\label{sec.Intro}
Aperiodic tilings of Euclidean space, a topic once regarded as an esoteric pursuit, are now widely recognized as a subject of central interest. Beginning with work of Hao Wang and Robert Berger in the 1960's studying a relationship between aperiodic tilings and the Halting problem for Turing machines, results surrounding aperiodic tilings have had substantial impacts in the last century within mathematics, computer science, the natural sciences, and even the liberal arts. Excellent summaries of many of these results, up until about 2010, can be found in \cite[Chapter 1]{Gard1997}, \cite[Chapters 1, 2]{Sene1995}, \cite[Forward by Roger Penrose]{BaakGrim2013}, and \cite[Chapter 1]{BaakGrim2013}. There has also been significant and highly publicized progress in this subject in the last decade.

One landmark, which has also become an inspiration in art and architecture, was Roger Penrose's discovery of a pair of polygonal tiles which could tile the plane, but could only do so aperiodically. Penrose published a first version of his tilings in 1974 \cite{Penr1974} (see also \cite{Penr1979}). Soon after, they were introduced to the general public in an article by Martin Gardner in the Scientific American \cite{Gard1977}.

\begin{figure}[h]
 \caption{Tiles used to construct rhombic Penrose tilings.\\}\label{fig.PenTiles}
 \centering
 \vspace*{-30bp}
 \includegraphics[width=0.75\textwidth]{PenTiles1mod}
\end{figure}

There are several related collections of tilings which are referred to as Penrose tilings. In this article we will consider \textit{rhombic Penrose tilings} (RPTs), which are tilings of the plane by copies of two rhombuses with angles $72^\circ$ and $36^\circ$, pictured in Figure \ref{fig.PenTiles}. The rhombuses are allowed to be rotated, but they must fit together edge to edge, and the matching rules indicated by the arrows must also be enforced. A small patch of an RPT is pictured in Figure \ref{fig.PenPatch} (see also \cite[Figures 1.2, 6.44]{BaakGrim2013}). When the rhombuses are chosen to have side length 1, we call the resulting tilings \textit{unit length RPTs}.

The main problem that we wish to investigate in this paper is that of estimating the number of vertices of an RPT which lie in $B_R$, the closed Euclidean ball of radius $R$, centered at some point in the plane which we fix to be the origin. RPTs can be defined using primitive inflation rules (see \cite[Section 6.2]{BaakGrim2013}), which guarantee existence of limiting frequencies of tile types in large regions. Although it is not obvious, it is not too difficult to deduce (see for example \cite[Remark 5]{SchmTrev2018}) that if $V$ is the vertex set of a unit length RPT then, for $R\ge 1$,
\begin{equation}\label{eqn.PenVertTrivEst}
 \#(V\cap B_R)=\pi C_P R^2 + O(R),
\end{equation}
where
\[C_P=\frac{\phi}{5}\sqrt{10+2\sqrt{5}}\approx 1.231,\]
with $\phi=(1+\sqrt{5})/2$. The error term on the right hand side of \eqref{eqn.PenVertTrivEst} comes from counting tiles which intersect the boundary of $B_R$.
Since there are in fact $\gg R$ such tiles, it may appear that this error term is essentially best possible (cf. results of Solomon from \cite{Solo2011} on counting tiles in \textit{supertiles} of primitive substitutions). However, we will show that for this special class of tilings, one can actually do better.

\begin{equation}\label{eqn.PenVertTrivEst}
	\#(V\cap B_R)=\pi C_P R^2 + O(R),
\end{equation}

\begin{figure}[h]
	\caption{A patch of an RPT constructed using the cut and project method detailed in Section \ref{sec.ModelSets}. The 4 colors of vertices are translates of the cut and project sets produced using the different windows.\\}\label{fig.PenPatch}
	\centering
	\includegraphics[width=0.75\textwidth]{PenPatch2}
\end{figure}

\begin{figure}[h]
	\caption{Tiles used to construct rhombic Penrose tilings.\\}\label{fig.PenTiles}
	\centering
	\vspace*{-30bp}
	\includegraphics[width=0.75\textwidth]{PenTiles1mod}
\end{figure}

