lambda)/2}\lVert
v\rVert_{L^{2}(\Omega)}^{\lambda/2}.$

Using Young’s inequality with $\varepsilon$ for $p=2/\lambda$ and $q=p/(p-1)$
gives

$\lVert v\rVert_{C^{\alpha}(\overline{\Omega})}\leq C(\varepsilon\lVert
v\rVert_{C^{2,\alpha}(\overline{\Omega})}^{q(2-\lambda)/2}+C_{\varepsilon}\lVert
v\rVert_{L^{2}(\Omega)})=(\varepsilon\lVert
v\rVert_{C^{2,\alpha}(\overline{\Omega})}+C_{\varepsilon}\lVert
v\rVert_{L^{2}(\Omega)})$

since $q=\frac{2}{2-\lambda}$. Using this in (3.7) and choosing
$\varepsilon>0$ sufficiently small finally gives

$\lVert v\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq C\left(\lVert
v\rVert_{C^{2,\alpha}(\partial\Omega)}+\lVert v\rVert_{H^{1}(\Omega)}\right).$

Combining the last estimate with (3.6) proves the result. ∎

##  4\. A smooth solution map with prescribed Cauchy data

As mentioned previously, in order to prove the main results, we need to
construct two smooth solution maps for the nonlinear equations

(4.1) $\Delta u+a_{i}(x,u)=0\text{ in }\Omega$

for $i=1,2$. In Section 2 we constructed the first one. One reason why we
cannot use the solution map $S_{a_{i},w_{i}}$ for both $i\in\\{1,2\\}$ is that
we need to control the Cauchy data. If $u_{1}=S_{a_{1},w_{1}}(v_{1})$ then we
would need to find a solution $u_{2}=S_{a_{2},w_{2}}(v_{2})$ such that
$u_{1},u_{2}$ have the same Cauchy data. But the solution maps
$S_{a_{i},w_{i}}$ don’t provide enough control of the Neumann data to
guarantee that this is possible. Another issue, in particular when identifying
the first derivatives $\partial_{u}a_{i}(x,w_{i})$, is that the method of
linearization relies on differentiating both solution maps
$S_{a_{1},w_{1}},S_{a_{2},w_{2}}$ in the same direction $v$. But in order to
use the same parameter $v$ for both operators, $v$ needs to solve both
linearized equations

(4.2) $\Delta v+\partial_{u}a_{i}(x,w_{i})v=0\text{ in }\Omega$

for $i\in\\{1,2\\}$. However, before having identified the first derivatives,
$\partial_{u}a_{1}(x,w_{1})=\partial_{u}a_{2}(x,w_{2})$, we don’t know that
such functions $v$ exist. So the goal of this section is to construct a new
solution map $T_{a_{i},w_{i}}$ that resolves these two issues. That is, we aim
to construct a smooth solution map $T_{a_{2},w_{2}}$ for

(4.3) $\Delta u+a_{2}(x,u)=0\text{ in }\Omega$

parametrized on solutions $v$ of

$\Delta v+\partial_{u}a_{1}(x,w_{1})v=0\text{ in }\Omega$

such that $T_{a_{2},w_{2}}(v)$ and $S_{a_{1},w_{1}}(v)$ have the same Cauchy
data.

Before constructing $T_{a_{i},w_{i}}$, we establish some preliminary results.
The construction of $T_{a_{i},w_{i}}$ is based on the implicit function
theorem. In order to properly define the function to which the implicit
function theorem is applied, we require the existence of a certain projection
mapping and existence of a bounded inverse of the Schrödinger operator
$\Delta+q$. We first establish these two results and then proceed to construct
$T_{a_{i},w_{i}}$.

######  Lemma 4.1.

Let $q\in C^{\alpha}(\overline{\Omega})$. Then the spaces

$\displaystyle Y$ $\displaystyle=\\{u\in C^{2,\alpha}(\overline{\Omega})\colon
u|_{\partial\Omega}=\partial_{\nu}u|_{\partial\Omega}=0\\}$ $\displaystyle Z$
$\displaystyle=\\{(\Delta+q)u\colon u\in Y\\}$

are Banach spaces.

###### Proof.

It follows from the continuity of the mappings
$C^{2,\alpha}(\overline{\Omega})\ni u\mapsto u|_{\partial\Omega}\in
C^{2,\alpha}(\partial\Omega)$ and $C^{2,\alpha}(\overline{\Omega})\ni
u\mapsto\partial_{\nu}u|_{\partial\Omega}\in C^{1,\alpha}(\partial\Omega)$
that $Y$ is a Banach space. To see that $Z$ is a Banach space, let
$v_{n}=\Delta w_{n}+qw_{n}\in Z$ be a sequence converging to some $v\in
C^{\alpha}(\overline{\Omega})$. Using Lemma 3.2, we have

$\lVert w_{n}-w_{m}\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq C\lVert\Delta
w_{n}+qw_{n}-\Delta w_{m}-qw_{m}\rVert_{C^{\alpha}(\overline{\Omega})},$

so that $w_{n}$ is a Cauchy sequence in $Y$. Hence there is some $w\in Y$ with
$w_{n}\to w$ in $C^{2,\alpha}(\overline{\Omega})$. Next,

$\lVert\Delta w_{n}+qw_{n}-\Delta
w-qw\rVert_{C^{\alpha}(\overline{\Omega})}\leq C\lVert
w_{n}-w\rVert_{C^{2,\alpha}(\overline{\Omega})}$

So for $v=\Delta w+qw$, we have $v_{n}\to v$ in
$C^{\alpha}(\overline{\Omega})$ and $Z$ is a Banach space. ∎

The following result shows that there is a bounded projection
$P:C^{2,\alpha}(\overline{\Omega})\to Z$. If $C^{2,\alpha}(\overline{\Omega})$
were a Hilbert space, the existence of a projection would follow from an
orthogonal decomposition $C^{2,\alpha}(\overline{\Omega})=Z\oplus W$. Since
$Z$ is the image of $\Delta+q$ acting on functions whose Cauchy data vanishes,
the orthocomplement $W$ would be the set of suitable functions $w$ with
$(\Delta+q)w=0$. Thus any $u\in C^{2,\alpha}(\overline{\Omega})$ could be
written as $u=(\Delta+q)y+w$, where $y\in Y$ and $(\Delta+q)w=0$. This shows
that $y$ needs to satisfy $(\Delta+q)^{2}y=(\Delta+q)u$. This formal argument
turns out to work also in our case.

