\begin{lemma}\label{absorption}
Let $u\colon\Sigma\times(T_0, T)\to \R$ be a solution to the graphical mean curvature flow  \eqref{gmcf2} with $\norm{u(\cdot,t)}_{C^1(\Sigma)}<r_0$ for all $t\in(T_0,T)$, then for any $t\in (T_0+2, T)$
\begin{equation}\label{gmcf-Schauder}
\|u\|_{C^{2,\a}(\Sigma\times(t-1,t))}\le C(\|u\|_{L^2(\Sigma\times(t-2,t))}+\|u\|_{C^{1,\alpha}(\Sigma\times(t-2,t))}\|u\|_{C^{2,\alpha}(\Sigma\times(t-2,t))})\,,
\end{equation}
where $C$ depends only on $\Sigma$, $M$ and $r_0$.

Moreover, there exists an $\varepsilon>0$, depending only on $\Sigma$ and $M$, such that, if $\|u\|_{C^{1,\a}(\Sigma\times(T-R, T))}<\varepsilon$ for some $T-R\ge T_0$, then
\begin{equation}\label{gmcf-SchauderII}
\|u\|_{C^{2,\a}(\Sigma\times (T-R/2,T))}\le C \|u\|_{L^2(\Sigma\times(T-R, T))} \,,
\end{equation}
where $C>0$ depends only on $\Sigma$ and $M$.
\end{lemma}

Morse theory provides a powerful framework for studying the topology of a space through the analysis of differentiable functions defined on it. Remarkably, many of its central ideas extend to infinite-dimensional settings, although their implementation requires substantially more delicate analysis. Foundational contributions in this direction include \cite{Smale1961,Palais1963,Chang1993,AbbondandoloMajer2001}.
More recently, Morse-theoretic methods have played an important role in the study of geometric variational problems, and in particular of the area functional on submanifolds of a fixed ambient manifold; see for example \cite{Tromba1977,Pitts1981,White1991,MarquesNeves2014,ChenGaspar2025}.

In this paper, we focus on Morse theory for the area functional restricted to \emph{smooth} submanifolds.
The gradient flow of this functional is the mean curvature flow, which can be described analytically as a one-parameter family of submanifolds $\Sigma_t$ in a Riemannian ambient manifold $(M,g)$ evolving by
\[
\frac{\partial X}{\partial t} = \vec H(X,t)\,, \quad X \in \Sigma_t \,,
\]
where $\vec H(X,t)$ denotes the mean curvature vector of $\Sigma_t$ at $X$.
This flow arises naturally in physics as a model for evolutionary processes governed by surface tension, such as the evolution of grain boundaries in annealing metals \cite{Mullins1956}. From a mathematical point of view, mean curvature flow was first studied systematically by Brakke \cite{Brakke1978} using geometric measure theory, and later by Huisken \cite{Huisken1984} via a more classical PDE approach. Since then, the subject has developed into a very active area of research.

Here, we are interested in the case of mean curvature flow of hypersurfaces in the presence of a boundary. The natural Neumann boundary value problem for mean curvature flow, known as the \emph{free boundary problem}, prescribes that the evolving hypersurface has boundary constrained to move on a fixed barrier hypersurface and meets it orthogonally. This problem was introduced by Huisken \cite{Huisken1989} in the non-parametric setting and further developed by Stahl \cite{Stahl1996Convergence,Stahl1996Regularity}, and later by \cite{Freire2010, Wheeler2014,Edelen2016,HirschLi2023,LangfordZhu2023}.

A fundamental role in free boundary mean curvature flow is played by \emph{free boundary minimal hypersurfaces}, which arise as stationary solutions of the flow equation. Recently, these objects have been the subject of intense research activity, due to fascinating new existence results, such as \cite{FraserSchoen2016,FranzSchulz2026,KarpukhinKusnerMcGrathStern2024}.
Most of these recent examples consist of \emph{unstable} critical points of the area functional. Therefore, by analogy with classical Morse theory, one expects the existence of flow lines emanating from these free boundary minimal hypersurfaces. More precisely, one expects \emph{ancient solutions} of mean curvature flow, namely solutions that have existed for all times in the past, which backward converge to unstable free boundary minimal hypersurfaces.

Morse-theoretic considerations also suggest that the dimension of the space of ancient solutions backward converging to $\Sigma$ is equal to the Morse index of $\Sigma$, i.e., the dimension of the negative directions of the area functional at $\Sigma$ at second order.
Our first main result makes this heuristic precise and may be summarized informally as follows.

\begin{corollary} \label{COR} If $\Sigma$ is nondegenerate, then \cref{uniqueness} holds without the assumption \eqref{tu:assume2}.
\end{corollary}

While such behavior is conjectured by analogy with classical Morse theory, the analysis of ancient solutions to free boundary mean curvature flow is far from trivial and was largely unexplored, due to the presence of the boundary interacting with the infinite dimensional nature of the problem and the potential degeneracy of the critical points. With the exception of recent classification results for convex solutions in dimension one and in highly symmetric higher-dimensional settings \cite{BourniLangford2023,BourniLangford2025,BourniBurnsCatron2025}, there has been essentially no systematic study of ancient solutions.
The present work constitutes, to the best of our knowledge, the first classification theory for general higher-dimension ancient solutions to the free boundary mean curvature flow.

\begin{alphatheorem}[cf.\ \cref{MCconstruction,COR}] \label{thmB}
Let $\Sigma$ be an unstable free boundary minimal hypersurface in a Riemannian manifold $(M,g)$ with convex boundary. Then there exists a mean-convex ancient solution emanating from $\Sigma$, converging exponentially fast at a rate determined by the first eigenvalue. Moreover, if $\Sigma$ is nondegenerate, this solution is unique, up to time translation, among mean-convex ancient solutions converging to $\Sigma$ in the $C^{1,\alpha}$-topology.
\end{alphatheorem}

For the uniqueness statement of \cref{thmA} (cf.\ \cref{uniqueness}), we require that the ancient solution can be expressed as a graph over a free boundary minimal hypersurface. Moreover, we assume the graph to have small parabolic $C^{1,\a}$-norm and to converge to zero sublinearly in the $C^0$-norm. When the underlying minimal hypersurface is nondegenerate, the sublinear decay assumption can be dropped.
We conjecture that the sublinear decay assumption can be dropped in the mean-convex case as well, namely there exists a unique ancient solution converging in $C^{1,\a}$ to any given unstable (free boundary) minimal hypersurface.

\begin{theorem}\label{uniqueness}
Let $u\colon\Sigma\times(-\infty, 0]\to\R$ be a smooth solution to the free boundary mean curvature flow \eqref{gmcf2}.
There exists a constant $\varepsilon>0$, depending only on $\Sigma$ and $M$
, such that if
\begin{equation} \label{tu:assume1}
\norm{u}_{C^{1,\a}(\Sigma\times(-\infty,0])} < \varepsilon \,,
\end{equation}
and
\begin{equation} \label{tu:assume2}
\liminf_{t\to-\infty}\ \abs{t} \norm{u(\cdot,t)}_{C^0(\Sigma)} = 0 \,,
\end{equation}
then, possibly after a time translation, $u$ coincides with one of the solutions constructed in \cref{construction}.
\end{theorem}

The idea of the proposition is that, if $\Sigma$ is minimal, we expect the existence of an ancient mean curvature flow whose graph over $\Sigma$, at first order, looks like $\varphi_k e^{-\lambda_kt}$ for $t\in(-\infty,T)$, with mean curvature, at first order, being $-\lambda_k\varphi_k e^{-\lambda_kt}$. Our proposition does not directly give this ancient solution, but it adds a correction $u$ to the first order approximation to satisfy the orthogonality condition at the boundary. Indeed, notice that a priori the graph of $\varphi_k e^{-\lambda_kt}$ over $\Sigma$ is not necessarily orthogonal to the boundary of~$M$.
\end{remark}
\begin{remark}
Assume that $\Sigma$ is an unstable free boundary minimal hypersurface.
Note that $\varphi_k$ changes sign for all $k>1$, while $\varphi_1>0$. Therefore, as observed in \cite[Section~2.2]{HaslhoferKetover2025}, the graph of $\varphi_1 e^{-\l_1 t}$ is a mean-convex foliation of a neighborhood of $\Sigma$. However, this is not necessarily free boundary, as remarked above. In \cref{prop:SurfKthEigenf}, we find a lower order correction such that the hypersurfaces $\Sigma_t$ form a smooth \emph{free boundary} mean-convex foliation of a neighborhood of $\Sigma$.
\end{remark}
\begin{remark}\label{w-norm}
The function $w$ constructed in \cref{prop:SurfKthEigenf} satisfies
\[
\|w(\cdot, t)\|_{C^{2,\a}(\Sigma)}\le Ce^{-\l_k t}\quad\forall t\in (-\infty, T)\,.
\]
Indeed, since $J_\Sigma(w)+ E(w)=-\l_k\phi_k e^{-\l_k t}-\rho u$, the elliptic Schauder estimates, together with \cref{E-estimates}, yield
\[
\begin{split}
\|w\|_{C^{2,\a}(\Sigma)}&\le C(\|w\|_{C^0(\Sigma)}+\|E(w)+\l_k\phi_ke^{-\l_k t}+\rho u\|_{C^{0,\a}(\Sigma)}+\|\e(w)\|_{C^{1,\a}(\partial\Sigma)})\\
&\le C(\|w\|_{C^0(\Sigma)}+\|w\|_{C^{1,\a}(\Sigma)}\|w\|_{C^{2,\a}(\Sigma)}+e^{-\l_k t}+\rho\| u\|_{C^{0,\a}(\Sigma)})\\
&\le C(e^{-\l_k t} + (\|w\|_{C^{1,\a}(\Sigma)} + \rho)\|w\|_{C^{2,\a}(\Sigma)}) \,.
\end{split}
\]
Moreover, recall that $\norm{w}_{C^{2,\a}(\Sigma)}\le C e^{-\l_kt} + \norm{u}_{C^{2,\a}(\Sigma)} < C e^{-\l_kt} + \delta$. As a result, for $\d>0$ and $\rho>0$ sufficiently small, and for time $t\in(-\infty,T)$ sufficiently small, we can absorb the term $(\|w\|_{C^{1,\a}(\Sigma)} + \rho)\|w\|_{C^{2,\a}(\Sigma)}$ on the left-hand side and obtain the result.
\end{remark}
\begin{proof}
Observe that $\rho\in(\l_I,0)$ is not an eigenvalue of the Jacobi operator on $\Sigma$, namely the following problem does not admit nontrivial solutions
\[
\begin{cases}
-\jac_\Sigma v = \rho v & \text{on $\Sigma$}\\
\frac{\partial v}{\partial \eta} = \II^{\partial M} (\nu,\nu) v & \text{on $\partial \Sigma$} \,.
\end{cases}
\]