There are several related collections of tilings which are referred to as Penrose tilings. In this article we will consider \textit{rhombic Penrose tilings} (RPTs), which are tilings of the plane by copies of two rhombuses with angles $72^\circ$ and $36^\circ$, pictured in Figure \ref{fig.PenTiles}. The rhombuses are allowed to be rotated, but they must fit together edge to edge, and the matching rules indicated by the arrows must also be enforced. A small patch of an RPT is pictured in Figure \ref{fig.PenPatch} (see also \cite[Figures 1.2, 6.44]{BaakGrim2013}). When the rhombuses are chosen to have side length 1, we call the resulting tilings \textit{unit length RPTs}.

\begin{figure}[h]
 \caption{A patch of an RPT constructed using the cut and project method detailed in Section \ref{sec.ModelSets}. The 4 colors of vertices are translates of the cut and project sets produced using the different windows.\\}\label{fig.PenPatch}
 \centering
 \includegraphics[width=0.75\textwidth]{PenPatch2}
\end{figure}

The work of de Bruijn, et al., mentioned at the beginning of this section (see references listed above), implies the following result.
\begin{theorem}\label{thm.PenVerts}
Every non-singular point set $\Lambda_\omega$, constructed as above, is the set of vertices of a unit length rhombic Penrose tiling. Furthermore, the vertex set of any unit length rhombic Penrose tiling can be obtained, after rotation, as the limit in an appropriate topology (e.g. the Chabauty-Fell topology) of a sequence of non-singular point sets $\Lambda_\omega$.
\end{theorem}
An example of a RPT obtained using the cut and project method described above is shown in Figure \ref{fig.PenPatch}. Theorem \ref{thm.PenVerts} will allow us to derive Theorem \ref{thm.Pen} from the following result.
\begin{theorem}\label{thm.PenCPVersion}
For the cut and project sets $\Lambda_{m,\omega}$ defined above, there is a constant $C>0$ with the property that, for any $\omega\in\C$, any integer $1\le m\le 4$, and any $R\ge 2$,
\[\left|\#\left(\Lambda_{m,\omega}\cap B_R\right)-\pi R^2\mr{dens}(\cL)\left|W_{m,\omega}\right|\right|\le CR^{2/3}(\log R)^{2/3},\]
where $|W_{m,\omega}|$ denotes the area of $W_{m,\omega}$, and $\mr{dens}(\cL)$ is the reciprocal of the volume of a fundamental domain for $\R^4/\cL$.
\end{theorem}
An important point is that the error term in this theorem is uniform in $\omega$, which allows us to take limits of non-singular point sets, to account for all possible vertex sets of RPTs, in order to establish Theorem \ref{thm.Pen}. There are a few other small details that have to be ironed out, but they are not difficult to justify, so we postpone their discussion until the end of Section \ref{sec.PenProof}. The majority of the remainder of the paper will be focused on proving Theorem \ref{thm.PenCPVersion}.