######  Lemma 4.2.

Let $q\in C^{\alpha}(\overline{\Omega})$ and let $Y$ and $Z$ be as in Lemma
4.1. Then there exists a bounded projection $P\colon
C^{2,\alpha}(\overline{\Omega})\to Z$ such that $P(u)=(\Delta+q)y$ where $y\in
C^{4,\alpha}(\overline{\Omega})$ is the unique solution of

$\begin{cases}(\Delta+q)^{2}y=(\Delta+q)u&\text{in }\Omega\\\
y=\partial_{\nu}y=0&\text{on }\partial\Omega.\end{cases}$

###### Proof.

We first show that there is a unique solution $y\in
C^{4,\alpha}(\overline{\Omega})$. If $y$ and $\tilde{y}$ are solutions, then
$(\Delta+q)^{2}(y-\tilde{y})=0$, and integrating this equation against
$y-\tilde{y}$ gives $(\Delta+q)(y-\tilde{y})=0$. Since $y-\tilde{y}$ has
vanising Cauchy data, we see that $y=\tilde{y}$ and solutions are unique.
Existence of weak solutions $y\in H^{2}_{0}(\Omega)$ for the equation
$(\Delta+q)^{2}y+\gamma y=F\in H^{-2}(\Omega)$, where $\gamma>0$ is a constant
chosen sufficiently large depending on $q$, follows by using the Riesz
representation theorem with the coercive bilinear form
$B(y,w)=((\Delta+q)y,(\Delta+q)w)_{L^{2}(\Omega)}+\gamma(y,w)_{L^{2}(\Omega)}$
for $y,w\in Y$. Fredholm theory shows that there is a countable set of
eigenvalues where unique solvability could fail, but our uniqueness argument
above shows that one has solvability for $(\Delta+q)^{2}y=F$. Elliptic
regularity shows that for $u\in C^{2,\alpha}(\overline{\Omega})$, one has
$y\in C^{4,\alpha}(\overline{\Omega})$.

Now that we know that the equation is uniquely solvable, let $u\in
C^{2,\alpha}(\overline{\Omega})$, and let $y\in
C^{4,\alpha}(\overline{\Omega})$ be the solution. Then $P(u)=(\Delta+q)y$.
Then $P(P(u))=P((\Delta+q)y)=(\Delta+q)v$ for the unique solution $v$ of

$\begin{cases}(\Delta+q)^{2}v=(\Delta+q)^{2}y&\text{in }\Omega\\\
v=\partial_{\nu}v=0&\text{on }\partial\Omega.\end{cases}$

Since $y$ has $0$ Cauchy data, $w=v-y$ satisfies

$\begin{cases}(\Delta+q)^{2}w=0&\text{in }\Omega\\\
w=\partial_{\nu}w=0&\text{on }\partial\Omega.\end{cases}$

The unique solution to this equation is $w=0$. It follows that $v=y$. Thus
$P(P(u))=P(u)$ and $P$ is indeed a projection. ∎

######  Lemma 4.3.

Let $q\in C^{\alpha}(\overline{\Omega})$ and let $Y$ and $Z$ be as in Lemma
4.1. Then $\Delta+q\colon Y\to Z$ is bounded and bijective and has a bounded
inverse $G\colon Z\to Y$.

###### Proof.

By definition of $Z$, $\Delta+q$ is surjective. To see injectivity, suppose
$u,v\in Y$ and $(\Delta+q)u=(\Delta+q)v$. Then $w=v-u$ satisfies

$\begin{cases}(\Delta+q)w=0&\text{in }\Omega\\\ w=0&\text{on
}\partial\Omega\\\ \partial_{\nu}w=0&\text{on }\partial\Omega.\end{cases}$

It follows by the unique continuation principle that $w=0$. Hence $\Delta+q$
is injective. Lastly, we have

$\lVert(\Delta+q)u\rVert_{C^{\alpha}(\overline{\Omega})}\leq\lVert\Delta
u\rVert_{C^{\alpha}(\overline{\Omega})}+\lVert
qu\rVert_{C^{\alpha}(\overline{\Omega})}\leq C\lVert
u\rVert_{C^{2,\alpha}(\overline{\Omega})}$

so that $\Delta+q$ is bounded. Now it follows from the open mapping theorem
that there exists a bounded inverse $G$ of $\Delta+q$. ∎

Below we will use the ball $V_{q,\delta}$ in the space of solutions,

$V_{q,\delta}=\\{v\in C^{2,\alpha}(\overline{\Omega})\,:\,\Delta v+qv=0\text{
and }\lVert v\rVert_{C^{2,\alpha}(\overline{\Omega})}<\delta\\}.$

######  Lemma 4.4.

Let $a_{1},a_{2}\in C^{k+1}(\mathbb{R},C^{1,\alpha}(\overline{\Omega}))$ with
$k\geq 2$ and let $w_{1},w_{2}$ have the same Cauchy data and solve $\Delta
w_{i}+a_{i}(x,w_{i})=0$ in $\Omega$. Write $q_{i}=\partial_{u}a_{i}(x,w_{i})$.
Let $S_{a_{1}}\colon V_{q_{1},\delta_{1}}\to C^{2,\alpha}(\overline{\Omega})$
be the solution map from Lemma 2.4, for some $\delta_{1}>0$. Suppose
$u_{1,v}=S_{a_{1}}(v)$ and that $C_{a_{1}}^{w_{1},\delta}\subseteq
C_{a_{2}}^{0,C}$. Then there exists a $\delta_{2}>0$ and a $C^{k}$ map
$T_{a_{2}}\colon V_{q_{1},\delta_{2}}\to C^{2,\alpha}(\overline{\Omega})$,
$T_{a_{2}}(v)=u_{2,v}$, where $u_{2,v}$ has the same Cauchy data as $u_{1,v}$
and solves $\Delta u_{2,v}+a_{2}(x,u_{2,v})=0$. Moreover, when
$\partial_{u}a_{1}(x,w_{1})=\partial_{u}a_{2}(x,w_{2})$ then
$T_{a_{2}}^{\prime}(0)v=v$.