From now on we assume that $\Sigma$ is a free boundary minimal hypersurface with index $I>0$.
We construct sub- and supersolutions to mean curvature flow that are close to the family $\{w(\cdot,t)\}_t$ constructed in \cref{prop:SurfKthEigenf} for the case $k=1$, associated to the first eigenfunction $\phi_1$. Note that this $w$ satisfies
\begin{equation}\label{wt}
\begin{cases}
\displaystyle w_t= -\l_1\phi_1 e^{-\l_1t}+ u_t= \frac{H_{w}}{\langle \nu_w, \overline\nu\rangle}+\rho u+ u_t & \text{on $\Sigma$}\\
\sk{\nu_w}{\bar\nu} = 0 & \text{on $\partial\Sigma$} \,,
\end{cases}
\end{equation}
and therefore, comparing with \eqref{gmcf}, the evolving hypersurfaces $\Sigma_{w(\cdot, t)}$ ``almost" satisfy mean curvature flow.
Let us now construct sub- and supersolutions by slightly reparametrizing the time.
\begin{lemma}\label{sub/super} Given $\l\in(\l_I, 0)$, we define the families
\[
w^\pm(x,t)= w(x, t\pm e^{-\l t})\,,
\]
where $\{w(\cdot,t)\}_t$ is the function constructed in the \cref{prop:SurfKthEigenf} for the case $k=1$, associated to the eigenfunction $\phi_1>0$.
Then there exists $t_\l\in\R$ such that $w^+$ is a supersolution and $w^-$ is a subsolution of mean curvature flow for all $t<t_\l$, that is
\[
\frac{d w^+}{dt}\ge \frac{H_{w^+}}{\langle \nu_{w^+}, \overline\nu\rangle}\quad \text{ and }\quad \frac{d w^-}{dt}\le \frac{H_{w^-}}{\langle \nu_{w^-}, \overline\nu\rangle}\,.
\]
\end{lemma}
\begin{remark}\label{sub/sup rmk} Note that \cref{sub/super} is also true for $\phi_1<0$, reversing the roles of $w^+$ and $w^-$.
\end{remark}
\begin{proof}
Let $w_f(x,t)= w(x, t+f(t))$. Then, using \eqref{wt},
\[
\begin{split}
\frac{d w_f}{dt}= \frac{\partial w}{\partial t} (1+f'(t))= \left(\frac{H_{w_f}}{\langle \nu_{w_f}, \overline\nu\rangle}+\rho u+\frac{\partial u}{\partial t}\right)(1+f'(t))\,,
\end{split}
\]
which implies
\begin{equation}\label{w-f}
\frac{d w_f}{dt}-\frac{H_{w_f}}{\langle \nu_{w_f}, \overline\nu\rangle}=-\l_1\phi_1 e^{-\l_1 (t+f(t))} f'(t)+ \left(\rho u+\frac{\partial u}{\partial t}\right)(1+f'(t))\,.
\end{equation}
Recall that, by \cref{prop:SurfKthEigenf}, $|u|+|u_t|\le Ce^{-2\l_1 t}$ and $\phi_1>0$. Therefore, for any $\l\in (\l_1, 0)$, if we choose $f(t)= \pm e^{-\l t}$, we obtain that for
 all $t$ sufficiently small the right-hand side of \eqref{w-f} is positive and negative respectively.
Hence, the result follows for $w^\pm=w_{\pm e^{-\l t}}$.
\end{proof}

While such behavior is conjectured by analogy with classical Morse theory, the analysis of ancient solutions to free boundary mean curvature flow is far from trivial and was largely unexplored, due to the presence of the boundary interacting with the infinite dimensional nature of the problem and the potential degeneracy of the critical points. With the exception of recent classification results for convex solutions in dimension one and in highly symmetric higher-dimensional settings \cite{BourniLangford2023,BourniLangford2025,BourniBurnsCatron2025}, there has been essentially no systematic study of ancient solutions.
The present work constitutes, to the best of our knowledge, the first  classification theory for general higher-dimension ancient solutions to the free boundary mean curvature flow.

Analogous classification results have been obtained in the boundaryless case by Choi--Mantoulidis \cite{ChoiMantoulidis2022} and extended to the noncompact case by Choi--Huang--Lee \cite{ChoiHuangLee2025}. Earlier results of this type have been known to be true for nonlinear parabolic PDEs in various settings (see e.g.\ \cite{Lunardi1995}).
Our results are inspired by these papers, primarily by \cite{ChoiMantoulidis2022}.
However, compared to the boundaryless case, the presence of a free boundary introduces substantial analytical difficulties, especially because the boundary condition does not lead to a linear boundary term. The underlying parabolic PDE becomes significantly more involved, the associated stability operator requires a much more delicate analysis, and the linearization procedures are considerably more intricate due to the interaction between interior geometry and boundary behavior.

We also note that deforming a free boundary minimal hypersurface by its first (or any) eigenfunction does not preserve the orthogonality condition at the boundary. The ``correct'' deformation is far from trivial and requires more delicate handling. We achieve this via the implicit function theorem and show that one can indeed deform to first order by  keeping the orthogonality at the boundary. In fact, we construct a smooth free boundary mean-convex foliation around unstable free boundary minimal hypersurfaces. We then use this foliation to obtain a different construction of mean-convex ancient solutions that also provides a more detailed geometric description.

\begin{alphatheorem}[cf.\ \cref{MCconstruction,COR}] \label{thmB}
Let $\Sigma$ be an unstable free boundary minimal hypersurface in a Riemannian manifold $(M,g)$ with convex boundary. Then there exists a mean-convex ancient solution emanating from $\Sigma$, converging exponentially fast at a rate determined by the first eigenvalue. Moreover, if $\Sigma$ is nondegenerate, this solution is unique, up to time translation, among mean-convex ancient solutions converging to $\Sigma$ in the $C^{1,\alpha}$-topology.
\end{alphatheorem}

\subsection{Further literature}

As described above, ancient solutions naturally appear when taking a Morse-theoretic perspective on (free boundary) minimal hypersurfaces, and they provide guidance for variational problems (see e.g.\ \cite[Section~1.1]{ChuLi2024} for a discussion related to min-max theory).
Historically, they were first studied because they play a central role in the analysis of singularity formation \cite{Hamilton1994}, and they have an intrinsic geometric interest due to their strong rigidity and symmetry properties (see, for example, \cite{BourniLangfordTinaglia2022} and the references therein).

In the boundaryless setting, extensive classification results are known in the convex regime. Under assumptions such as uniform convexity, bounded eccentricity, type-I curvature decay, or bounded isoperimetric ratio, the only compact convex ancient solutions are shrinking spheres \cite{HuiskenSinestrari2015}; see also \cite{DaskalopoulosHamiltonSesum2010, HaslhoferHershkovits2016, Langford2017}. When the ambient space is the sphere, the only geodesically convex ancient solutions are shrinking hemispheres \cite{BryanLouie2016, HuiskenSinestrari2015}.

In Euclidean space, however, shrinking spheres are not the only compact convex ancient solutions. There exist families of solutions that contract to round points as $t \to 0$ but become increasingly eccentric as $t \to -\infty$ \cite{AngenentDaskalopoulosSesum2019, HaslhoferHershkovits2016, White2003}. These exhaust all compact convex ancient solutions that are non-collapsed, or equivalently entire \cite{AngenentDaskalopoulosSesum2020, BourniLangfordLynch2023, BrendleNaff2024}. In the collapsed setting, a unique rotationally symmetric example is known \cite{BourniLangfordTinaglia2021}, while without symmetry assumptions a rich family of examples already appears in dimension two \cite{BourniLangfordTinaglia2022}.

Outside the convex regime, classification becomes significantly more difficult. Under strong decay assumptions as $t \to -\infty$, ancient solutions converge backward to minimal hypersurfaces, motivating the problem of classifying ancient solutions emanating from minimal hypersurfaces from another perspective.

\subsection{Future directions}

For the uniqueness statement of \cref{thmA} (cf.\ \cref{uniqueness}), we require that the ancient solution can be expressed as a graph over a free boundary minimal hypersurface. Moreover, we assume the graph to have small parabolic $C^{1,\alpha}$-norm and to converge to zero sublinearly in the $C^0$-norm. When the underlying minimal hypersurface is nondegenerate, the sublinear decay assumption can be dropped.
We conjecture that the sublinear decay assumption can be dropped in the mean-convex case as well, namely there exists a unique ancient solution converging in $C^{1,\alpha}$ to any given unstable (free boundary) minimal hypersurface.

A natural further problem is to classify ancient solutions in a fixed ambient manifold without assuming backward convergence to a minimal hypersurface.
In this context, it is interesting to ask whether it is possible to construct solutions whose backward limit is singular, particularly in the mean-convex setting. Similarly, one may investigate whether ancient solutions with unbounded area can be mean convex. In an extreme scenario, it would be interesting to construct ancient solutions that ``fill up'' the ambient manifold.

\subsection{Plan of the paper} In \cref{sec:definitions}, we recall basic definitions and we set the notation. In \cref{sec:GraphsOverFBMS}, we obtain several elliptical and parabolic estimates for graphs over a free boundary minimal hypersurface. In \cref{sec:construction}, we prove existence of an $I$-parameter family of ancient solutions emanating from a free boundary minimal hypersurface, by careful application of a fix point theorem.  In \cref{sec:rigidity}, we obtain uniqueness of the constructed family of ancient solutions, by use of an ODE lemma due to Merle--Zaag \cite{MerleZaag1998}, as adapted by \cite{ChoiMantoulidis2022}. Finally, in \cref{sec:meanconvex}, we provide a different more geometric proof of the existence of  mean-convex ancient solutions, by the use of barriers constructed via the implicit function theorem.

\subsection*{Acknowledgements}
We would like to thank Kyeongsu Choi and Christos Mantoulidis for clarifying certain points in their paper \cite{ChoiMantoulidis2022}, especially for pointing out the use of the ``absorption lemma'' in the proof of \cref{absorption} together with the Schauder estimates.
We would also like to thank Lucas Ambrozio for pointing us to his paper \cite{Ambrozio2015}.

T.\,B.\ was supported by NSF grant DMS-2405007, and G.\,F.~was supported by NSF grant DMS-2405361.
Moreover, part of this work was performed while the authors were in residence at the Simons Laufer Mathematical Sciences Institute (formerly MSRI) during the Fall 2024 semester, supported by NSF grant DMS-1928930.

A proof outline for Theorem~\ref{thmA} is given in the “Plan of the paper”: in \cref{sec:GraphsOverFBMS} the authors “obtain several elliptical and parabolic estimates for graphs over a free boundary minimal hypersurface.” Then, in \cref{sec:construction}, they “prove existence of an $I$-parameter family of ancient solutions emanating from a free boundary minimal hypersurface, by careful application of a fix point theorem.” Next, in \cref{sec:rigidity}, they “obtain uniqueness of the constructed family of ancient solutions, by use of an ODE lemma due to Merle--Zaag \cite{MerleZaag1998}, as adapted by \cite{ChoiMantoulidis2022}.” They also note (in connection with handling the free boundary condition) that the “\`\`correct'' deformation is far from trivial,” and that they “achieve this via the implicit function theorem” to “deform to first order by keeping the orthogonality at the boundary,” constructing “a smooth free boundary mean-convex foliation around unstable free boundary minimal hypersurfaces,” which is used for a “different more geometric proof” in \cref{sec:meanconvex} via “barriers constructed via the implicit function theorem.”

A proof outline for \begin{alphatheorem}[cf.\ \cref{construction,uniqueness}, \cref{COR}] \label{thmA}
Let $\Sigma$ be a free boundary minimal hypersurface of Morse index $I$ in a Riemannian manifold $(M,g)$ with convex boundary. Then there exists an $I$-parameter family of ancient free boundary mean curvature flows emanating from $\Sigma$. Moreover, any ancient solution with sufficiently fast decay to $\Sigma$ belongs to this family.
\end{alphatheorem} is given in the “Plan of the paper”: next the authors “obtain several elliptical and parabolic estimates for graphs over a free boundary minimal hypersurface.” Then they “prove existence of an $I$-parameter family of ancient solutions emanating from a free boundary minimal hypersurface, by careful application of a fix point theorem.” Next they “obtain uniqueness of the constructed family of ancient solutions, by use of an ODE lemma due to Merle--Zaag \cite{MerleZaag1998}, as adapted by \cite{ChoiMantoulidis2022}.” They also note (in connection with handling the free boundary condition) that the “\`\`correct'' deformation is far from trivial,” and that they “achieve this via the implicit function theorem” to “deform to first order by keeping the orthogonality at the boundary,” constructing “a smooth free boundary mean-convex foliation around unstable free boundary minimal hypersurfaces,” which is used for a “different more geometric proof” later via “barriers constructed via the implicit function theorem.”

[cf.\ construction,uniqueness, \begin{corollary} \label{COR} If $\Sigma$ is nondegenerate, then \cref{uniqueness} holds without the assumption \eqref{tu:assume2}.
\end{corollary}] \label{thmA}
Let $\Sigma$ be a free boundary minimal hypersurface of Morse index $I$ in a Riemannian manifold $(M,g)$ with convex boundary. Then there exists an $I$-parameter family of ancient free boundary mean curvature flows emanating from $\Sigma$. Moreover, any ancient solution with sufficiently fast decay to $\Sigma$ belongs to this family.

Question: Let $(M,g)$ be a Riemannian manifold whose boundary $\partial M$ is convex, and let $\Sigma$ be a free boundary minimal hypersurface in $M$ of Morse index $I$ (that is, the area functional has exactly $I$ negative directions at second order along $\Sigma$). An ancient free boundary mean curvature flow is a one-parameter family of hypersurfaces $\{\Sigma_t\}_{t< T}$, defined for all sufficiently negative times, evolving by mean curvature flow, with $\partial \Sigma_t \subset \partial M$ and meeting $\partial M$ orthogonally; saying that such a flow emanates from $\Sigma$ means it converges backward in time to $\Sigma$ as $t\to -\infty$. Under these assumptions, which statement holds?