If $W\subseteq\R^2$ then we write $\chi_W$ for the indicator function of $W$. The similarity in notation to $\chi_R$, as defined above, will not cause confusion in what follows. Observe that
\[\widehat{\chi}_W(y)=\int_We(-x\cdot y)\mr{d}x=\frac{-1}{(2\pi|y|)^2}\int_W\nabla_x^2\left( e(-x\cdot y)\right)\mr{d}x,\]
where $\nabla_x^2=\nabla_x\cdot\nabla_x$ is the Laplacian. By the Divergence Theorem, under reasonable hypotheses on $W$, this allows us to express the Fourier transform as a line integral over the boundary of $W$. Applying this argument in the case when $W$ is a polygon leads to a particularly nice formula for the Fourier transform, which we reproduce here in the form given in \cite{BranColzTrav1997}.
\begin{proposition}\label{prop.FourTranPoly}
 Let $P\subseteq\R^2$ be a polygon with counterclockwise oriented vertices $\{a_j\}_{j=1}^m$. Denote by $\sigma_j$ the unit vector parallel to the side $[a_j,a_{j+1}]$ and by $v_j$ the outward unit normal to this side. Then, with $a_{m+1}:=a_1$, we have
 \[\widehat{\chi}_P(y)=\frac{-1}{(2\pi|y|)^2}\sum_{j=1}^m\left(\frac{e(-y\cdot a_{j+1})-e(-y\cdot a_{j})}{y\cdot \sigma_j}\right)y\cdot v_j.\]
\end{proposition}
This proposition is proved as \cite[Lemma 2.2]{BranColzTrav1997}. The quantity in parenthesis in the sum is interpreted to equal
\[-\left(2\pi i |a_{j+1}-a_j|\right)e\left(\frac{-y\cdot(a_j+a_{j+1})}{2}\right),\]
when $y\cdot\sigma_j=0$. Proposition \ref{prop.FourTranPoly} implies that
\begin{equation}\label{eqn.PolyFourEst}
 \widehat{\chi}_P(y)\ll_P \frac{1}{|y|}\sum_{j=1}^m\min\left\{1,\frac{1}{|y\cdot\sigma_j|}\right\}.
\end{equation}
One thing to note is that these Fourier coefficients decay in most directions like $|y|^{-2}$. However, in directions perpendicular to the sides of $P$, they decay only like $|y|^{-1}$. This turns out to be a source of some difficulty, which is given special attention in the proofs below.

\noindent\emph{Bound for $E_2$:} To estimate $E_2$, first write
\[\mc{S}_{m,n}=\left\{\theta\in L^\circledast : \frac{2^{m-1}}{R}<|\theta|\le\frac{2^m}{R},~\frac{nR}{M2^m}\le|\theta^\star|<\frac{(n+1)R}{M2^m}\right\},\]
with $M$ chosen as in \eqref{eqn.MChoice}. Then we have that
\begin{align}
 E_2&\ll R^{1/2}\sum_{m=1}^{\lceil \log_2R\rceil}\sum_{n=1}^\infty\sum_{\theta\in\mc{S}_{m,n}}\frac{1}{|\theta|^{3/2}}|\widehat{\chi}_W(\theta^\star)|\left(1+\delta^2|\theta^\star|^2\right)^{-N}\nonumber\\
 &\ll R^2 \sum_{m=1}^{\lceil\log_2R\rceil}2^{-3m/2}\sum_{n=1}^\infty\left(1+\frac{\delta^2n^2R^2}{2^{2m}}\right)^{-N}\sum_{\theta\in\mc{S}_{m,n}}|\widehat{\chi}_W(\theta^\star)|.\label{eqn.E2Bound1}
\end{align}
Here we will use the following lemma.
\begin{lemma}\label{lem.ChiWEst2}
 With notation as above, for $m,n\in\N$ with $m\le\lceil \log_2R\rceil$, we have that
 \[\sum_{\theta\in\mc{S}_{m,n}}|\widehat{\chi}_W(\theta^\star)|\ll\frac{2^{2m}\log n}{nR^2}.\]
\end{lemma}
\begin{proof}
 Let us write
 \[\tilde{\mc{S}}=\left\{\lambda\in \frac{1}{25}L : \frac{2^{m-1}}{R}<|\lambda|\le\frac{2^m}{R},~\frac{nR}{M2^m}\le|\lambda^\star|<\frac{(n+1)R}{M2^m},~|\mr{arg}(\lambda^\star)|\le \frac{\pi}{5}\right\}.\]
 By the argument used in the proof of Lemma \ref{lem.ChiWEst1} leading up to equation \eqref{eqn.ChiWEst1}, it is sufficient for this lemma to prove that
 \begin{equation*}
 \sum_{\lambda\in\tilde{\mc{S}}}\frac{1}{|\lambda^\star|^2|\sin(\mathrm{arg}(\lambda^\star))|}\ll\frac{2^{2m}\log n}{nR^2}.
 \end{equation*}
 Label the elements of $\star(\tilde{\mc{S}})$ as
 \[\lambda_j^\star=\rho_je^{it_j},\]
 for $1\le j\le J:=|\tilde{\mc{S}}|$, with each $\rho_j\in\R^+$ and with $0< |t_1|\le |t_2|\le\cdots \le |t_J|\le\pi/5$. Also, label the element of $\tilde{\mc{S}}$ corresponding to each $\lambda_j^\star$ as $\lambda_j$.