###### Proof.

First we use $C_{a_{1}}^{w_{1},\delta}\subseteq C_{a_{2}}^{0,C}$ to find, for
any $v\in V_{q_{1},\delta_{1}}$, a function $u_{2,v}$ with the same Cauchy
data as $u_{1,v}$ and solving $\Delta u_{2,v}+a_{2}(x,u_{2,v})=0$. Note that
$u_{1,0}=w_{1}$. Moreover, both $u_{2,0}$ and $w_{2}$ solve the equation
$\Delta u+a_{2}(x,u)=0$ and they have the same Cauchy data, so by Lemma 3.3
one has $u_{2,0}=w_{2}$. By (2.17) we have

(4.4) $\lVert u_{1,v}-w_{1}\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq
C\lVert v\rVert_{C^{2,\alpha}(\overline{\Omega})},$

Using this, Lemma 3.3,
$(u_{2,v}-w_{2})|_{\partial\Omega}=(u_{1,v}-w_{1})|_{\partial\Omega}$,
$\partial_{\nu}(u_{2,v}-w_{2})|_{\partial\Omega}=\partial_{\nu}(u_{1,v}-w_{1})|_{\partial\Omega}$
and the fact that $\lVert u_{2,v}\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq
C$, we have

(4.5) $\lVert u_{2,v}-w_{2}\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq
C\lVert v\rVert_{C^{2,\alpha}(\overline{\Omega})}.$

Let $r_{v}=u_{1,v}-u_{2,v}$. Then $r_{v}$ satisfies

$(\Delta+q_{2})r_{v}=q_{2}r_{v}+a_{2}(x,u_{2,v})-a_{1}(x,u_{1,v})=q_{2}r_{v}+a_{2}(x,u_{1,v}-r_{v})-a_{1}(x,u_{1,v}).$

Let $G$ be the inverse of $\Delta+q_{2}:Y\to Z$ provided by Lemma 4.3. Then
$r_{v}$ solves the fixed point equation

(4.6) $r_{v}=G(q_{2}r_{v}+a_{2}(x,u_{1,v}-r_{v})-a_{1}(x,u_{1,v})).$

We would like to show that $r_{v}$ depends smoothly on $v$ by applying the
implicit function theorem to (4.6). However, for a general function $r$ the
expression $q_{2}r+a_{2}(x,u_{1,v}-r)-a_{1}(x,u_{1,v})$ might not be in the
domain of $G$. For this reason we introduce the projection $P\colon
C^{2,\alpha}(\overline{\Omega})\to Z$ from Lemma 4.2. Now define the map
$F\colon V_{q_{1},\delta_{1}}\times C^{2,\alpha}(\overline{\Omega})\to
C^{2,\alpha}(\overline{\Omega})$ by

$F(v,r)=r-GP(q_{2}r+a_{2}(x,u_{1,v}-r)-a_{1}(x,u_{1,v})).$

Next we compute $F(0,w_{1}-w_{2})$ and $D_{r}F(0,w_{1}-w_{2};h)$ and find

$\displaystyle F(0,w_{1}-w_{2})$
$\displaystyle=w_{1}-w_{2}-GP(q_{2}(w_{1}-w_{2})+a_{2}(x,w_{1}-(w_{1}-w_{2}))-a_{1}(x,w_{1}))$
$\displaystyle=w_{1}-w_{2}-GP(q_{2}(w_{1}-w_{2})+\Delta(-w_{2}+w_{1}))$
$\displaystyle=w_{1}-w_{2}-GP((\Delta+q_{2})(w_{1}-w_{2}))$
$\displaystyle=w_{1}-w_{2}-G((\Delta+q_{2})(w_{1}-w_{2}))=0$

and

$D_{r}F(0,w_{1}-w_{2};h)=h-GP(q_{2}h-\partial_{u}a_{2}(x,w_{2})h)=h.$

Since $h\mapsto D_{r}F(0,w_{1}-w_{2};h)$ is bijective, it follows from the
implicit function theorem [HG27, Theorem 4] that there exists a $\delta_{2}$
with $0<\delta_{2}\leq\delta_{1}$ and a $C^{k}$ map $R\colon
V_{q_{1},\delta_{2}}\to C^{2,\alpha}(\overline{\Omega})$ such that
$\tilde{r}=R(v)$ is the unique solution to

(4.7)
$\tilde{r}=GP(q_{2}\tilde{r}+a_{2}(x,u_{1,v}-\tilde{r})-a_{1}(x,u_{1,v})).$

for $\tilde{r}$ close to $w_{1}-w_{2}$. Choosing $v\in V_{q_{1},\delta_{2}}$
in $u_{1,v}=S_{a_{1}}(v)$, we find that $r_{v}$ is in the range of $R$ and
that $r_{v}$ satisfies (4.7). Moreover, by (4.4) and (4.5) we also have

$\lVert r_{v}-(w_{1}-w_{2})\rVert_{C^{2,\alpha}(\overline{\Omega})}\leq
C\lVert v\rVert_{C^{2,\alpha}(\overline{\Omega})}.$

By the uniqueness of $\tilde{r}=R(v)$ near $w_{1}-w_{2}$ we have $r_{v}=R(v)$
for $v\in B_{\delta_{2}}$. Thus the map $v\mapsto r_{v}$ is indeed $C^{k}$.