There exists an $I$-parameter family of ancient free boundary mean curvature flows emanating from $\Sigma$. Moreover, any ancient solution that decays to $\Sigma$ sufficiently fast belongs to this family.

Choices:
B. There exists a unique ancient free boundary mean curvature flow emanating from $\Sigma$. Moreover, every ancient solution converging to $\Sigma$ as $t\to -\infty$ coincides with it, up to time translation.
C. There exists an $I$-parameter family of ancient free boundary mean curvature flows emanating from $\Sigma$.
D. There exists an $I$-parameter family of ancient free boundary mean curvature flows emanating from $\Sigma$. Moreover, any ancient solution that converges to $\Sigma$ as $t\to -\infty$ belongs to this family.
E. There exists an ancient free boundary mean curvature flow emanating from $\Sigma$ if and only if $\Sigma$ is nondegenerate; in that case the space of such ancient solutions is $I$-dimensional, and any sufficiently fast decaying ancient solution belongs to this family.


\label{prob:main}
Let $S\in \mathbb S_n$ be a degree $n$ singular surface in $\mathbb P^3$. In the closure of which Severi varieties $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta$ the point [S] corresponding to $S$
is contained? In particular, compute $\delta_S$.

Let $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta\subset |\mathcal O_{\mathbb P^3}(n)|$ be the \emph{Severi variety} of surfaces $S$ of degree $n$ with $\delta\geqslant 1$ ordinary double points (called \emph{nodes}) and no other singularity. Then $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta$ is a locally closed subvariety of $|\mathcal O_{\mathbb P^3}(n)|\cong \mathbb P^{{{n+3}\choose 3}-1}$. Of course $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_1$ is irreducible of codimension 1 in $|\mathcal O_{\mathbb P^3}(n)|$ and its closure $\mathbb S_n\subset |\mathcal O_{\mathbb P^3}(n)|$ is the set of all singular surfaces of degree $n$. If $S\in \mathbb S_n$ is a singular surface, it makes sense to consider the maximum $\delta_S$ of the integers $\delta$ such that the point [S] corresponding to $S$ belongs to the closure of $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta$.

In this article we will consider the following:

Let $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta\subset |\mathcal O_{\mathbb P^3}(n)|$ be the \emph{Severi variety} of surfaces $S$ of degree $n$ with $\delta\geq 1$ ordinary double points (called \emph{nodes}) and no other singularity. Then $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta$ is a locally closed subvariety of $|\mathcal O_{\mathbb P^3}(n)|\cong \mathbb P^{{{n+3}\choose 3}-1}$. Of course $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_1$ is irreducible of codimension 1 in $|\mathcal O_{\mathbb P^3}(n)|$ and its closure $\mathbb S_n\subset |\mathcal O_{\mathbb P^3}(n)|$ is the set of all singular surfaces of degree $n$. If $S\in \mathbb S_n$ is a singular surface, it makes sense to consider the maximum $\delta_S$ of the integers $\delta$ such that the point [S] corresponding to $S$ belongs to the closure of $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta$.

In this article we will consider the following:

Though this question is rather basic, it does not seem to have received much attention so far.

\begin{problem}\label{prob:main2}
Let $V$ be an irreducible component of a Severi variety $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta$. Let $[S]\in V$ be general and let $C$ be a general plane curve section of the surface $S$. When does it occur that $C$ is a general plane curve of degree $n$?
\end{problem}

\begin{proposition}\label{prop:frt} In the above set--up, suppose that $C$ has $\delta_C$ nodes, that $S_A$ has $\delta_A$ nodes and that $S_\Theta$ has $\delta_\Theta$ nodes all off $E$. Let us denote by $N$ the set of these $\delta:=\delta_A+\delta_C+\delta_\Theta$ nodes. Suppose that
$h^1(\mathcal X_0, \mathcal O_{\mathcal X_0} (nH-m\Theta)\otimes I_N)=0$,
that is equivalent to say that the points in $N$ give independent conditions to surfaces in $|\mathcal O_{\mathcal X_0} (nH-m\Theta)|$. Then $S_0$ can be deformed to a $\delta$--nodal surface $S_t\subset \mathcal X_t\cong \mathbb P^3$ of degree $n$, for $t\neq 0$ and the point corresponding to $S$ sits in the closure of a regular component of the Severi variety $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta$.
\end{proposition}

\begin{theorem}\label{thm:ord} Let $\delta$ be such that there is a very regular irreducible component $V$ of the Severi variety $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(m)|}_\delta$. Then the variety $\mathcal T_{n,m}$ sits in the Zariski closure of a regular component of $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_{\delta}$. In other words, the general degree $n$ surface $S$ with a general ordinary singularity of multiplicity $m$ is the limit of surfaces of degree $n$ with $\delta$ nodes tending to the multiplicity $m$ point.
\end{theorem}

\begin{theorem}\label{thm:ord1} Let $S$ be a surface of degree $n$ in $\mathbb P^3$ with a unique {ordinary singularity} $p$ of multiplicity $m$. Consider the minimal desingularization $f: S'\longrightarrow S$ of $S$ with the exceptional divisor of $S'$ over $p$ a plane curve $C$ of degree $m$. Let $\delta$ be such that there is a surface $X$ of degree $m$ in $\mathbb P^3$ having $C$ as a plane section and belonging to a regular component of
$V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(m)|}_{\delta}$. Then the point corresponding to $S$ belongs to the closure of a regular component of $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_{\delta}$.
\end{theorem}

\begin{theorem}\label{thm:m3} Let $S\subset \mathbb P^3$ be a general surface of degree $n\geq 3$ that has only {ordinary singularities} along a line $R$. Then $S$ belongs to the closure of a regular component of $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta$, with $\delta=3n-4-\epsilon$, where $0\leq \epsilon\leq 1$ has the same parity of $n$. In particular $\mathcal F_n$ is contained in $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta$.
\end{theorem}

Though this question  is  rather basic, it does not seem to have received much attention so far.

One can ask a similar question for plane curves, and it seems to  have been  considered for the first time  by G. Albanese in \cite{Alb}, where he claimed that an irreducible plane curve of degree $n$ and genus $g$ sits in the closure of the Severi variety $V^{\mathbb P^2, |\mathcal O_{\mathbb P^2}(n)|}_\delta$, with
$$\delta={{n-1}\choose 2}-g$$
and $\delta$ is, of course, the maximum integer for which this happens.
However Albanese's arguments do not meet the present standard of rigor, and this theorem has been proved in more recent times by Arbarello--Cornalba \cite {AC} and Zariski \cite {Za}.

In the case of surfaces in $\mathbb P^3$ the problem is much more complicated and we have no idea of what could be a general answer to it. In the present paper, which should be regarded as having an experimental flavor, we focus on two study cases, namely: (a) the case in which $S$ has a single ordinary or quasi--ordinary singularity of multiplicity $m$ (for the definitions  see the beginning of Section \ref {sec:ord}) and, (b) the case in which $S$ has ordinary singularities along a line, i.e., at the general point of the line $S$ has normal crossings, and at finitely many points of the line it has \emph{pinch--points} (i.e., the tangent cone consists there of a plane with multiplicity 2).

Our methods are based on deformation theory of  \emph{regular} components of the Severi variety $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta$ whose closure may contain the point [S] corresponding to $S$.

Recall that an irreducible component $V$ of a Severi variety $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta$ is said to be \emph{regular} if for the general point $[S]\in V$ one has $h^1(\mathbb P^3, \mathcal O_{\mathbb P^3}(n)\otimes \mathcal I_N)=0$, where $N$ is the reduced scheme of lenght $\delta$ of the singular points of $S$. This means that the points in $N$ give $\delta$ independent conditions to the surfaces in $|\mathcal O_{\mathbb P^3}(n)|$. If $V$ is regular, then $V$ is reduced and
$$
\dim (V)= h^0(\mathbb P^3, \mathcal O_{\mathbb P^3}(n))-1-\delta={{n+3}\choose 3}-1-\delta.
$$
Note that if  [S] is in the closure of an irreducible regular component of $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta$, then it sits also in the closure of an irreducible regular component of $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_{\delta'}$ with $\delta'\leqslant \delta$, because the nodes in a regular component of the Severi variety can be independently smoothed.

Case (a) is considered in Section \ref {sec:ord}. In Corollary \ref {cor:gtp} we give an answer to Problem \ref {prob:main} for $3\leqslant m\leqslant 5$.  This is the best possible answer for $m=3$. For $4\leqslant m\leqslant 5$ this is the best possible answer provided one looks only at deformations in regular components of the Severi variety.  In Corollary \ref {cor:gtp}  we  also provide a partial answer for any $m$. In particular we prove that if $S\subset \mathbb P^3$ is surface  of degree $n$ with a unique \emph{general} ordinary singularity of multiplicity $m$, then  the point corresponding to $S$ is in the closure of a regular component of $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_{{m-1}\choose 2}$. An ordinary singularity of multiplicity $m$ is said to be \emph{general} if its tangent cone is the cone over a general plane curve $C$ of degree $m$. In any case, we show that the answer to Problem \ref {prob:main} in case (a)  definitely depends on the fact that the singularity is ordinary or quasi--ordinary.

In order to attack case (a) we first need to consider in Section \ref {sec:curvesec} a preliminary, equally interesting problem, i.e., Problem \ref {prob:main2}: given  an irreducible component $V$ of a Severi variety $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta$, and given  $[S]\in V$  a general point and $C$  a general plane curve section of $S$,  when is it the case that $C$ is a general plane curve of degree $n$? Again we are unable to solve this problem in general, but we prove some partial results, of independent interest (see, e.g., Proposition \ref {prop:good} and its consequences)   that are useful to prove the aforementioned results in case (a).

Case (b) is treated in Section \ref {sec:line}. Our main result here is Theorem \ref {thm:m3}, that says that if $S$ is a general surface of degree $n\geqslant 3$ that has only {ordinary singularities} along a line, then $S$ belongs to the closure of a regular component of $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_{3n-4-\epsilon}$, with  $0\leqslant \epsilon\leqslant 1$ having the same parity of $n$. In particular we prove that we can choose $\ell$, with $3n-4=2\ell+\epsilon$, pinch points of $S$ on the line so that $S$ deforms to a $2\ell$--nodal surface in a regular component  of the corresponding Severi variety, so that each chosen pinch point deforms to two nodes. The integer $3n-4-\epsilon$ is the maximum  for which this happens in the $n$ even case, whereas in the $n$ odd case the maximum could be $3n-4$.

To prove this result we have to rely on the existence of suitable rational surfaces of degree $n$ with a line $r$ of points of multiplicity $n-2$ and with $3n-4-\epsilon$ nodes off $r$, that impose independent conditions to surfaces of degree $n$ in $\mathbb P^3$ that have points of multiplicity $n-2$ along $r$. This is done in Section \ref {sec:rat}.

The method of the proof in both cases (a) and (b) is the same, and it is based on a deformation argument. In both cases we construct a flat family $\mathcal X\longrightarrow \mathbb D$ parametrized by a disk $\mathbb D$, such that for $t\neq 0$, the fibre $\mathcal X_t$ over $t\in \mathbb D$ is isomorphic to $\mathbb P^3$, whereas the central fibre $\mathcal X_0$ is the union of two smooth irreducible components $A\cup \Theta$, that intersect transversally along a surface $E$. The starting surface $S$ for which we consider Problem \ref {prob:main} in the two cases (a) and (b) has a minimal resolution $f: S'\longrightarrow S$, with a curve $C$ mapped by $f$ to the singular locus of $S$. What we  do is to realize $S'$ as a surface $S_A$ in the threefold $A$ in such a way that $S_A\cap E=C$. Then we construct a surface $S_\Theta\subset \Theta$ such that $S_\Theta\cap E=C$ so that $S_0=S_A\cup S_\Theta$ is a Cartier divisor in $\mathcal X_0$. Finally we can do this so that $S_\Theta$ has $\delta$ nodes off $E$, that impose independent conditions to the surfaces in the linear system $|S_0|$ on $\mathcal X_0$. If this is the case then we can apply \cite [Thm. 4.6]{CG} that ensures that $S_0$ can be deformed to a surface $S_t\subset \mathcal X_t\cong \mathbb P^3$ of degree $n$ with $\delta$ nodes that give independent conditions to surfaces of degree $n$. From this it follows  that $S$ sits in the closure of a regular component of $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta$.