\noindent\emph{Proof of Theorem \ref{thm.Pen}:} Let $V$ be the vertex set of a unit length RPT. If $V$ can be constructed as a cut and project set, as $V=\Lambda_\omega$ for some $\omega\in\C$, then the result of Theorem \ref{thm.Pen} follows immediately from Theorem \ref{thm.PenCPVersion}. Otherwise, by Theorem \ref{thm.PenVerts}, there is a sequence of complex numbers $\{\omega_j\}_{j=1}^\infty$ with the property that, after a possible rotation,
\[V=\lim_{j\rar\infty}\Lambda_{\omega_j},\]
where the limit is taken in the Chabauty-Fell topology. For $R\ge 2$, the number of points in $\R^2$ which lie on the boundary of $B_R$, and which are the limits of sequences of points $\lambda_j\in\Lambda_{\omega_j}$, as $j\rar\infty,$ must be $\ll R^{2/3}(\log R)^{2/3}$. This follows from the uniformity over $\omega\in \C$ of the error term in Theorem \ref{thm.PenCPVersion}. It then follows, again by uniformity, that
\begin{align}
 \#(V\cap B_R)&=\lim_{j\rar\infty}\#\left(\Lambda_{\omega_j}\cap B_R\right)+O(R^{2/3}(\log R)^{2/3})\nonumber\\
 &=\pi R^2\mr{dens}(\cL)\sum_{m=1}^4|W_{m,\omega+m}|+O( R^{2/3}(\log R)^{2/3}).\label{eqn.ChabLim1}
\end{align}
Finally, we compute that
\[\mr{dens}(\cL)=\mr{disc}(L)^{-1/2}=\frac{4}{25\sqrt{5}}.\]
Substituting this in \eqref{eqn.ChabLim1}, together with the areas of the pentagonal windows, completes the proof of Theorem \ref{thm.Pen}.

Readers who are familiar with the analogous counting problem for integer lattice points in the plane, called the Gauss circle problem, may at this point recognize some similarities. Let us write
\[\#(\mathbb{Z}^2\cap B_R)=\pi R^2 + E(R).\]
Gauss himself observed that $|E(R)|\le 2\sqrt{2}\pi R$, and he conjectured that it should be possible to improve the exponent on $R$  to obtain an error term of $\ll R^{1/2}$. The next substantial result was published in 1906 by Sierpi\'nski \cite{Sier1906}, who showed that $E(R)\ll R^{2/3}$ (see also \cite{Schi1972}). Subsequent improvements have been made by many authors (the introduction of \cite{Huxl2003} contains a list), and currently the best known upper bound, due to Martin Huxley \cite{Huxl2003}, is $E(R)\ll R^{131/208}(\log R)^\tau$, with $\tau=18627/8320$.

The important link which allows us to employ techniques used in the Gauss circle problem in order to study vertices of Penrose tilings is a remarkable observation of de Bruijn, that the vertex set of any RPT can be obtained from a cut and project set, by projecting `slices' of lattice points in a higher dimensional space onto a 2 dimensional subspace. We describe this connection in detail in Section \ref{sec.ModelSets}. With this as a general framework, we are able in Sections \ref{sec.Fourier}-\ref{sec.PenProof} to bring in tools from Fourier analysis and to approach our point counting problem through the lens of the Poisson summation formula (PSF). Results of a similar flavor, estimating discrepancies of points in growing patches of cut and projects sets, have recently been explored by Koivusalo and Lagac{\'e} in \cite{KoivLagacBjorHart2025}, and by R\"uhr, Smilansky, and Weiss in \cite{RuhrSmilWeis2024}.