Since $r_{v}=u_{1,v}-u_{2,v}$ we can define the $C^{k}$ map

$T_{a_{2}}(v)\coloneqq S_{a_{1}}(v)-R(v).$

It remains to show that $T_{a_{2}}^{\prime}(0)v=v$, provided
$\partial_{u}a_{1}(x,w_{1})=\partial_{u}a_{2}(x,w_{2})$. To do this, we use
the implicit function theorem to compute $R^{\prime}(0)$,

$R^{\prime}(0)v=-[D_{r}F(0,R(0))]^{-1}D_{v}F(0,R(0))v.$

Since $D_{r}F(0,R(0))v=v$ and $D_{v}F(0,R(0))v=0$ it follows that
$R^{\prime}(0)v=0$. Now we have

$T_{a_{2}}^{\prime}(0)v=S_{a_{1}}^{\prime}(0)v+R^{\prime}(0)v=S_{a_{1}}^{\prime}(0)v=v.\qed$

##  5\. First linearization

Throughout this section, we let $a_{1},a_{2}\in
C^{3,\alpha}(\mathbb{R},C^{1,\alpha}(\overline{\Omega}))$ and let $w\in
C^{2,\alpha}(\overline{\Omega})$ be a fixed solution of $\Delta
w+a_{1}(x,w)=0$ in $\Omega$. Write $q=\partial_{u}a_{1}(x,w)$ and consider the
sets

$\displaystyle V_{q}$ $\displaystyle=\\{v\in
C^{2,\alpha}(\overline{\Omega})\,:\,\Delta v+qv=0\text{ in $\Omega$}\\},$
$\displaystyle V_{q,\delta}$ $\displaystyle=\\{v\in V_{q}\,:\,\lVert
v\rVert_{C^{2,\alpha}(\overline{\Omega})}<\delta\\}.$

Assume $C_{a_{1}}^{w,\delta}\subseteq C_{a_{2}}^{0,C}$. For any $v\in
V_{q,\delta}$ with $\delta$ small, we let $u_{1,v}=S_{a_{1},w}(v)$ and
$u_{2,v}=T_{a_{2},w}(v)$ be the solutions of $\Delta
u_{j,v}+a_{j}(x,u_{j,v})=0$ given by Lemmas 2.4 and 4.4.

######  Lemma 5.1.

Suppose that $C_{a_{1}}^{w,\delta}\subseteq C_{a_{2}}^{0,C}$. There is
$\delta_{1}>0$ such that for any $v\in V_{q,\delta_{1}}$ one has

$\partial_{u}a_{1}(x,u_{1,v}(x))=\partial_{u}a_{2}(x,u_{2,v}(x)),\qquad
x\in\overline{\Omega}.$

###### Proof.

Let $v\in V_{q,\delta}$ and let $v_{t}=v+th$ where $h$ solves $\Delta h+qh=0$
and $t$ is small. Consider the solutions $u_{1,v_{t}}=S_{a_{1},w}(v_{t})$ and
$u_{2,v_{t}}=T_{a_{2},w}(v_{t})$ of

$\Delta u_{j,v_{t}}+a_{j}(x,u_{j,v_{t}})=0.$

The solutions $u_{j,v_{t}}$ are $C^{2}$ with respect to $t$ and have the same
Cauchy data. Differentiating the above equation in $t$ and writing
$\dot{u}_{j}=\partial_{t}u_{j,v_{t}}|_{t=0}$, we obtain

$\Delta\dot{u}_{j}+\partial_{u}a_{j}(x,u_{j,v})\dot{u}_{j}=0.$

Subtracting the equations for $j=1,2$ and rewriting yields

(5.1)
$(\Delta+\partial_{u}a_{2}(x,u_{2,v}))(\dot{u}_{1}-\dot{u}_{2})+(\partial_{u}a_{1}(x,u_{1,v})-\partial_{u}a_{2}(x,u_{2,v}))\dot{u}_{1}=0.$

Suppose that $\tilde{v}_{2}$ solves
$(\Delta+\partial_{u}a_{2}(x,u_{2,v}))\tilde{v}_{2}=0$. Integrating (5.1)
against $\tilde{v}_{2}$ and using that $\dot{u}_{1}-\dot{u}_{2}$ has zero
Cauchy data gives

$\int_{\Omega}(\partial_{u}a_{1}(x,u_{1,v})-\partial_{u}a_{2}(x,u_{2,v}))\dot{u}_{1}\tilde{v}_{2}\,dx=0.$

It remains to study $\dot{u}_{1}=DS_{a_{1},w}(v)h$. By Lemma 2.5, when $v\in
V_{q}$ is sufficiently small any solution $\tilde{v}_{1}$ of
$(\Delta+\partial_{u}a_{1}(x,u_{1,v}))\tilde{v}_{1}=0$ can be written as
$DS_{a_{1},w}(v)h$ for a suitable $h$. It follows that

$\int_{\Omega}(\partial_{u}a_{1}(x,u_{1,v})-\partial_{u}a_{2}(x,u_{2,v}))\tilde{v}_{1}\tilde{v}_{2}\,dx=0$

for any solutions $\tilde{v}_{j}$ of
$(\Delta+\partial_{u}a_{j}(x,u_{j,v}))\tilde{v}_{j}=0$. Now it follows from
the density of products of solutions as in the standard Calderón problem (see
[SU87] for $n\geq 3$ and Bukhgeim, btw20 for $n=2$) that
$\partial_{u}a_{1}(x,u_{1,v})=\partial_{u}a_{2}(x,u_{2,v})$. ∎

######  Lemma 5.2.

In the setting of Lemma 5.1, the function

$\varphi_{v}=u_{2,v}-u_{1,v}$

is independent of $v\in V_{q,\delta_{1}}$.

###### Proof.

Write $\psi_{t}=\varphi_{tv}$. The function $\psi_{t}$ is $C^{2}$ in $t$, has
zero Cauchy data on $\partial\Omega$, and satisfies

$\Delta\psi_{t}=a_{1}(x,u_{1,tv})-a_{2}(x,u_{2,tv}).$

Thus the derivative $z_{t}=\partial_{t}\psi_{t}$ satisfies

$\Delta
z_{t}=\partial_{u}a_{1}(x,u_{1,tv})\partial_{t}u_{1,tv}-\partial_{u}a_{2}(x,u_{2,tv})\partial_{t}u_{2,tv}.$

Combining this with Lemma 5.1 yields

$\Delta z_{t}=-\partial_{u}a_{1}(x,u_{1,tv})z_{t}.$

Since $z_{t}$ has zero Cauchy data, it follows that $z_{t}=0$ and consequently
$\psi_{t}$ is independent of $t$. In particular, $\varphi_{v}=\varphi_{0}$. ∎

We can now give the proofs of Theorem 1.1 and 1.3.