We notice that the strength of this method consists in studying  deformations of a surface $S\subset \mathbb P^3$ corresponding to a (presumably) singular point of a Severi variety in terms of deformations of a surface $S_A\cup S_\Theta$ corresponding to a smooth point in  a relative Severi variety.

\medskip

{\bf Aknowledgements:} The authors are members of GNSAGA of INdAM. In particular, the second author acknowledges
funding from the GNSAGA of INdAM and the European Union - NextGenerationEU under the National Recovery and Resilience
Plan (PNRR) - Mission 4 Education and research - Component 2 From research to business - Investment 1.1, Prin 2022
``Geometry of algebraic structures: moduli, invariants, deformations", DD N. 104, 2/2/2022, proposal code 2022BTA242 - CUP
J53D23003720006.

The authors also thank Fabrizio Catanese, Barbara Fantechi and Rita Pardini for useful discussions on related problems to the subject of this paper. \medskip

The post-theorem text does not give a proof of Problem~\ref{prob:main} (it is posed as a question), but it does outline the deformation-theoretic method used to obtain answers in the two study cases.

Key steps of the deformation argument (cases (a) and (b)):
\begin{itemize}
\item Work with \emph{regular} components of $V^{\mathbb P^3,|\mathcal O_{\mathbb P^3}(n)|}_\delta$, i.e. those for which for a general $[S]\in V$ one has $h^1(\mathbb P^3,\mathcal O_{\mathbb P^3}(n)\otimes\mathcal I_N)=0$ (the $\delta$ singular points give independent conditions). Note: if $[S]$ is in the closure of a regular component for $\delta$, then it is also in the closure for any $\delta'\le \delta$ since “the nodes in a regular component ... can be independently smoothed.”
\item Construct a flat family $\mathcal X\to\mathbb D$ such that for $t\neq 0$, $\mathcal X_t\cong\mathbb P^3$, and the central fiber is a normal-crossing union $\mathcal X_0=A\cup\Theta$ of two smooth components meeting transversally along a surface $E$.
\item Let $f:S'\to S$ be the minimal resolution of the starting surface, with a curve $C\subset S'$ mapped to the singular locus of $S$. Realize $S'$ as a surface $S_A\subset A$ with $S_A\cap E=C$.
\item Construct a surface $S_\Theta\subset\Theta$ with $S_\Theta\cap E=C$ so that $S_0=S_A\cup S_\Theta$ is a Cartier divisor on $\mathcal X_0$.
\item Arrange that $S_\Theta$ has $\delta$ nodes off $E$ “that impose independent conditions to the surfaces in the linear system $|S_0|$ on $\mathcal X_0$.”
\item Apply \cite[Thm.~4.6]{CG}: if those $\delta$ nodes impose independent conditions, then $S_0$ deforms to a surface $S_t\subset\mathcal X_t\cong\mathbb P^3$ of degree $n$ with $\delta$ nodes giving independent conditions. Conclude that $S$ lies in the closure of a \emph{regular} component of $V^{\mathbb P^3,|\mathcal O_{\mathbb P^3}(n)|}_\delta$.
\end{itemize}

For case (b), the outline adds that one “rel[ies] on the existence of suitable rational surfaces of degree $n$ with a line $r$ of points of multiplicity $n-2$ and with $3n-4-\epsilon$ nodes off $r$, that impose independent conditions ...,” which is established in Section~\ref{sec:rat}.

The post-theorem text does not give a proof of Problem~\ref{prob:main} (it is posed as a question), but it does outline the deformation-theoretic method used to obtain answers in the two study cases.

Key steps of the deformation argument (cases (a) and (b)):
\begin{itemize}
\item Work with \emph{regular} components of $V^{\mathbb P^3,|\mathcal O_{\mathbb P^3}(n)|}_\delta$, i.e. those for which for a general $[S]\in V$ one has $h^1(\mathbb P^3,\mathcal O_{\mathbb P^3}(n)\otimes\mathcal I_N)=0$ (the $\delta$ singular points give independent conditions). Note: if $[S]$ is in the closure of a regular component for $\delta$, then it is also in the closure for any $\delta'\le \delta$ since “the nodes in a regular component ... can be independently smoothed.”
\item Construct a flat family $\mathcal X\to\mathbb D$ such that for $t\neq 0$, $\mathcal X_t\cong\mathbb P^3$, and the central fiber is a normal-crossing union $\mathcal X_0=A\cup\Theta$ of two smooth components meeting transversally along a surface $E$.
\item Let $f:S'\to S$ be the minimal resolution of the starting surface, with a curve $C\subset S'$ mapped to the singular locus of $S$. Realize $S'$ as a surface $S_A\subset A$ with $S_A\cap E=C$.
\item Construct a surface $S_\Theta\subset\Theta$ with $S_\Theta\cap E=C$ so that $S_0=S_A\cup S_\Theta$ is a Cartier divisor on $\mathcal X_0$.
\item Arrange that $S_\Theta$ has $\delta$ nodes off $E$ “that impose independent conditions to the surfaces in the linear system $|S_0|$ on $\mathcal X_0$.”
\item Apply \cite[Thm.~4.6]{CG}: if those $\delta$ nodes impose independent conditions, then $S_0$ deforms to a surface $S_t\subset\mathcal X_t\cong\mathbb P^3$ of degree $n$ with $\delta$ nodes giving independent conditions. Conclude that $S$ lies in the closure of a \emph{regular} component of $V^{\mathbb P^3,|\mathcal O_{\mathbb P^3}(n)|}_\delta$.
\end{itemize}

For case (b), the outline adds that one “rel[ies] on the existence of suitable rational surfaces of degree $n$ with a line $r$ of points of multiplicity $n-2$ and with $3n-4-\epsilon$ nodes off $r$, that impose independent conditions ...,” which is established later.

\label{prob:main}
Let $S\in \mathbb S_n$ be a degree $n$ singular surface in $\mathbb P^3$. In the closure of which Severi varieties $V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta$ is the point $[S]$ corresponding to $S$ contained? In particular, compute $\delta_S$.

Question:
Let \(V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta\subset |\mathcal O_{\mathbb P^3}(n)|\) denote the Severi variety of degree \(n\) surfaces in \(\mathbb P^3\) having exactly \(\delta\ge 1\) ordinary double points (nodes) and no other singularities. Let \(\mathbb S_n\) be the closure of \(V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_1\), so \(\mathbb S_n\) is the set of all singular degree \(n\) surfaces, and for \(S\in \mathbb S_n\) define
\[
\delta_S:=\max\{\delta : [S]\in \overline{V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta}\}.
\]
For a singular degree \(n\) surface \(S\in \mathbb P^3\), which statement is to be determined?

Determine precisely for which integers \(\delta\) the point \([S]\) lies in the closure of the Severi variety \(V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta\); in particular, compute \(\delta_S=\max\{\delta : [S]\in \overline{V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta}\}."

Choices:
B. Determine precisely for which integers \(\delta\) the point \([S]\) lies in the closure of a regular component of the Severi variety \(V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta\), and require moreover that the corresponding \(\delta\) nodes impose independent conditions on \(|\mathcal O_{\mathbb P^3}(n)|\); in particular, compute the largest such \(\delta\).
C. Determine at least the maximal integer \(\delta_S\) such that \([S]\in \overline{V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta}\).
D. Determine precisely for which integers \(\delta\ge 1\) the point \([S]\) lies in the closure of the Severi variety \(V^{\mathbb P^3, |\mathcal O_{\mathbb P^3}(n)|}_\delta\), knowing that if this holds for some \(\delta\), then it holds only for that maximal value \(\delta=\delta_S\).
E. Determine precisely for which integers \(\delta\) there exists a flat family of degree \(n\) surfaces specializing to \(S\) whose general member has exactly \(\delta\) nodes, all tending to the singular locus of \(S\); in particular, compute \(\delta_S\).


Let $X\subseteq\mathbb{C}$ be an open connected set and $f:X\to\mathbb{C}$ be an elementary meromorphic function with infinitely many critical values. Then the equation $f(x)=a$ is unsolvable in elementary functions.

The question of solvability in explicit form, as a finite composition of admissible operations over a base field, is profoundly significant to the development of algebra. After the general solutions by radicals for polynomial equations of degree up to $4$ were discovered, numerous unsuccessful attempts at formulating a closed-form solution for equations of higher degree led to the belief that such solutions simply do not exist. This was proven by Ruffini (with refinements by Cauchy) in $1813$ and also by Abel in $1824$.

\smallskip

Pondering on this question, Abel laid the foundations of the theory of algebraic curves, while Galois was the first to notice that solvability by radicals is dependent on an associated group, now called the Galois group of an algebraic equation. Liouville continued Abel's work on algebraic and differential forms and made significant improvements regarding expressibility by elementary functions, that is, compositions of arithmetic operations, the exponential function $\exp$ and the logarithmic function $\ln$. He rigorously proved the unsolvability in elementary functions of some differential equations \cite{Khovanskii2014}, e.g. $f'(x) = e^{-x^2}$. Later work by Chebyshev, Ritt, Risch, Rosenlicht, and others extended this aspect of Galois theory further.

\smallskip

Arnold discovered topological reasons for the unsolvability of equations -- for instance, see his proof for the quintic \cite{GoldmakherArnoldQuintic}. His student Khovanskii \cite{khovanskii2019dimensionaltopologicalgaloistheory,Khovanskii2014} formalized the methods of the one-dimensional version of topological Galois theory around 1970. The fundamental concepts of topological Galois theory are similar to classical Galois theory, but are suitable for analyzing transcendental functions -- the former one works with extensions of the rational numbers $\mathbb{Q}$, while the latter one is with extensions of the field of rational functions $\mathbb{C}(x)$. The topological criterion for solvability is also based on a group defined through $f$, known as the monodromy group.

\smallskip

Consider the equation $f(x)=a$, where $f:X\to\mathbb{C}$ is a meromorphic elementary function defined on a connected open set $X\subseteq\mathbb{C}$, and $a\in\mathbb{C}$ is a parameter. We aim to determine when the inverse $f^{-1}(a)$ is also an elementary function. All approaches so far rely on analyzing the specific monodromy group and the structure of the roots, hence results up to now have been on individual equations, such as $\tan(x)-x=a$ \cite{Belov_Kanel_2020}, $e^x + x = a$ and $x^x=a$ \cite{kanelbelov2024insolvabilityxxa} by Kanel-Belov, Malistov, Zaytsev.

\smallskip

Other recent developments are by Zelenko \cite{zelenko2021genericmonodromygroupriemann}, proving unsolvability in quadratures, i.e. elementary functions, integration, differentiation and taking the solutions of an algebraic equation, when $f$ is holomorphic, surjective, no two critical points share the same critical value, and for some positive integer $\rho$ \[\lim_{R\to\infty}\sup_{z:|z|\geq R}\frac{\ln|f(z)|}{|z|^\rho}\] is bounded.

\smallskip

We extend the unsolvability to all elementary meromorphic functions for which the set of critical values
    \[B=\{a\in\mathbb{C}:\ \exists x\in X:f(x)=a,f'(x)=0\}\]
is infinite.

Pondering on this question, Abel laid the foundations of the theory of algebraic curves, while Galois was the first to notice that solvability by radicals is dependent on an associated group, now called the Galois group of an algebraic equation. Liouville continued Abel's work on algebraic and differential forms and made significant improvements regarding expressibility by elementary functions, that is, compositions of arithmetic operations, the exponential function $\exp$ and the logarithmic function $\ln$. He rigorously proved the unsolvability in elementary functions of some differential equations \cite{Khovanskii2014}, e.g. $f'(x) = e^{-x^2}$. Later work by Chebyshev, Ritt, Risch, Rosenlicht, and others extended this aspect of Galois theory further.

Arnold discovered topological reasons for the unsolvability of equations -- for instance, see his proof for the quintic \cite{GoldmakherArnoldQuintic}. His student Khovanskii \cite{khovanskii2019dimensionaltopologicalgaloistheory,Khovanskii2014} formalized the methods of the one-dimensional version of topological Galois theory around 1970. The fundamental concepts of topological Galois theory are similar to classical Galois theory, but are suitable for analyzing transcendental functions -- the former one works with extensions of the rational numbers $\mathbb{Q}$, while the latter one is with extensions of the field of rational functions $\mathbb{C}(x)$. The topological criterion for solvability is also based on a group defined through $f$, known as the monodromy group.