We remark that our proof, although structurally similar, is not a na\"ive application of known methods (e.g. Huxley's Fourier analytic approach to lattice point counting in regions with smooth boundary, using Van der Corput's method). Here, there are two new difficulties which are not encountered in the classical versions of the Gauss circle problem. The first is that, for the cut and project sets that we consider, the problem amounts to counting points in a lattice in a $4$-dimensional cylinder which is only growing radially. Since the height of the cylinder stays fixed, the smoothing necessary to apply the PSF must shrink faster in the direction of the height than in the radial direction. Secondly, the lattices from which we derive RPTs are irrational. This has the effect that, depending on how well approximable the lattice points are by rational points, smoothing our regions may pick up points which are unusually close to their boundaries. This is a real phenomenon, which can potentially cause a blow up in the Fourier coefficients after using the PSF. It is one of the main issues that we must confront using the specific geometric and algebraic data which define RPTs.

\begin{figure}[h]
	\caption{A patch of an RPT constructed using the cut and project method detailed in Section \ref{sec.ModelSets}. The 4 colors of vertices are translates of the cut and project sets produced using the different windows.\\}\label{fig.PenPatch}
	\centering
	\includegraphics[width=0.75\textwidth]{PenPatch2}
\end{figure}

Finally, there is the question of how these results could be generalized. Several parts of the proofs below can be adapted quite well to deal with much wider classes of cut and project sets. However, there are some crucial details in the proof, in particular those contained in Section \ref{sec.PenProof}, which form a narrow gate that any more general result would need to pass through. The cut and project sets used to define RPTs are extremely special, both in the algebraic properties of the lattice involved and in the exact shape, symmetry, and orientation of the windows involved, with respect to the specific underlying lattice. Any attempt to modify the lattice, or change the windows (even by a small rotation) will destroy important features of the argument and make it difficult or even impossible to control the sizes of the Fourier coefficients involved.

On the other hand, it seems likely that the arguments given here should apply equally well, with the appropriate modifications, to vertex sets of Ammann-Beenker tilings (see \cite[Section 10.4]{GrunShep1987} and \cite{Been1982}). We leave it to the interested reader to work out the details.

To prove Theorem~\ref{thm.Pen}, the authors connect vertex counting in a unit length RPT to the Gauss circle problem by using “a remarkable observation of de Bruijn, that the vertex set of any RPT can be obtained from a cut and project set, by projecting `slices' of lattice points in a higher dimensional space onto a 2 dimensional subspace” (detailed in Section~\ref{sec.ModelSets}). With this framework, they “bring in tools from Fourier analysis” and “approach [the] point counting problem through the lens of the Poisson summation formula (PSF)” (Sections~\ref{sec.Fourier}--\ref{sec.PenProof}).

They emphasize two main technical obstacles (and thus key components of the argument): (1) the counting reduces to “counting points in a lattice in a $4$-dimensional cylinder which is only growing radially,” so because “the height of the cylinder stays fixed, the smoothing necessary to apply the PSF must shrink faster in the direction of the height than in the radial direction”; and (2) the relevant lattices are “irrational,” so “depending on how well approximable the lattice points are by rational points, smoothing our regions may pick up points which are unusually close to their boundaries,” which “can potentially cause a blow up in the Fourier coefficients after using the PSF.” Controlling this requires “the specific geometric and algebraic data which define RPTs,” and the authors note that “crucial details in the proof, in particular those contained in Section~\ref{sec.PenProof}, … make it difficult or even impossible to control the sizes of the Fourier coefficients involved” under general perturbations of lattice/windows.

To prove the main theorem, the authors connect vertex counting in a unit length RPT to the Gauss circle problem by using “a remarkable observation of de Bruijn, that the vertex set of any RPT can be obtained from a cut and project set, by projecting `slices' of lattice points in a higher dimensional space onto a 2 dimensional subspace” (detailed later). With this framework, they “bring in tools from Fourier analysis” and “approach [the] point counting problem through the lens of the Poisson summation formula (PSF)” (proved later).