###### Proof of Theorem 1.3.

Let $w_{1}$ solve $\Delta w_{1}+a_{1}(x,w_{1})=0$ and assume that
$C_{a_{1}}^{w_{1},\delta}\subseteq C_{a_{2}}^{0,C}$. Using Lemma 5.2, we have

$\displaystyle\Delta\varphi$ $\displaystyle=\Delta u_{2,v}-\Delta
u_{1,v}=a_{1}(x,u_{1,v})-a_{2}(x,u_{2,v})$
$\displaystyle=a_{1}(x,u_{1,v})-a_{2}(x,u_{1,v}+\varphi).$

This can be rewritten as

$a_{1}(x,u_{1,v}(x))=T_{\varphi}a_{2}(x,u_{1,v}(x)).$

It is enough to show that there is $\varepsilon>0$ such that for any
$\bar{x}\in\overline{\Omega}$ and $\lambda\in[-\varepsilon,\varepsilon]$, one
can find a small solution $v$ such that

(5.2) $u_{1,v}(\bar{x})=w_{1}(\bar{x})+\lambda.$

Fix $x_{0}\in\overline{\Omega}$, and use Runge approximation (Lemma A.1) to
generate a solution $v=v_{x_{0}}$ of $\Delta v+\partial_{u}a_{1}(x,w_{1})v=0$
with $v(x_{0})=4$. Let $U_{x_{0}}$ be a neighborhood of $x_{0}$ so that
$v(x)\geq 2$ for $x\in\overline{U}_{x_{0}}\cap\overline{\Omega}$. In the
notation of Lemma 2.4 one has

$u_{1,tv}=w_{1}+tv+Q_{a_{1},w_{1}}(tv)$

where

$\lVert Q_{a_{1},w_{1}}(tv)\rVert\leq C_{a_{1},w_{1}}t^{2}\lVert
v\rVert_{C^{2,\alpha}(\overline{\Omega})}^{2}.$

Thus for $x\in\overline{U}_{x_{0}}\cap\overline{\Omega}$ one has

$|u_{1,tv}(x)-w_{1}(x)|\geq 2|t|-C_{a_{1},w_{1}}t^{2}\lVert
v\rVert_{C^{2,\alpha}(\overline{\Omega})}^{2}.$

Set $\varepsilon_{x_{0}}=1/(C_{a_{1},w_{1}}\lVert
v\rVert_{C^{2,\alpha}(\overline{\Omega})}^{2})$. Then for
$|t|\leq\varepsilon_{x_{0}}$

$|u_{1,tv}(x)-w_{1}(x)|\geq|t|.$

The next step is to use compactness to find a finite cover
$\\{U_{x_{1}},\ldots,U_{x_{N}}\\}$ of $\overline{\Omega}$ and to set

$\varepsilon=\min\\{\varepsilon_{x_{1}},\ldots,\varepsilon_{x_{N}},\delta_{0}\\}.$

Here $\delta_{0}$ is chosen so that $\lVert
tv_{x_{j}}\rVert_{C^{2,\alpha}}\leq\delta$ whenever $|t|\leq\delta_{0}$ and
$1\leq j\leq N$.

Now fix any $\bar{x}\in\overline{\Omega}$ and
$\lambda\in[-\varepsilon,\varepsilon]$, and choose $j$ so that $\bar{x}\in
U_{j}$. Define

$\eta(t)=u_{1,tv_{x_{j}}}(\bar{x})-w_{1}(\bar{x}).$

Then $\eta:[-\varepsilon,\varepsilon]\to\mathbb{R}$ is continuous with
$\eta(\varepsilon)\geq\varepsilon$ and $\eta(-\varepsilon)\leq-\varepsilon$.
By continuity, there is $\bar{t}\in[-\varepsilon,\varepsilon]$ such that
$\eta(\bar{t})=\lambda$. This proves that one has (5.2) for some choice of
$v$, which proves the theorem. ∎

###### Proof of Theorem 1.1.

Since $w$ is a common solution for nonlinearities $a_{1}$ and $a_{2}$, we have
$w_{1}=w_{2}=w$ in Theorem 1.3. Consequently $\varphi=0$ and
$T_{\varphi}a_{2}=a_{2}$. The result now follows from Theorem 1.3. ∎

##  6\. Higher order linearization

In this section we prove theorem 1.4. We use the higher order linearization
method with the smooth solution maps from Sections 2 and 4. Essentially the
method is to show that the derivatives of order $k$ of the solution maps
satisfy a certain partial differential equation and have the same Cauchy data,
provided $\partial_{u}^{l}a_{1}(x,w_{1})=\partial_{u}^{l}a_{2}(x,w_{2})$ for
$l\leq k-1$. Then Theorem 1.4 follows from an integration by parts argument.
We start by proving that the derivatives of order $k$ of the solution maps
satisfy a certain differential equation in the following Lemma.

######  Lemma 6.1.

Let $a_{1},a_{2}\in
C^{k+2,\alpha}(\mathbb{R};C^{1,\alpha}(\overline{\Omega}))$ with $k\geq 1$.
Let $S_{a_{1}},T_{a_{2}}$ be the solution operators $\Delta u+a_{i}(x,u)=0$
from Lemma 2.4 and Lemma 4.4. Suppose that
$\partial_{u}^{l}a_{1}(x,w_{1})=\partial_{u}^{l}a_{2}(x,w_{2})$ for all $1\leq
l\leq k$, then $f\coloneqq
D^{k+1}S_{a_{1}}(0;v_{1},\ldots,v_{k+1})-D^{k+1}T_{a_{2}}(0;v_{1},\ldots,v_{k+1})$
satisfies

$\begin{cases}\Delta
f+qf=[\partial_{u}^{k+1}a_{2}(x,w_{2})-\partial_{u}^{k+1}a_{1}(x,w_{1})]\prod_{i=1}^{k+1}v_{i}&\text{in
}\Omega\\\ f=0&\text{on }\partial\Omega\\\ \partial_{\nu}f=0&\text{on
}\partial\Omega\end{cases}$

for any solutions $v_{i}$ of $\Delta v+qv=0$ where
$q=\partial_{u}a_{1}(x,w_{1})=\partial_{u}a_{2}(x,w_{2})$.