Consider the equation $f(x)=a$, where $f:X\to\mathbb{C}$ is a meromorphic elementary function defined on a connected open set $X\subseteq\mathbb{C}$, and $a\in\mathbb{C}$ is a parameter. We aim to determine when the inverse $f^{-1}(a)$ is also an elementary function. All approaches so far rely on analyzing the specific monodromy group and the structure of the roots, hence results up to now have been on individual equations, such as $\tan(x)-x=a$ \cite{Belov_Kanel_2020}, $e^x + x = a$ and $x^x=a$ \cite{kanelbelov2024insolvabilityxxa} by Kanel-Belov, Malistov, Zaytsev.

Other recent developments are by Zelenko \cite{zelenko2021genericmonodromygroupriemann}, proving unsolvability in quadratures, i.e. elementary functions, integration, differentiation and taking the solutions of an algebraic equation, when $f$ is holomorphic, surjective, no two critical points share the same critical value, and for some positive integer $\rho$ \[\lim_{R\to\infty}\sup_{z:|z|\geq R}\frac{\ln|f(z)|}{|z|^\rho}\] is bounded.

\smallskip

We extend the unsolvability to all elementary meromorphic functions for which the set of critical values
 \[B=\{a\in\mathbb{C}:\ \exists x\in X:f(x)=a,f'(x)=0\}\]
is infinite.

\begin{abstract}
 An equation $f(x)=a$, where $f$ is a complex meromorphic function and $a\in\mathbb{C}$ is a parameter, is solvable in elementary functions if the inverse map $x=f^{-1}(a)$ can be expressed as a finite composition of arithmetic operations (addition, subtraction, multiplication, and division), the exponential function, the complex logarithm, and constants. Specific functions such as $\tan x - x$, $\exp x + x$, $x^x$ have been proven to be unsolvable by Kanel-Belov, Malistov, Zaytsev, while almost all entire surjective functions of at most exponential growth have been covered by Zelenko. All these rely on one-dimensional topological Galois theory, developed by Khovanskii. We generalize to provide a proof for the unsolvability of all elementary meromorphic functions $f$ such that the derivative of $f$ has infinitely many roots $x_i$ and the set of distinct values $f(x_i)$ is infinite.
\end{abstract}

Consider the equation $f(x)=a$, where $f:X\to\mathbb{C}$ is a meromorphic elementary function defined on a connected open set $X\subseteq\mathbb{C}$, and $a\in\mathbb{C}$ is a parameter. We aim to determine when the inverse $f^{-1}(a)$ is also an elementary function. All approaches so far rely on analyzing the specific monodromy group and the structure of the roots, hence results up to now have been on individual equations, such as $\tan(x)-x=a$ \cite{Belov_Kanel_2020}, $e^x + x = a$ and $x^x=a$ \cite{kanelbelov2024insolvabilityxxa} by Kanel-Belov, Malistov, Zaytsev.

We extend the unsolvability to all elementary meromorphic functions for which the set of critical values
 \[B=\{a\in\mathbb{C}:\ \exists x\in X:f(x)=a,f'(x)=0\}\]
is infinite.

The provided proof combines a result of Wielandt on primitive group actions on infinite sets (Theorem \ref{thm:wielandt}), and a relation between primitivity and decompositions (Theorem \ref{prop:imprimitive-equiv-decomposition}) stated, for example, in the doctoral work of Burda on algebraic functions.

\section{\Large Topological unsolvability in elementary functions}
Consider the equation $f(x)=a$, where $f:X\to\mathbb{C}$ is a meromorphic non-constant function defined on a connected open set $X\subseteq\mathbb{C}$, and $a\in\mathbb{C}$ is a parameter. By Theorem~\ref{theorem:identity-theorem} for any $a$ the set of roots is discrete, hence the inverse $f^{-1}$ is an $\mathscr{S}$-function.

\begin{proposition}
\label{prop:wielandt-corollary}
 Let $X \subseteq \mathbb{C}$ be an open connected set and $f:X\to\mathbb{C}$ be a meromorphic map with infinite monodromy group $G$. Suppose $G$ is primitive. Then $G$ contains an unsolvable subgroup, and $f(x)=a$ is unsolvable in elementary functions.
\end{proposition}

\begin{theorem}
\label{thm:unsolvability-infinite-branch-locus}
 Let $X \subseteq \mathbb{C}$ be an open connected set and $f:X\to\mathbb{C}$ be an elementary meromorphic function with an infinite branch locus and infinite monodromy group $G$. Then the equation $f(x)=a$ is unsolvable in elementary functions.
\end{theorem}

\begin{conjecture}
\label{conj:any-solvable-tower-core}
 Let $X\subseteq\mathbb{C}$ be a connected open set and $f: X \to \mathbb{C}$ be a meromorphic elementary function. If the equation $f(x)=a$ is solvable in elementary functions, then there exists a finite sequence $g_1,g_2,\ldots,g_\ell$ of solvable functions of depth at most $1$, such that $f(x)=(g_1\circ g_2\circ\ldots\circ g_\ell)(x)\text{.}$
\end{conjecture}

\begin{proposition} \emph{\cite[Theorem 27]{Burda2012Topological}}
\label{prop:imprimitive-equiv-decomposition}
    Let $X \subseteq \mathbb{C}$ be an open connected set and $f:X\to\mathbb{C}$ be a meromorphic map with monodromy group $G$. Then $G$ is imprimitive if and only if $f$ can be expressed as a composition $(h\circ g)(x)$ of two meromorphic functions $g:X\to Y$ and $h:Y\to\mathbb{C}$ for some open connected set $Y\subseteq\mathbb{C}$. Moreover, the monodromy groups of $g$ and $h$ are quotients of $G$.
\end{proposition}

\begin{theorem}\emph{(Wielandt, \cite[Theorem 29.2]{neumann2023infinitepermutationgroups})}
\label{thm:wielandt}
    Let $S$ be an infinite set and $G$ be a primitive group acting on $S$. If $G$ contains a non-identity permutation with finite support, then $\alt(S)\leq G$.
\end{theorem}

The provided proof combines a result of Wielandt on primitive group actions on infinite sets (Theorem \ref{thm:wielandt}), and a relation between primitivity and decompositions (Theorem \ref{prop:imprimitive-equiv-decomposition}) stated, for example, in the doctoral work of Burda on algebraic functions.

\medskip

The provided proof combines a result of Wielandt on primitive group actions on infinite sets. We first use the following theorem.

\begin{theorem}\emph{(Wielandt, \cite[Theorem 29.2]{neumann2023infinitepermutationgroups})}
\label{thm:wielandt}
    Let $S$ be an infinite set and $G$ be a primitive group acting on $S$. If $G$ contains a non-identity permutation with finite support, then $\alt(S)\leq G$.
\end{theorem}

It also uses a relation between primitivity and decompositions. More precisely, we use the following proposition.

\begin{proposition} \emph{\cite[Theorem 27]{Burda2012Topological}}
\label{prop:imprimitive-equiv-decomposition}
    Let $X \subseteq \mathbb{C}$ be an open connected set and $f:X\to\mathbb{C}$ be a meromorphic map with monodromy group $G$. Then $G$ is imprimitive if and only if $f$ can be expressed as a composition $(h\circ g)(x)$ of two meromorphic functions $g:X\to Y$ and $h:Y\to\mathbb{C}$ for some open connected set $Y\subseteq\mathbb{C}$. Moreover, the monodromy groups of $g$ and $h$ are quotients of $G$.
\end{proposition}

This relation is stated, for example, in the doctoral work of Burda on algebraic functions.

Question:
Let \(X\subseteq\mathbb{C}\) be an open connected set, and let \(f:X\to\mathbb{C}\) be an elementary meromorphic function. Assume that the set of critical values
\[
B=\{a\in\mathbb{C}:\exists x\in X\text{ such that }f(x)=a\text{ and }f'(x)=0\}
\]
is infinite. Here, saying that the equation \(f(x)=a\) is solvable in elementary functions means that the inverse relation \(x=f^{-1}(a)\) can be expressed by a finite composition of arithmetic operations, the exponential function, the complex logarithm, and constants. Which statement holds?

The equation \(f(x)=a\) is unsolvable in elementary functions; equivalently, the inverse \(x=f^{-1}(a)\) cannot be expressed as a finite composition of arithmetic operations, \(\exp\), the complex logarithm, and constants.

Choices:
B. The equation \(f(x)=a\) is unsolvable in elementary functions provided, in addition, that the monodromy group of \(f\) is infinite and primitive.
C. There exists at least one parameter \(a\in B\) for which the equation \(f(x)=a\) is unsolvable in elementary functions.
D. The equation \(f(x)=a\) is unsolvable in elementary functions whenever \(f\) has infinitely many critical points, regardless of whether the corresponding critical values are distinct.
E. If the equation \(f(x)=a\) is solvable in elementary functions, then \(f\) admits a decomposition \(f=h\circ g\) into meromorphic functions on connected open sets such that the monodromy groups of \(g\) and \(h\) are solvable.

primitive-case hypothesis promoted to theorem hypothesis

universal unsolvability conclusion weakened to existence of an unsolvable parameter

infinite critical values replaced by infinite critical points

quotients of monodromy in decomposition upgraded to solvable factors under solvable inverse


\label{Lions-theo}
For any $s$ in $\mathbb R$, the space $\mathcal H_s(\Omega) $  is a RKHS and its two-point kernel $K_s$ is given,  for any pair of $x$ and $y$ in $\Omega$,  by
 \begin{equation} \label{l-formu}
 K_s (x,y)=  \int_{\partial\Omega} \Big((M^{-\frac{s}{2}} \, \gamma_N \big((G_1(x,\cdot)\big)\Big)(z) \, \Big(M^{-\frac{s}{2}} \, \gamma_N \big((G_1(y,\cdot)\big) \Big)(z)\,d\sigma(z) .
 \end{equation}

\begin{thm} \label{Lions-theo}
For any $s$ in $\R$, the space $\mathcal H_s(\Omega) $  is a RKHS and its two-point kernel $K_s$ is given,  for any pair of $x$ and $y$ in $\Omega$,  by
 \begin{equation} \label{l-formu}
 K_s (x,y)=  \int_{\partial\Omega} \Big((M^{-\frac{s}{2}} \, \gamma_N \big((G_1(x,\cdot)\big)\Big)(z) \, \Big(M^{-\frac{s}{2}} \, \gamma_N \big((G_1(y,\cdot)\big) \Big)(z)\,d\sigma(z) .
 \end{equation}
\end{thm}

\begin{Definition} \label{defRKHS}
We say that a Hilbert space $H$ of  real-valued functions defined on  $\Omega$ is a reproducing kernel Hilbert space (RKHS for short)
 if for any $x$ in $\Omega$, the evaluation mapping $E_x$ defined for any $u$ in $H$ by $E_x (u) = u(x)$
is linear and continuous from $H$ to $\R$. Then, we denote by  $K_x$ the unique element in $H$ such that for any $u$ in $H$,  $E_x (u) = \langle u,K_x\rangle $, the inner product of $u$ and $K_x$ in $H$, given by Riesz' theorem. Finally, the two-point kernel is defined for any pair of $x$ and $y$ in $\Omega$  by $K(x,y)= \langle K_x,K_y\rangle $.
\end{Definition}

\subsection{Reproducing kernel Hilbert space}

The notion of  reproducing kernel Hilbert space was first introduced in 1907 by Stanislaw Zaremba for boundary value problems for harmonic and biharmonic functions, and simultaneously by James Mercer in the theory of integral equations, before being more systematically tackled by Nachman Aronszajn and Stefan Bergman. These spaces have various applications in complex analysis, harmonic analysis, quantum mechanics, and statistical learning theory.
They are now defined as Hilbert spaces of functions in which pointwise evaluations are continuous linear functionals. It then follows from Riesz' theorem that each of these functionals can be represented as an inner product with an element of this Hilbert space. One defines a two-point kernel by considering the inner products of any pair of such elements. \ \par \
More precisely, we have the following definition, where  $\Omega$, as in all the paper, is a bounded connected open set of class $C^\infty$ in $\mathbb R^N$.
\begin{Definition} \label{defRKHS}
We say that a Hilbert space $H$ of  real-valued functions defined on  $\Omega$ is a reproducing kernel Hilbert space (RKHS for short)
 if for any $x$ in $\Omega$, the evaluation mapping $E_x$ defined for any $u$ in $H$ by $E_x (u) = u(x)$
is linear and continuous from $H$ to $\mathbb R$. Then, we denote by  $K_x$ the unique element in $H$ such that for any $u$ in $H$,  $E_x (u) = \langle u,K_x\rangle $, the inner product of $u$ and $K_x$ in $H$, given by Riesz' theorem. Finally, the two-point kernel is defined for any pair of $x$ and $y$ in $\Omega$  by $K(x,y)= \langle K_x,K_y\rangle $.
\end{Definition}
Observe that it follows from the properties of the inner product that the kernel $K$ is symmetric and positive definite.
Actually, a famous theorem by  E. H. Moore and N. Aronszajn, see \cite{Aronszajn},  states that every symmetric, positive definite kernel defines a unique reproducing kernel Hilbert space.