They emphasize two main technical obstacles (and thus key components of the argument): (1) the counting reduces to “counting points in a lattice in a $4$-dimensional cylinder which is only growing radially,” so because “the height of the cylinder stays fixed, the smoothing necessary to apply the PSF must shrink faster in the direction of the height than in the radial direction”; and (2) the relevant lattices are “irrational,” so “depending on how well approximable the lattice points are by rational points, smoothing our regions may pick up points which are unusually close to their boundaries,” which “can potentially cause a blow up in the Fourier coefficients after using the PSF.” Controlling this requires “the specific geometric and algebraic data which define RPTs,” and the authors note that “crucial details in the proof, in particular those contained in the later argument, … make it difficult or even impossible to control the sizes of the Fourier coefficients involved” under general perturbations of lattice/windows.

\label{thm.Pen}
Let $V$ be the set of vertices of any unit length RPT. Then, for $R\ge 2$,
\[\#(V\cap B_R)=\pi C_P R^2 + O(R^{2/3}(\log R)^{2/3}).\],

Question:
Let $V$ be the vertex set of any unit length rhombic Penrose tiling, meaning a tiling of the plane by copies of the two Penrose rhombi of side length $1$ with angles $72^\circ$ and $36^\circ$, placed edge-to-edge and satisfying the usual matching rules. Let $B_R$ denote the closed Euclidean ball of radius $R$ centered at the origin, and define
\[
C_P=\frac{\phi}{5}\sqrt{10+2\sqrt{5}}, \qquad \phi=\frac{1+\sqrt{5}}{2}.
\]
Which quantitative estimate holds for the number of vertices of the tiling in $B_R$ when $R\ge 2$?

For every such $V$ and all $R\ge 2$,
\[
\#(V\cap B_R)=\pi C_P R^2 + O\!\left(R^{2/3}(\log R)^{2/3}\right).
\]

Choices:
B.
For every such $V$ and all $R\ge 2$,
\[
\#(V\cap B_R)=\pi C_P R^2 + O\!\left(R^{1/2}(\log R)^{2/3}\right).
\]
C.
For every such $V$ and all $R\ge 2$,
\[
\#(V\cap B_R)=\pi C_P R^2 + o(R).
\]
D.
For every such $V$ there exists a constant $K_V>0$ such that for all $R\ge 2$,
\[
\#(V\cap B_R)=\pi C_P R^2 + O\!\left(K_V R^{2/3}(\log R)^{2/3}\right).
\]
E.
For every such $V$ and all $R\ge 2$,
\[
\#(V\cap B_R)=\pi C_P R^2 + O\!\left(R^{2/3}\right).
\]


Question:
Let $V$ be the vertex set of any unit length rhombic Penrose tiling of the plane, meaning a tiling by copies of the two Penrose rhombi with angles $72^\circ$ and $36^\circ$, placed edge-to-edge with the usual arrow matching rules, and with each side of length $1$. For $R\ge 2$, let $B_R$ denote the closed Euclidean ball of radius $R$ centered at the origin. Write
\[
C_P=\frac{\phi}{5}\sqrt{10+2\sqrt{5}},\qquad \phi=\frac{1+\sqrt5}{2}.
\]
As $R\to\infty$, which asymptotic statement holds for the number of vertices of the tiling contained in $B_R$?

For every unit length rhombic Penrose tiling, one has, for $R\ge 2$,
\[
\#(V\cap B_R)=\pi C_P R^2+O\big(R^{2/3}(\log R)^{2/3}\big).
\]

Choices:
B.
For every unit length rhombic Penrose tiling, one has, for $R\ge 2$,
\[
\#(V\cap B_R)=\pi C_P R^2+O(R^{1/2+\varepsilon})\qquad\text{for every }\varepsilon>0.
\]
C.
For every unit length rhombic Penrose tiling, one has, for $R\ge 2$,
\[
\#(V\cap B_R)=\pi C_P R^2+o(R).
\]
D.
For every unit length rhombic Penrose tiling, there exists a constant $K=K(V)$ such that, for $R\ge 2$,
\[
\#(V\cap B_R)=\pi C_P R^2+O\big(KR^{2/3}(\log R)^{2/3}\big).
\]
E.
For every unit length rhombic Penrose tiling, one has, for $R\ge 2$,
\[
\#(V\cap B_R)=\pi C_P R^2+O\big(R^{2/3}\big).
\]