###### Proof.

We start with the boundary conditions. We have by construction that
$S_{a_{1}}(v)|_{\partial\Omega}=T_{a_{2}}(v)|_{\partial\Omega}$. Let
$\iota\colon C^{2,\alpha}(\overline{\Omega})\to C^{2,\alpha}(\partial\Omega)$
be the natural injection. Then the operator
$v\mapsto\iota(S_{a_{1}}(v)-T_{a_{2}}(v))$ is identically equal to $0$. Hence
its derivatives are also $0$,
$D^{l}S_{a_{1}}(0;v_{1},v_{2})|_{\partial\Omega}=D^{l}T_{a_{2}}(0;v_{1},v_{2})|_{\partial\Omega}$
for all $l\in\mathbb{N}$, or equivalently $f|_{\partial\Omega}=0$. Similarly,
$\partial_{\nu}f|_{\partial\Omega}=0$

The proof for the differential equation is by induction on $k$ and we start
with the base case $k=1$, where we assume that
$\partial_{u}a_{1}(x,w_{1})=\partial_{u}a_{2}(x,w_{2})$. Since $S_{a_{1}}(v)$
and $T_{a_{2}}(v)$ are solution maps for $\Delta u+a_{1}(x,u)=0$ and $\Delta
u+a_{2}(x,u)=0$, respectively, the operators $v\mapsto\Delta
S_{a_{1}}(v)+a_{1}(x,S_{a_{1}}(v))$ and $v\mapsto\Delta
T_{a_{2}}(v)+a_{2}(x,T_{a_{2}}(v))$ are identically equal to $0$. Hence their
derivatives are also $0$. The second derivatives being $0$ can be rewritten as

$\Delta
D^{2}S_{a_{1}}(0;v_{1},v_{2})+qD^{2}S_{a_{1}}(0;v_{1},v_{2})=-\partial_{u}^{2}a_{1}(x,S_{a_{2}}(0))DS_{a_{1}}(0;v_{1})DS_{a_{1}}(0;v_{2})\quad\text{in
}\Omega$

and

$\Delta
D^{2}T_{a_{2}}(0;v_{1},v_{2})+qD^{2}T_{a_{2}}(0;v_{1},v_{2})=-\partial_{u}^{2}a_{2}(x,T_{a_{2}}(0))DT_{a_{2}}(0;v_{1})DT_{a_{2}}(0;v_{2})\quad\text{in
}\Omega$

From $\partial_{u}a_{1}(x,w_{1})=\partial_{u}a_{2}(x,w_{2})$ we get
$DS_{a_{1}}(0;v)=DT_{a_{2}}(0;v)=v$. Using this and $S_{a_{1}}(0)=w_{1}$,
$T_{a_{2}}(0)=w_{2}$ we get, by subtracting the equations, that

$\Delta
f+qf=[\partial_{u}^{2}a_{2}(x,w_{1})-\partial_{u}^{2}a_{2}(x,w_{2})]v_{1}v_{2}\quad\text{in
}\Omega$

where $f\coloneqq
D^{2}S_{a_{1}}(0;v_{1},v_{2})-D^{2}T_{a_{2}}(0;v_{1},v_{2})$. Now suppose
that, for some $m\geq 2$, the statement holds for $k=m-1$. That is, suppose
for $l\leq m-1$ that
$\partial_{u}^{l}a_{1}(x,w_{1})=\partial_{u}^{l}a_{2}(x,w_{2})$ and that
$f\coloneqq
D^{m}S_{a_{1}}(0;v_{1},\ldots,v_{m})-D^{m}T_{a_{2}}(0;v_{1},\ldots,v_{m})$
solves

(6.1) $\begin{cases}\Delta
f+qf=[\partial_{u}^{m}a_{2}(x,w_{2})-\partial_{u}^{m}a_{1}(x,w_{1})]\prod_{i=1}^{m}v_{i}&\text{in
}\Omega\\\ f=0&\text{on }\partial\Omega\\\ \partial_{\nu}f=0&\text{on
}\partial\Omega.\end{cases}$

To show that it holds also for $k=m$ assume additionally that
$\partial_{u}^{m}a_{1}(x,w_{1})=\partial_{u}^{m}a_{2}(x,w_{2})$. Then (6.1)
simplifies to

$\begin{cases}\Delta f+qf=0&\text{in }\Omega\\\ f=0&\text{on
}\partial\Omega\\\ \partial_{\nu}f=0&\text{on }\partial\Omega\end{cases}$

and we conclude that $f=0$. Let $F(v)\coloneqq S_{a_{1}}(v)-T_{a_{2}}(v)$, so
that $D^{m}F(0;v_{1},\ldots,v_{m})=f=0$ and let
$g=D^{m+1}F(0;v_{1},\ldots,v_{m+1})$. Let $\delta_{ij}$ denote the Kronecker
delta. Then we have by differentiating (6.1) in direction $v_{m+1}$ that

$\displaystyle\Delta g+qg=$
$\displaystyle-\partial_{u}^{2}a_{1}(x,w_{1})\underbrace{D^{m}F(0;v_{1},\ldots,v_{m})}_{=0}v_{m+1}$
$\displaystyle+\underbrace{[\partial_{u}^{m}a_{2}(x,w_{2})-\partial_{u}^{m}a_{1}(x,w_{1})]}_{=0}\sum_{i=1}^{m}D^{2}S_{a_{1}}(0;v_{i},v_{m+1})\prod_{j=1}^{m}(1-\delta_{ij})v_{j}$
$\displaystyle+[\partial_{u}^{m+1}a_{2}(x,w_{2})-\partial_{u}^{m+1}a_{1}(x,w_{1})]\prod_{i=1}^{m+1}v_{i}$
$\displaystyle=$
$\displaystyle[\partial_{u}^{m+1}a_{2}(x,w_{2})-\partial_{u}^{m+1}a_{1}(x,w_{1})]\prod_{i=1}^{m+1}v_{i}$

and this proves the lemma. ∎

###### Proof of Theorem 1.4.

Let $l\geq 2$ be an arbitrary integer and suppose that
$\partial^{j}_{u}a_{1}(x,w_{1})=\partial^{j}_{u}a_{2}(x,w_{2})$ for $1\leq
j\leq l-1$. Then we have from Lemma 6.1 that