\subsection{Lions' formula for a reproducing kernel Hilbert space of harmonic functions}
In \cite{Lions}, J.L. Lions has established an interesting formula for certain reproducing kernel Hilbert spaces of harmonic functions.
To recall his result, let us introduce the following differential operator on functions defined on the boundary $\partial \Omega$.
\begin{Definition} \label{DefBel}
Let $L$ be the negative of the Laplace-Beltrami operator on $\partial \Omega$, and let $M  := L+ 1$. This operator $M$ is called  the  smoothed Laplace-Beltrami operator on $\partial \Omega$.
\end{Definition}
Let $L^2 (\partial \Omega)$ the Hilbert space of square integrable functions on $\partial \Omega$
with respect to the $d-1$-dimensional measure $d\sigma$ on $\partial \Omega$.
Then the  smoothed Laplace-Beltrami operator $M$ is a positive self-adjoint operator in $L^2 (\partial \Omega)$, so that, by Borel  functional calculus, it makes sense to speak of its powers of  any real order. This allows us to define some Sobolev spaces on $\partial \Omega$ for any $s$ in $\mathbb R$ as follows.
\begin{Definition}\label{def1.3}
For any $s$ in $\mathbb R$,  the Sobolev space $H^s(\partial \Omega)$ is the closure of the smooth functions on $\partial \Omega$ for the norm associated with the scalar product
\begin{equation} \label{inpr}
\langle u,v\rangle_s :=
\big\langle M^{\frac{s}{2}} \, u,M^{\frac{s}{2}} \,\,v\big\rangle_{L^2 (\partial \Omega)} .
\end{equation}
\end{Definition}
The Dirichlet trace $\gamma_D$ on the boundary $\partial \Omega$, defined by
\begin{equation} \label{gammad}
    \gamma_D: \, u \in C^\infty (\overline \Omega) \mapsto  u\vert_{\partial \Omega} \in C^\infty (\partial \Omega)
\end{equation}
extends
in a continuous linear map from the  Sobolev space $H^s(\Omega)$
to  the Sobolev space $H^{s-\frac12} (\partial \Omega)$ for all $s > \frac12$. It does not extend to the negative Sobolev spaces $H^{t} (\partial \Omega)$, with $t\leqslant 0$; with such properties.
However, it extends to the subspace of harmonic functions of the Sobolev space $H^s(\Omega)$ for any $s \in \mathbb R$, again as a continuous linear map  with values in  $H^{s-\frac12} (\partial \Omega)$.
Actually, this holds not only for harmonic functions but also for much more general
solutions of elliptic equations, as established by the works of  Lions and Magenes, see \cite{LM63,LM68}, by establishing some
 regularity properties in the case of Sobolev spaces with positive indexes first, then by using a duality argument and finally interpolation theory.
 Another approach, based on the theory of pseudodifferential operators for boundary value problems,
 emphasized that such a property can be seen as a variant of the partial hypoellipticity of the elliptic equations and originated from H\"ormander in \cite[Theorem 2.5.6]{H63} and Boutet de Monvel \cite{B71}, see also  \cite[Theorem 11.4]{Gr} for a more recent account of the topic.
With a slight abuse of notation, we keep the same notation $\gamma_D$ for this trace map, as a continuous linear map from the subspace of harmonic functions of the Sobolev space $H^s(\Omega)$ to $H^{s-\frac12} (\partial \Omega)$
for any $s \in \mathbb R$.

Now, the space of harmonic distributions considered by Lions is the following.
\begin{Definition} \label{def-Hs}
For any $s \in \mathbb R$, we define $\mathcal H_s(\Omega) $ as the space of harmonic functions $u$ in $ H^{s+\frac12} (\Omega)$  whose  trace $\gamma_D (u)$  on $\partial \Omega$ belongs to $H^s(\partial \Omega)$.
\end{Definition}
It follows from the references above that for any
$s$ in $\mathbb R$,  any $u$ in  $\mathcal H_s (\Omega)$ is actually a $C^\infty$ function in $\Omega$, and for any $x$ in $\Omega$,
 we have the Poisson formula:
\begin{equation} \label{Cani1}
    u(x)
    = \Big\langle \gamma_D (u),\, M^{-{s}} \,\gamma_N \big( G_1 (x,\cdot)\big) \Big\rangle_s ,
\end{equation}
where $G_1$ is the Green function, that is the function such that for any $x$ in $\Omega$,
$G_1(x,\cdot)$  is the classical Green function with singularity at $x$, that is the solution to
\begin{equation} \label{DIR}
    -\Delta G_1(x,\cdot)=\delta_x \quad\text{in}\quad \mathcal D'(\Omega)\qquad\&\qquad G_1(x,\cdot)=0\quad\text{on}\quad \partial\Omega,
\end{equation}
and where $\gamma_N \big(G_1(x,\cdot)\big)$ denotes its Neumann trace, that is
$$\gamma_N \big(G_1(x,\cdot)\big)(z):= \nabla_z G_1(x,z)\cdot\nu(z) \quad \text{for}\quad  z\in\partial\Omega,$$ where $\nu(z)$ is the outer unit normal at $z$.
In \eqref{DIR}, $\delta_x$ denotes the Dirac delta distribution at the position $x$.
In particular, it follows from elliptic regularity that
\begin{equation} \label{REg}
    \forall x \in \Omega, \quad   \forall s \in \mathbb R, \quad M^{-\frac{s}{2}} \gamma_N \big(G_1(x,\cdot)\big)  \in L^2 (\partial\Omega) ,
\end{equation}
 so that  the
right-hand side of \eqref{Cani1}
makes sense. Moreover, for any $s$ in $\mathbb R$, the mapping that associates any $g$ in  $H^s(\partial \Omega)$ to the function $\mathcal P_s [g]$ defined for any  $x$  in $\Omega$, by
\begin{equation} \label{poisson-iso}
 \mathcal P_s[g] (x) :=   \Big\langle g, \,M^{-{s}} \,\gamma_N \big( G_1 (x,\cdot)\big) \Big\rangle_s ,
\end{equation}
is a one-to-one isometry onto $\mathcal H_s(\Omega) $. This allows us to transport the Hilbert structure from $H^s(\partial \Omega)$ to $\mathcal H_s(\Omega) $, that is we
 equip $\mathcal H_s(\Omega) $  with the scalar product, defined for any $ u,v$ in $\mathcal H_s(\Omega) $, by
 $\langle \gamma_D (u),\gamma_D (v) \rangle_s $.
Actually, Lions' result, see \cite[Formula (2.13)]{Lions}, is that it is a RKHS with a  two-point kernel given as a boundary integral involving the
Green function $G_1$.

\subsection{Reproducing kernel Hilbert space}

\subsection{Lions' formula for a reproducing kernel Hilbert space of harmonic functions}
In \cite{Lions}, J.L. Lions has established an interesting formula for certain reproducing kernel Hilbert spaces of harmonic functions.
To recall his result, let us introduce the following differential operator on functions defined on the boundary $\partial \Omega$.
\begin{Definition} \label{DefBel}
Let $L$ be the negative of the Laplace-Beltrami operator on $\partial \Omega$, and let $M := L+ 1$. This operator $M$ is called the smoothed Laplace-Beltrami operator on $\partial \Omega$.
\end{Definition}
Let $L^2 (\partial \Omega)$ the Hilbert space of square integrable functions on $\partial \Omega$
with respect to the $d-1$-dimensional measure $d\sigma$ on $\partial \Omega$.
Then the smoothed Laplace-Beltrami operator $M$ is a positive self-adjoint operator in $L^2 (\partial \Omega)$, so that, by Borel functional calculus, it makes sense to speak of its powers of any real order. This allows us to define some Sobolev spaces on $\partial \Omega$ for any $s$ in $\mathbb R$ as follows.
\begin{Definition}\label{def1.3}
For any $s$ in $\mathbb R$, the Sobolev space $H^s(\partial \Omega)$ is the closure of the smooth functions on $\partial \Omega$ for the norm associated with the scalar product
\begin{equation} \label{inpr}
\langle u,v\rangle_s :=
\big\langle M^{\frac{s}{2}} \, u,M^{\frac{s}{2}} \,\,v\big\rangle_{L^2 (\partial \Omega)} .
\end{equation}
\end{Definition}
The Dirichlet trace $\gamma_D$ on the boundary $\partial \Omega$, defined by
\begin{equation} \label{gammad}
 \gamma_D: \, u \in C^\infty (\overline \Omega) \mapsto u\vert_{\partial \Omega} \in C^\infty (\partial \Omega)
\end{equation}
extends
in a continuous linear map from the Sobolev space $H^s(\Omega)$
to the Sobolev space $H^{s-\frac12} (\partial \Omega)$ for all $s > \frac12$. It does not extend to the negative Sobolev spaces $H^{t} (\partial \Omega)$, with $t\leqslant 0$; with such properties.
However, it extends to the subspace of harmonic functions of the Sobolev space $H^s(\Omega)$ for any $s \in \mathbb R$, again as a continuous linear map with values in $H^{s-\frac12} (\partial \Omega)$.
Actually, this holds not only for harmonic functions but also for much more general
solutions of elliptic equations, as established by the works of Lions and Magenes, see \cite{LM63,LM68}, by establishing some
 regularity properties in the case of Sobolev spaces with positive indexes first, then by using a duality argument and finally interpolation theory.
 Another approach, based on the theory of pseudodifferential operators for boundary value problems,
 emphasized that such a property can be seen as a variant of the partial hypoellipticity of the elliptic equations and originated from H\"ormander in \cite[Theorem 2.5.6]{H63} and Boutet de Monvel \cite{B71}, see also \cite[Theorem 11.4]{Gr} for a more recent account of the topic.
With a slight abuse of notation, we keep the same notation $\gamma_D$ for this trace map, as a continuous linear map from the subspace of harmonic functions of the Sobolev space $H^s(\Omega)$ to $H^{s-\frac12} (\partial \Omega)$
for any $s \in \mathbb R$.

Now, the space of harmonic distributions considered by Lions is the following.
\begin{Definition} \label{def-Hs}
For any $s \in \mathbb R$, we define $\mathcal H_s(\Omega) $ as the space of harmonic functions $u$ in $ H^{s+\frac12} (\Omega)$ whose trace $\gamma_D (u)$ on $\partial \Omega$ belongs to $H^s(\partial \Omega)$.
\end{Definition}
It follows from the references above that for any
$s$ in $\mathbb R$, any $u$ in $\mathcal H_s (\Omega)$ is actually a $C^\infty$ function in $\Omega$, and for any $x$ in $\Omega$,
 we have the Poisson formula:
\begin{equation} \label{Cani1}
 u(x)
 = \Big\langle \gamma_D (u),\, M^{-{s}} \,\gamma_N \big( G_1 (x,\cdot)\big) \Big\rangle_s ,
\end{equation}
where $G_1$ is the Green function, that is the function such that for any $x$ in $\Omega$,
$G_1(x,\cdot)$ is the classical Green function with singularity at $x$, that is the solution to
\begin{equation} \label{DIR}
 -\Delta G_1(x,\cdot)=\delta_x \quad\text{in}\quad \mathcal D'(\Omega)\qquad\&\qquad G_1(x,\cdot)=0\quad\text{on}\quad \partial\Omega,
\end{equation}
and where $\gamma_N \big(G_1(x,\cdot)\big)$ denotes its Neumann trace, that is
$$\gamma_N \big(G_1(x,\cdot)\big)(z):= \nabla_z G_1(x,z)\cdot\nu(z) \quad \text{for}\quad z\in\partial\Omega,$$ where $\nu(z)$ is the outer unit normal at $z$.
In \eqref{DIR}, $\delta_x$ denotes the Dirac delta distribution at the position $x$.
In particular, it follows from elliptic regularity that
\begin{equation} \label{REg}
 \forall x \in \Omega, \quad \forall s \in \mathbb R, \quad M^{-\frac{s}{2}} \gamma_N \big(G_1(x,\cdot)\big) \in L^2 (\partial\Omega) ,
\end{equation}
 so that the
right-hand side of \eqref{Cani1}
makes sense. Moreover, for any $s$ in $\mathbb R$, the mapping that associates any $g$ in $H^s(\partial \Omega)$ to the function $\mathcal P_s [g]$ defined for any $x$ in $\Omega$, by
\begin{equation} \label{poisson-iso}
 \mathcal P_s[g] (x) := \Big\langle g, \,M^{-{s}} \,\gamma_N \big( G_1 (x,\cdot)\big) \Big\rangle_s ,
\end{equation}
is a one-to-one isometry onto $\mathcal H_s(\Omega) $. This allows us to transport the Hilbert structure from $H^s(\partial \Omega)$ to $\mathcal H_s(\Omega) $, that is we
 equip $\mathcal H_s(\Omega) $ with the scalar product, defined for any $ u,v$ in $\mathcal H_s(\Omega) $, by
 $\langle \gamma_D (u),\gamma_D (v) \rangle_s $.
Actually, Lions' result, see \cite[Formula (2.13)]{Lions}, is that it is a RKHS with a two-point kernel given as a boundary integral involving the
Green function $G_1$.