error-exponent strengthened to near-Gauss-circle strength

explicit quantitative error term dropped to a weaker sublinear remainder

uniformity over all Penrose tilings replaced by tiling-dependent constant

logarithmic loss from anisotropic smoothing/Fourier control removed


\begin{thm}
  \label{main1}
  Let $X$ be a tropical variety in $\mathbb R^n$.
  Then $\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is a rational affine linear subspace of $\mathbb R^n$.
\end{thm}

Tropical geometry is a subject of mathematics developed in recent decades, which studies geometric objects called tropical varieties.
Sometimes tropical geometry is called \textquotedblleft the algebraic geometry over $\mathbb T$\textquotedblright, where $\mathbb T$ is a semifield called the tropical semifield.
Since $\mathbb T$ is not a field, tropical geometry is not included in classical algebraic geometry.
However, for many albedo-geometric theories, their tropical versions are studied.
For instance, the tropical intersection theory \cite{allermann2010first,jensen2013computing,jensen2016stable,katz2012tropical,rau2016intersections}, the theory of divisors on tropical curves \cite{amini2013reduced,an2014canonical,haase2012linear,hladky2013rank,len2014brill}, and the theory of tropical homology \cite{amini2020hodge,gross2019sheaf,itenberg2019tropical,ruddat2021homology} have been developed.

One of the basic objects in algebraic geometry is the ideals of rings.
In tropical geometry, there are several analogous objects.
The simplest one is the concept of ideal of a semiring, which is defined as a submodule of the semiring itself.
Another one is the concept of tropical ideal, which is defined by Maclagan and Rinc\'{o}n in \cite{maclagan2018tropical}.
It is an analogue of an ideal of a ring since the quotient semiring $R/E$ is defined.
Moreover, there are several specific class of congruences.
In \cite{giansiracusa2016equations}, J. Giansiracusa and N. Giansiracusa define \textit{bend congruences} to develop the theory of scheme theoretic tropicalization.
Another important class is the congruences of the form $\mathbf{E}(Z)$ for some $Z \subset \mathbb R^n$.
Here,
$$\mathbf{E}(Z) = \{ (f,g) \in \mathbb T[x_1, \ldots, x_n] \ | \ f|_Z = g|_Z \}.$$
The semiring $\mathbb T[x_1, \ldots, x_n]$ is sometimes replaced by another one such as the Laurent polynomial semiring $\mathbb T[x_1^{\pm}, \ldots, x_n^{\pm}]$, the rational function semifield $\mathbb T(x_1, \ldots, x_n)$, the polynomial ring $\mathbb B[x_1, \ldots, x_n]$ over the Boolean semifield, or the polynomial function semiring $\mathbb T[x_1, \ldots, x_n]_{\mathrm{fcn}}$.
This is a simple analogue of the concept of vanishing ideal in algebraic geometry.
The congruences of this form are studied by Bertram and Easton in \cite{bertram2017tropical}, Jo\'{o} and Mincheva in \cite{joo2018prime}, Song in \cite{song2024congruences}, and the author in \cite{ito2022local}.
Also, \textit{prime congruence} are defined in \cite{joo2018prime}, which are of course analogues of the prime ideals of rings.

Almost all rings appeared in classical algebraic geometry are Noetherian, and their ideals are finitely generated.
This fact is used to develop the dimension theory.
Hence it is natural that we consider whether a given ideal (or congruence) is finitely generated.
There are few known results.
In \cite[Example 3.10]{maclagan2018tropical}, an example of tropical ideal which is not finitely generated is given.
Also, it holds by definition that a bend relation defined by a polynomial is finitely generated.
In this paper, we consider the congruences on $\mathbb T[x_1^{\pm}, \ldots, x_n^{\pm}]$ of the form $\mathbf E(Z)$.
In particular, we consider the case that $Z$ is the support of a tropical variety.
The following is our main result.