(6.2) $\begin{cases}\Delta
f+qf=[\partial_{u}^{l}a_{2}(x,w_{2})-\partial_{u}^{l}a_{1}(x,w_{1})]\prod_{i=1}^{l}v_{i}&\text{in
}\Omega\\\ f=0&\text{on }\partial\Omega\\\ \partial_{\nu}f=0&\text{on
}\partial\Omega\end{cases}$

where $v_{j}$ solve $\Delta v_{j}+\partial_{u}a_{1}(x,w_{1})v_{j}=0$ for
$j\in\\{1,\ldots,l\\}$. Let $v_{l+1}$ solve $\Delta
v_{l+1}+\partial_{u}a_{1}(x,w_{1})v_{l+1}=0$. Multiplying the differential
equation (6.2) by $v_{l+1}$ and integrating by parts twice gives

$\int_{\Omega}[\partial_{u}^{l}a_{2}(x,w_{2})-\partial_{u}^{l}a_{1}(x,w_{1})]\prod_{i=1}^{l+1}v_{i}\,dx=0.\qed$

##  Appendix A Runge approximation

In the proof of Theorem 1.3 we need to find a solution of the linearized
equation which is nonzero in some fixed but arbitrary point of the domain. A
few ways to achieve this are described in [LLLS21a, Remark 2.2]. For the sake
of completeness, we give a proof based on Runge approximation that is valid in
our situation following [LLS20].

######  Lemma A.1.

Let $\Omega\subseteq\mathbb{R}^{n}$ be a bounded open set and let $q\in
C^{\alpha}(\overline{\Omega})$. For any $x_{0}\in\overline{\Omega}$, there is
$u\in C^{2,\alpha}(\overline{\Omega})$ solving $(-\Delta+q)u=0$ in $\Omega$
with $u(x_{0})\neq 0$.

###### Proof.

Let $\Omega_{2}$ be a large ball with $\overline{\Omega}\subseteq\Omega_{2}$,
and extend $q$ as a function in $C^{\alpha}_{c}(\Omega_{2})$. We may choose
$\Omega_{2}$ in such a way that $0$ is not a Dirichlet eigenvalue of
$-\Delta+q$ in $\Omega_{2}$ (see e.g. [Ste90, Lemma 3.2]). Now by [BJS64,
Theorem 1 in Section 5.4], there is a small ball $\Omega_{1}$ centered at
$x_{0}$ and a function $u_{0}\in C^{2,\alpha}(\overline{\Omega}_{1})$ solving
$(-\Delta+q)u_{0}=0$ in $\Omega_{1}$ with $u_{0}(x_{0})=1$. By Runge
approximation (see Lemma A.3 below), there is $u\in
C^{2,\alpha}(\overline{\Omega}_{2})$ solving $(-\Delta+q)u=0$ in $\Omega_{2}$
with $u(x_{0})$ arbitrarily close to $u_{0}(x_{0})=1$. This concludes the
proof. ∎

It remains to prove the Runge approximation result. Since the approximation is
in the $C(\overline{\Omega}_{1})$ norm, we need a notion of suitable weak
solutions with measure data in the duality argument. Let
$\Omega\subseteq\mathbb{R}^{n}$ be a bounded open set with smooth boundary,
let $q\in L^{\infty}(\Omega)$, and assume that $0$ is not a Dirichlet
eigenvalue of $-\Delta+q$ in $\Omega$. If $\mu$ is a bounded linear functional
on $C(\overline{\Omega})$ (i.e. $\mu$ is a measure), we say that $u\in
L^{1}(\Omega)$ is a _very weak solution_ of

(A.1) $(-\Delta+q)u=\mu\text{ in $\Omega$},\qquad u|_{\partial\Omega}=0,$

if

$\int_{\Omega}u(-\Delta+q)\varphi\,dx=\mu(\varphi)$

for any $\varphi\in C^{2}(\overline{\Omega})\cap H^{1}_{0}(\Omega)$.

######  Proposition A.2.

For any $p<\frac{n}{n-1}$ there is $C>0$ such that for any bounded linear
functional $\mu$ on $C(\overline{\Omega})$, there is a unique very weak
solution $u\in W^{1,p}(\Omega)$ of (A.1) satisfying

$\lVert u\rVert_{W^{1,p}(\Omega)}\leq C\lVert\mu\rVert,$

where
$\lVert\mu\rVert=\sup_{\lVert\varphi\rVert_{C(\overline{\Omega})}=1}\,\lvert\mu(\varphi)\rvert$.

###### Proof.

If $q\geq 0$ this follows from [Sta65, Theorem 9.1]. In general we may replace
$q$ by $q+\gamma$ where $\gamma>0$ is a large constant, and use the part of
[Sta65, Theorem 9.1] where $\lambda$ is away from the spectrum. ∎

We can now prove the Runge approximation result.

######  Lemma A.3.

Let $\Omega_{1}$, $\Omega\subseteq\mathbb{R}^{n}$ be bounded open sets so that
$\overline{\Omega}_{1}\subseteq\Omega$, $\Omega\setminus\overline{\Omega}_{1}$
is connected, and $\Omega$ has smooth boundary. Suppose that $q\in
C^{\alpha}_{c}(\Omega)$ and that $0$ is not a Dirichlet eigenvalue of
$-\Delta+q$ in $\Omega$. Consider the sets

$\displaystyle S_{1}$ $\displaystyle=\\{u\in
C^{2,\alpha}(\overline{\Omega}_{1}),\ (-\Delta+q)u=0\text{ in
$\Omega_{1}$}\\},$ $\displaystyle S$ $\displaystyle=\\{u\in
C^{2,\alpha}(\overline{\Omega}),\ (-\Delta+q)u=0\text{ in $\Omega$}\\}.$

For any $u\in S_{1}$ and any $\varepsilon>0$, there is $v\in S$ with $\lVert
u-v|_{\Omega_{1}}\rVert_{C(\overline{\Omega}_{1})}\leq\varepsilon$.