One quite explicit formula for the fractional Laplace operator of order $a$, with $a\in (0,1)$, in $N\in \N^*$ dimensions, reads
$$
(-\Delta)^a u (z) := c_{N,a}\, \pv \int_{\R^N}\frac{u(z)-u(y)}{|z-y|^{N+2a}}dy,
$$
with
$$c_{N,a} :=\pi^{N/2}a4^a\frac{\Gamma(\frac{N+2a}{2})}{\Gamma(1-a)},$$
where $\Gamma$ is the gamma function, and where $\pv$ refers to the Cauchy principal value.
Above $u$ is a function from $\R^N$ with real values.
The reason for the presence of the normalization constant $c_{N,a}$
 is to match with the Fourier definition which sets the fractional Laplace operator $(-\Delta)^a$ as the Fourier multiplier of symbol
$|\xi|^{2a}$.
 The Green function $G_a(x,y)$ associated to the operator $(-\Delta)^a$ and the homogeneous Dirichlet condition
is the solution to
\begin{equation} \label{DIRa}
 (-\Delta)^a G_a(x,\cdot)=\delta_x \quad\text{in}\quad \mathcal D'(\Omega)\qquad\&\qquad G_a(x,\cdot)=0\quad\text{in}\quad \R^N\setminus\Omega.
\end{equation}
Such a Green function plays the same role for the operator $(-\Delta)^a$ as the Green function $G_1$ for the classical Laplace operator.
In the companion paper \cite{sidy-franck_1}, we computed the following Hadamard-type variational formula for the fractional Green function $G_a(x,y)$: for any $x$ and $y$ in $\Omega$,
\begin{align}\label{var-green}
 DG_a (\alpha) (x,y) = \Gamma^2(1+a)\int_{\partial\Omega} \, \big(\gamma_0^a(G_a(x,\cdot))\big)(z) \, \big(\gamma^a_0(G_a(y,\cdot))\big)(z)
 \alpha(z) \;d\sigma(z).
\end{align}
Above, $ \Gamma^2(1+a)$ denotes the square of $ \Gamma(1+a)$ while $\gamma_0^a$ denotes the fractional trace of order $a$, that is
$$\gamma_0^a(G_a(x,\cdot):= \gamma_D \big(\frac{G_a(x,\cdot)}{d^a}\big),$$ where $d$ is the distance function to the boundary $\partial\Omega$.
The right-hand side of \eqref{var-green} makes sense because by boundary regularity, for any $x$ in $\Omega$, $\gamma_0^a(G_a(x,\cdot))$ is $C^\infty$ on ${\partial\Omega}$, see \cite[Equation (2.14)]{Grubb-2014}.
Let us point out that the analysis performed in \cite{sidy-franck_1} allows for more general perturbations.
We also refer to the earlier works \cite{DalibardVaret,DFW} on the subject.

\begin{proof}
Let $u$ in $\mathcal H_{a,s}(\Omega)$. Recalling that $\gamma^{a-1}_0(u)$ is in $H^{2\theta}(\partial\Omega)$,
by density, we have a sequence $(g_n)_{n\in\mathbb N}$ in $C^\infty(\partial\Omega)$ such that
\begin{equation} \label{ref-jeudi}
g_n \to \gamma^{a-1}_0(u) \quad \text{in } H^{2\theta}(\partial\Omega).
\end{equation}
By Theorem \ref{f}, for any $n$ in $\mathbb N$, we set
 $u_n := \mathcal P_{a,s}[g_n]$. This
is the unique $u_n$ in $H^{(a-1)(s+2a)}(\Omega)$
solution of
\begin{equation*}
(-\Delta)^a u_n = 0 \ \text{in }\Omega \ \ \& \ \ \gamma^{a-1}_0(u_n) = g_n \ \text{on }\partial\Omega.
\end{equation*}
We fix $x$ in $\Omega$. Let $\rho$ in $C^\infty_0(\mathbb R^N)$ with $\rho \ge 0$ and
$\int_{\mathbb R^N}\rho = 1$, and set $\rho_\varepsilon(z) := \varepsilon^{-N}\rho(z/\varepsilon)$.
For $\varepsilon>0$ small enough so that $\mathrm{supp}\,\rho_\varepsilon(\cdot-x)\subset \Omega$,
we define
\begin{equation} \label{*}
v_{x,\varepsilon}(z) := \int_\Omega G_a(z,y)\rho_\varepsilon(y-x)\,dy.
\end{equation}
Then $v_{x,\varepsilon}=0$ in $\mathbb R^N\setminus\Omega$, and in $\mathcal D'(\Omega)$ we have
\begin{equation} \label{**}
(-\Delta)^a v_{x,\varepsilon} = \rho_\varepsilon(\cdot-x).
\end{equation}
Moreover, since $\rho_\varepsilon(\cdot-x)$ is smooth with compact support in $\Omega$,
the function $v_{x,\varepsilon}$ is regular enough for the hypotheses of \cite[Theorem 5.1]{Grubb-2020}.
We apply Grubb's exact Green formula to the pair $(u_n,v_{x,\varepsilon})$.
Since $(-\Delta)^a u_n = 0$ in $\Omega$ and $(-\Delta)^a v_{x,\varepsilon}=\rho_\varepsilon(\cdot-x)$ in $\Omega$,
we obtain
\begin{equation} \label{***}
\int_\Omega u_n(y)\rho_\varepsilon(y-x)\,dy
= \Gamma(a)\Gamma(a+1)\int_{\partial\Omega} g_n(z)\,\gamma_0^a(v_{x,\varepsilon})(z)\,d\sigma(z).
\end{equation}

Our main result establishes a counterpart of Theorem \ref{Lions-theo} in the case of $a$-harmonic functions by showing that, for an appropriate range of indexes, the spaces $\mathcal H_{a,s}(\Omega)$ are RKHS with two-point kernels given by a boundary integral of some derivatives of the Green function $G_a(x,\cdot)$ of the fractional Laplacian with a singularity at $x$.
We recall that $M$ is the smoothed Laplace-Beltrami operator on $\partial \Omega$, see Definition \ref{DefBel}.
\begin{thm} \label{s-Lions-theo}
Let $a\in (0,1)$, $s>-a-\frac{1}{2}$, and
 $\theta$ be given by \eqref{theta}.
Then the space $\mathcal H_{a,s}(\Omega)$ is a RKHS, and its two-point kernel, which we denote $K_{a,s}$, is given, for any $x$ and $y$ in $\Omega$, by
 \begin{equation} \label{l-formu-s}
 K_{a,s}(x,y)= \Gamma^2(a) \Gamma^2(a+1) \int_{\partial \Omega} \Big( M^{-\theta} \,
 \gamma_0^{a}(G_a(x,\cdot))
 \Big) \Big(M^{-\theta} \, \gamma_0^{a}(G_a(y,\cdot)) \Big) d\sigma.
 \end{equation}
\end{thm}

We also recall the following classical Poisson formula: for any solution $(u,p)$ of the steady Stokes problem \eqref{eq_stoke} we have
\begin{equation} \label{eq_Green-St}
u(x)
= \int_{\partial \Omega} ( \Sigma (\mathfrak G(x,\cdot),\mathfrak P(x,\cdot)) n) \cdot u \,d\sigma.
\end{equation}
where
\begin{equation} \label{eq_Newt2}
\Sigma(u,p) = 2 D(u) - p \mathbb{I}_3 \quad \text{ where } 2 D(u):= \nabla u + (\nabla u)^T ,
\end{equation}
and $\mathbb{I}_3$ is the $3 \times 3$ identity matrix
and $(\mathfrak G(x,\cdot),\mathfrak P(x,\cdot))$ is the Green function associated with
 the Stokes system that is, the unique tensor such that for any $b \in \R^3$, $(u,p) := (\mathfrak G(x,\cdot)b,\mathfrak P(x,\cdot)b)$ is the unique solution to the problem:
\begin{equation*}
\left\{
\begin{array}{rcl}
- \Delta u + \nabla p &= b \delta_x & \, \\
\operatorname{div} u &= 0 &\,
\end{array}
\right. \quad \text{ in $\Omega$} , \end{equation*}
with \begin{equation*}
 u =0\quad\text{on}\quad \partial\Omega.
\end{equation*}
 A reasoning similar to the ones above leads to the conclusion that the space $\mathfrak H_s$ is a RKHS with the following
 Lions type formula for its $2$-points kernel $\mathfrak K_s$:
$$ \mathfrak K_s (x,y) = \int_{\partial\Omega} \Big((M^{-\frac{s}{2}} \, \gamma_D \big(\Sigma(\mathfrak G(x,\cdot),\mathfrak P(x,\cdot))n\big)\Big)(z) \cdot \Big(M^{-\frac{s}{2}} \, \gamma_D \big( \Sigma(\mathfrak G(y,\cdot),\mathfrak P(y,\cdot))n \big) \Big)(z)\,d\sigma(z) .
$$
 On the other hand, it is known since some works by Simon, see for instance \cite{Simon} and the recent works by \cite{KU,Oz},
 that
 the Hadamard variation formula in the case of the Stokes equations reads: for any $b \in \R^3$,
 $$D(\mathfrak G b) (\alpha) (x,y) =
 \int_{\partial\Omega} \gamma_N \big(\mathfrak G(x,\cdot) b\big)(z) \, \gamma_N \big(\mathfrak G(y,\cdot)b\big)(z) \, \alpha(z)d\sigma(z) , $$
where $\alpha\in C^\infty(\partial\Omega)$.

Let us point out that the integral in \eqref{l-formu} makes sense, and is finite, since  for any  $x$  in $\Omega$, for any $s$ in $\mathbb R$,
$M^{-\frac{s}{2}} \gamma_N \big((G_1(x,\cdot)\big)   $
is in $L^2 (\partial\Omega)$.
Actually, Lions dealt with some function spaces associated with second-order elliptic boundary value problems, see \cite{Lions}.
Lions used a variational approach and obtained his formula through a penalization limit.
Another, more direct approach to Theorem \ref{Lions-theo} was proposed later in \cite{ELPL}, together with some extensions to more general elliptic operators of even orders.
We reproduce below their proof of \eqref{l-formu} as a preparation for our extension to the case of fractional Laplace operators.

 \begin{proof}
 It follows from the Poisson formula \eqref{Cani1}, \eqref{REg}
and the Cauchy-Schwarz inequality that for any $x$ in $\Omega$, the evaluation mapping $E_x$, see Definition \ref{defRKHS},   is continuous from $\mathcal H_s(\Omega) $ to $\mathbb R$.
Since $\mathcal H_s(\Omega) $ is a Hilbert space,
 it follows from Riesz' theorem that  for any  $x$  in $\Omega$, there exists $K_x$ in
$\mathcal H_s(\Omega) $
such that for any $u$ in $\mathcal H_s(\Omega) $,
\begin{equation} \label{Cani2}
u(x) = \langle \gamma_D (u) ,\, K_x \rangle_s = \langle M^{\frac{s}{2}} \, \gamma_D (u) ,\,  M^{\frac{s}{2}} \, K_x \rangle_{L^2 (\partial \Omega)}.
\end{equation}
Gathering \eqref{Cani1} and \eqref{Cani2} we deduce that for any $u$ in  $\mathcal H_s (\Omega)$,
\begin{equation*}
  \Big\langle M^{\frac{s}{2}} \, \gamma_D (u) , \, M^{-\frac{s}{2}} \,\gamma_N \big( G_1 (x,\cdot)\big)  - M^{\frac{s}{2}} \, K_x \Big\rangle_{L^2 (\partial \Omega)}=0.
\end{equation*}
 Then by a density argument on $u$, we conclude that
 $$ M^{\frac{s}{2}} \, K_x = M^{-\frac{s}{2}} \,\gamma_N \big( G_1 (x,\cdot)\big)   ,$$
 almost everywhere on $\partial\Omega$.
 Therefore,   for any  $x$ and $y$ in $\Omega$,
 \begin{align*}
     K_s(x,y) &=       \Big\langle M^{\frac{s}{2}} \, K_x, \,M^{\frac{s}{2}} \, K_y  \Big\rangle _{L^2 (\partial \Omega) }
 \\ &=\Big\langle M^{-\frac{s}{2}} \, \gamma_N \big( G_1 (x,\cdot)\big) ,\, M^{-\frac{s}{2}} \, \gamma_N \big( G_1 (y,\cdot)\big) \Big\rangle _{L^2 (\partial \Omega) },
 \end{align*}
 which is the desired conclusion.
 \end{proof}

\subsection{Resemblance with the Hadamard variation formula}
An interesting observation in \cite{ELPL} is the resemblance of \eqref{l-formu} with the Hadamard variation formula
for the Green function of the classical Laplacian, which we now recall.
This formula expresses the derivative of the Green function $G_1$ with respect to the domain.
More precisely, one considers some perturbations $\Omega_t$ of the domain $\Omega$ in the normal direction by modifying the  boundary $\partial\Omega$ of $\Omega$ in
$$\partial\Omega_t=\Big\{y=x+t \alpha(x)\nu(x), \;x\in\partial\Omega\Big\},$$
where $\alpha\in C^\infty(\partial\Omega)$, and $t$ runs in a open interval containing $0$.
Let us rename $G_\Omega$ the Green function
 $G_1$ associated with the domain $\Omega$, as defined by \eqref{DIR}, to emphasize here the dependence on the domain. Then we define the derivative of $G_\Omega$ with respect to the domain in  $\alpha$ as
\begin{align}
     DG_\Omega (\alpha) (x,y):=\lim_{t\to 0, t \neq 0} \, \frac{G_{\Omega_t}(x,y)-G_\Omega(x,y)}{t} .
\end{align}
Then the pioneering discovery by Hadamard, see \cite{Hadamard}, is that not only does this limit exist, but it is even given by the following explicit integral formula: for any $x$ and $y$ in $\Omega$,
\begin{align} \label{Hadamard}
    DG_\Omega (\alpha) (x,y) =\int_{\partial\Omega} \gamma_N \big(G_1(x,\cdot)\big)(z) \, \gamma_N \big(G_1(y,\cdot)\big)(z) \,  \alpha(z)d\sigma(z) .
\end{align}
Let us mention that a rigorous proof of Hadamard's formula was given later on by  Garabedian who also considered more general perturbations. After these pioneering works, shape derivative computations have received significant interest and several extensions of Hadamard's formula for more general elliptic boundary value problems were obtained, with various techniques. We refer here to \cite{HenrotPierre,Sokolowski} for more.
\medskip

Following \cite{ELPL},
 it is now clear that  \eqref{l-formu} and \eqref{Hadamard} look very similar, in particular in the case where $s=0$ for which the only difference is the presence of the extra factor $\alpha$ in \eqref{Hadamard}.
 In \cite{ELPL}, it is said furthermore that the authors  "are of the opinion that the connection between Hadamard variation formula and the reproducing kernel just indicated, in the classical harmonic case, seems to be an isolated phenomenon peculiar to the second order case."

Using the Poisson formula \eqref{Cani1}, \eqref{REg} and the Cauchy--Schwarz inequality, one shows that for any $x\in\Omega$ the evaluation mapping $E_x$ is continuous from $\mathcal H_s(\Omega)$ to $\mathbb R$. Since $\mathcal H_s(\Omega)$ is a Hilbert space, Riesz' theorem gives for each $x\in\Omega$ an element $K_x\in\mathcal H_s(\Omega)$ such that for all $u\in\mathcal H_s(\Omega)$,
\[
 u(x)=\langle \gamma_D(u),K_x\rangle_s=\langle M^{\frac{s}{2}}\gamma_D(u),\,M^{\frac{s}{2}}K_x\rangle_{L^2(\partial\Omega)}. \tag{\eqref{Cani2}}
\]
Gathering \eqref{Cani1} and \eqref{Cani2} yields, for any $u\in\mathcal H_s(\Omega)$,
\[
\Big\langle M^{\frac{s}{2}}\gamma_D(u),\, M^{-\frac{s}{2}}\gamma_N\big(G_1(x,\cdot)\big)-M^{\frac{s}{2}}K_x\Big\rangle_{L^2(\partial\Omega)}=0.
\]
By a density argument on $u$, one concludes that $M^{\frac{s}{2}}K_x=M^{-\frac{s}{2}}\gamma_N\big(G_1(x,\cdot)\big)$ a.e. on $\partial\Omega$. Therefore, for any $x,y\in\Omega$,
\[
K_s(x,y)=\langle M^{\frac{s}{2}}K_x,\,M^{\frac{s}{2}}K_y\rangle_{L^2(\partial\Omega)}
=\big\langle M^{-\frac{s}{2}}\gamma_N\big(G_1(x,\cdot)\big),\,M^{-\frac{s}{2}}\gamma_N\big(G_1(y,\cdot)\big)\big\rangle_{L^2(\partial\Omega)},
\]
which is exactly \eqref{l-formu}.

Using the Poisson formula
\begin{equation} \label{Cani1}
    u(x)
    = \Big\langle \gamma_D (u),\, M^{-{s}} \,\gamma_N \big( G_1 (x,\cdot)\big) \Big\rangle_s ,
\end{equation}
\begin{equation} \label{REg}
    \forall x \in \Omega, \quad   \forall s \in \R, \quad M^{-\frac{s}{2}} \gamma_N \big(G_1(x,\cdot)\big)  \in L^2 (\partial\Omega) ,
\end{equation}
and the Cauchy--Schwarz inequality, one shows that for any $x\in\Omega$ the evaluation mapping $E_x$ is continuous from $\mathcal H_s(\Omega)$ to $\mathbb R$. Since $\mathcal H_s(\Omega)$ is a Hilbert space, Riesz' theorem gives for each $x\in\Omega$ an element $K_x\in\mathcal H_s(\Omega)$ such that for all $u\in\mathcal H_s(\Omega)$,
\begin{equation} \label{Cani2}\nu(x) = \langle \gamma_D (u) ,\, K_x \rangle_s = \langle M^{\frac{s}{2}} \, \gamma_D (u) ,\,  M^{\frac{s}{2}} \, K_x \rangle_{L^2 (\partial \Omega)}.
\end{equation}
Gathering the Poisson formula above and the preceding identity yields, for any $u\in\mathcal H_s(\Omega)$,
\[
\Big\langle M^{\frac{s}{2}}\gamma_D(u),\, M^{-\frac{s}{2}}\gamma_N\big(G_1(x,\cdot)\big)-M^{\frac{s}{2}}K_x\Big\rangle_{L^2(\partial\Omega)}=0.
\]
By a density argument on $u$, one concludes that $M^{\frac{s}{2}}K_x=M^{-\frac{s}{2}}\gamma_N\big(G_1(x,\cdot)\big)$ a.e. on $\partial\Omega$. Therefore, for any $x,y\in\Omega$,
\[
K_s(x,y)=\langle M^{\frac{s}{2}}K_x,\,M^{\frac{s}{2}}K_y\rangle_{L^2(\partial\Omega)}
=\big\langle M^{-\frac{s}{2}}\gamma_N\big(G_1(x,\cdot)\big),\,M^{-\frac{s}{2}}\gamma_N\big(G_1(y,\cdot)\big)\big\rangle_{L^2(\partial\Omega)},
\]
which is exactly
\begin{equation} \label{l-formu}
 K_s (x,y)=  \int_{\partial\Omega} \Big((M^{-\frac{s}{2}} \, \gamma_N \big((G_1(x,\cdot)\big)\Big)(z) \, \Big(M^{-\frac{s}{2}} \, \gamma_N \big((G_1(y,\cdot)\big) \Big)(z)\,d\sigma(z) .
 \end{equation}
.

Question:
Let \(\Omega\) be a domain with boundary \(\partial\Omega\). Let \(L\) be the negative Laplace--Beltrami operator on \(\partial\Omega\), set \(M:=L+1\), and for \(s\in\mathbb R\) define
\[
\mathcal H_s(\Omega):=\{u\in H^{s+\frac12}(\Omega): \Delta u=0 \text{ in }\Omega,\ \gamma_D(u)\in H^s(\partial\Omega)\},
\]
where \(\gamma_D\) is the Dirichlet trace. For \(x\in\Omega\), let \(G_1(x,\cdot)\) be the Dirichlet Green function, characterized by
\[
-\Delta G_1(x,\cdot)=\delta_x \text{ in }\mathcal D'(\Omega),\qquad G_1(x,\cdot)=0 \text{ on }\partial\Omega,
\]
and let \(\gamma_N(G_1(x,\cdot))\) denote its Neumann trace on \(\partial\Omega\). Which explicit two-point kernel is the reproducing kernel of the Hilbert space \(\mathcal H_s(\Omega)\) for every \(s\in\mathbb R\)?

The space \(\mathcal H_s(\Omega)\) is a reproducing kernel Hilbert space, and its reproducing kernel \(K_s\) is
\[
K_s(x,y)=\int_{\partial\Omega} \big(M^{-\frac s2}\gamma_N(G_1(x,\cdot))\big)(z)\,\big(M^{-\frac s2}\gamma_N(G_1(y,\cdot))\big)(z)\,d\sigma(z),\qquad x,y\in\Omega.
\]

Choices:
B.
The space \(\mathcal H_s(\Omega)\) is a reproducing kernel Hilbert space, and its reproducing kernel \(K_s\) is
\[
K_s(x,y)=\int_{\partial\Omega} \big(M^{-s}\gamma_N(G_1(x,\cdot))\big)(z)\,\big(M^{-s}\gamma_N(G_1(y,\cdot))\big)(z)\,d\sigma(z),\qquad x,y\in\Omega.
\]
C.
The space \(\mathcal H_s(\Omega)\) is a reproducing kernel Hilbert space, and its reproducing kernel can be written as an \(L^2(\partial\Omega)\)-inner product of the boundary functions \(M^{-\frac s2}\gamma_N(G_1(x,\cdot))\) and \(M^{-\frac s2}\gamma_N(G_1(y,\cdot))\); equivalently,
\[
K_s(x,y)=\big\langle M^{-\frac s2}\gamma_N(G_1(x,\cdot)),\,M^{-\frac s2}\gamma_N(G_1(y,\cdot))\big\rangle_{L^2(\partial\Omega)},\qquad x,y\in\Omega.
\]
D.
The space \(\mathcal H_s(\Omega)\) is a reproducing kernel Hilbert space, and its reproducing kernel \(K_s\) is
\[
K_s(x,y)=\int_{\partial\Omega} \big(M^{-\frac s2}\gamma_D(G_1(x,\cdot))\big)(z)\,\big(M^{-\frac s2}\gamma_D(G_1(y,\cdot))\big)(z)\,d\sigma(z),\qquad x,y\in\Omega.
\]
E.
For every \(s\in\mathbb R\) and every \(x\in\Omega\), there exists \(K_x\in\mathcal H_s(\Omega)\) such that
\[
K_s(x,y)=\int_{\partial\Omega} \big(M^{-\frac s2}\gamma_N(G_1(x,\cdot))\big)(z)\,\big(M^{\frac s2}\gamma_N(G_1(y,\cdot))\big)(z)\,d\sigma(z),\qquad x,y\in\Omega.
\]

half-power of M from the Hilbert pairing

dropped the explicit boundary integral measure form while keeping the equivalent L^2-inner-product identity

Neumann trace replaced by Dirichlet trace of the Green function

mismatched powers of M on the two kernel factors, violating the Riesz identification