###### Proof.

By the Hahn-Banach theorem [Con90, Corollary 3.15], it is enough to show that
any continuous linear functional on $C(\overline{\Omega}_{1})$ that vanishes
on $S|_{\Omega_{1}}$ must also vanish on $S_{1}$. Thus, let $\mu$ be a
continuous linear functional on $C(\overline{\Omega}_{1})$ that satisfies

(A.2) $\mu(v|_{\overline{\Omega}_{1}})=0\text{ for all $v\in S$.}$

We consider the extension defined by

$\bar{\mu}:C(\overline{\Omega})\to\mathbb{R},\ \
\bar{\mu}(u)=\mu(u|_{\overline{\Omega}_{1}}).$

By the Riesz representation theorem, $\bar{\mu}$ is a measure in
$\overline{\Omega}$ with
$\mathrm{supp}(\bar{\mu})\subseteq\overline{\Omega}_{1}$.

We use Proposition A.2 to find a very weak solution $w\in W^{1,p}(\Omega)$ of
the problem

(A.3) $(-\Delta+q)w=\bar{\mu}\text{ in $\Omega$},\qquad
w|_{\partial\Omega}=0.$

We use the assumption (A.2) and the unique continuation principle to prove
that

(A.4) $w=0$ in $\Omega\setminus\overline{\Omega}_{1}$.

Assuming (A.4), the proof can be concluded as follows. Since
$\mathrm{supp}(w)\subseteq\overline{\Omega}_{1}$, there exist $w_{j}\in
C^{\infty}_{c}(\Omega_{1})$ with $w_{j}\to w$ in $W^{1,p}(\Omega)$. Given any
$u\in S_{1}$, we let $\bar{u}$ be some function in $C^{2,\alpha}_{c}(\Omega)$
with $\bar{u}|_{\overline{\Omega}_{1}}=u$ and compute

$\displaystyle\mu(u)=\bar{\mu}(\bar{u})=\int_{\Omega}w(-\Delta+q)\bar{u}\,dx=\lim\int_{\Omega}w_{j}(-\Delta+q)\bar{u}\,dx=\lim\int_{\Omega_{1}}w_{j}(-\Delta+q)u\,dx=0.$

Thus $\mu|_{S_{1}}=0$ as required.

It remains to prove (A.4). We begin by studying the regularity of $w$ near
$\partial\Omega$. Choose a ball $\Omega_{2}$ with
$\overline{\Omega}\subseteq\Omega_{2}$ so that $0$ is not a Dirichlet
eigenvalue, and a very weak solution $\tilde{w}$ of

$(-\Delta+q)\tilde{w}=\tilde{\mu}\text{ in
$\Omega_{2}$},\qquad\tilde{w}|_{\partial\Omega_{2}}=0,$

where $\tilde{\mu}$ is the extension of $\bar{\mu}$ by zero to
$\overline{\Omega}_{2}$. Using the definition of very weak solutions and the
facts that $q\in C^{\alpha}_{c}(\Omega)$ and
$\mathrm{supp}(\bar{\mu})\subseteq\overline{\Omega}_{1}$, we see that
$\Delta\tilde{w}=0$ near $\partial\Omega$ in the sense of distributions. Hence
$\tilde{w}$ is $C^{\infty}$ near $\partial\Omega$. Let $g\in
C^{2,\alpha}(\overline{\Omega})$ be the solution of

$(-\Delta+q)g=0\text{ in $\Omega$},\qquad
g|_{\partial\Omega}=-\tilde{w}|_{\partial\Omega}.$

Then both $w$ and $\tilde{w}|_{\Omega}+g$ are very weak solutions of (A.3),
and by uniqueness one has

$w=\tilde{w}|_{\Omega}+g.$

It follows that $w$ is $C^{2,\alpha}$ near $\partial\Omega$. Moreover, since
$w$ is a $W^{1,p}$ solution of $(-\Delta+q)w=0$ in
$\Omega\setminus\overline{\Omega}_{1}$, it follows from [HR72, see section 4.
Concluding remarks] that $w$ is $W^{1,2}$ and consequently $C^{2,\alpha}$ in
$\Omega\setminus\overline{\Omega}_{1}$.

We now let $v\in S$ and choose $\chi\in C^{\infty}_{c}(\Omega)$ such that
$\chi=1$ near $\overline{\Omega}_{1}$ and $w$ is $C^{2,\alpha}$ in
$\mathrm{supp}(1-\chi)\cap\overline{\Omega}$. Then

$\int_{\Omega}w(-\Delta+q)v\,dx=\int_{\Omega}w(-\Delta+q)(\chi
v)\,dx+\int_{\Omega}w(-\Delta+q)((1-\chi)v)\,dx.$

We use the definition of very weak solutions in the first term, and since $w$
is regular in $\mathrm{supp}(1-\chi)$ we may integrate by parts in the second
term. This yields

$\int_{\Omega}w(-\Delta+q)v\,dx=\bar{\mu}(\chi
v)+\int_{\partial\Omega}(\partial_{\nu}w)v\,dS=\mu(v|_{\overline{\Omega}_{1}})+\int_{\partial\Omega}(\partial_{\nu}w)v\,dS.$

Since $v\in S$, we have $\mu(v|_{\overline{\Omega}_{1}})=0$ by the assumption
(A.2). Since we can vary the Dirichlet data of $v\in S$, it follows that
$\partial_{\nu}w|_{\partial\Omega}=0$. Thus $w$ in particular satisfies

$(-\Delta+q)w=0\text{ in $\Omega\setminus\overline{\Omega}_{1}$},\qquad
w|_{\partial\Omega}=\partial_{\nu}w|_{\partial\Omega}=0.$

Since $w$ is $C^{2,\alpha}$ in $\Omega\setminus\overline{\Omega}_{1}$ and this
set is connected, the unique continuation principle yields (A.4). This
finishes the proof. ∎

## References

  * [AF03] Robert A. Adams and John J.. Fournier  “Sobolev spaces” 140, Pure and Applied Mathematics (Amsterdam)  Elsevier/Academic Press, Amsterdam, 2003, pp. xiv+305
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