Almost all rings appeared in classical algebraic geometry are Noetherian, and their ideals are finitely generated.
This fact is used to develop the dimension theory.
Hence it is natural that we consider whether a given ideal (or congruence) is finitely generated.
There are few known results.
In \cite[Example 3.10]{maclagan2018tropical}, an example of tropical ideal which is not finitely generated is given.
Also, it holds by definition that a bend relation defined by a polynomial is finitely generated.
In this paper, we consider the congruences on $\mathbb T[x_1^{\pm}, \ldots, x_n^{\pm}]$ of the form $\mathbf E(Z)$.
In particular, we consider the case that $Z$ is the support of a tropical variety.
The following is our main result.

Thus, for almost all tropical variety $X$, $\mathbf E(X)$ is not finitely generated.
The only if part of this theorem is a consequence of the following more general result.

\begin{thm}
 \label{main}
 Let $X$ be a tropical variety in $\mathbb R^n$.
 Then $\mathbf E(X)$ is finitely generated if and only if $|X|$ is a rational affine linear subspace of $\mathbb R^n$.
\end{thm}

Hence, for a subset $Z \subset W$, in order to show that $\mathbf E(Z)$ is not finitely generated, it is enough to show that $\mathbf E(\iota^{-1}(Z))$ is not finitely generated.
Thus we may assume that $Z$ is not included in any proper rational affine linear subspace of $\mathbb R^n$.
Note that if $Z$ is the support of a tropical variety in $\mathbb R^n$, then so is $\iota^{-1}(Z)$ by Lemma \ref{trop var reduction}.

Let $X$ be a tropical variety in $\mathbb R^n$.
Assume that $W := |X|$ is a $d$-dimensional rational affine linear subspace of $\mathbb R^n$.
By Lemma \ref{translation}, we may assume that $\mathbf 0 \in W$, i.e., $W$ is a linear subspace of $\mathbb R^n$.
Since $W$ is rational, the orthogonal complement $W'$ (with respect to the standard inner product) of $W$ is also rational.
We use the notations $\Lambda(W) = W \cap \mathbb Z^n$ and $\Lambda(W') = W' \cap \mathbb Z^n$.
Let $\{ \bm q_1, \ldots, \bm q_{n-d} \}$ be a basis of $\Lambda(W')$.
We see that the congruence $\mathbf E(W)$ is generated by the finite set $S := \left\{ \left(\overline{\bm x^{\bm q_i}}, 0 \right) \ | \ i=1, \ldots, n-d \right\}$.

\begin{prop}
 \label{trop unbounded}
 Let $X$ be a tropical variety in $\mathbb R^n$.
 Assume that $|X|$ is not included in a proper rational affine linear subspace of $\mathbb R^n$.
 Then $|X|$ is unbounded in all rational directions.
\end{prop}

\begin{proof}[Proof of \textquotedblleft Proposition \ref{trop unbounded in f} $\Longrightarrow$ Proposition \ref{trop unbounded}\textquotedblright]
 Assume that Proposition \ref{trop unbounded in f} holds.
 Let $X$ be a tropical variety in $\mathbb R^n$ whose support is not included in a proper rational affine linear subspace of $\mathbb R^n$.
 Take any $\bm p \in \mathbb Z^n \setminus \{ \mathbf 0 \}$.
 If the function $f: |X| \to \mathbb R, \bm a \mapsto \bm a \cdot \bm p$ is constant with the value $k$, then $|X|$ is included in the set $\{ \bm a \in \mathbb R^n \ | \ \bm a \cdot \bm p = k \}$, which is a proper rational affine linear subspace of $\mathbb R^n$.
 This contradicts to the assumption, hence $f$ is not constant.
 Thus $f$ is upper unbounded.
 Since $\bm p$ is an arbitrary element of $\mathbb Z^n \setminus \{ \mathbf 0 \}$, $|X|$ is unbounded in all rational directions.
\end{proof}

\begin{proof}[Proof of the only if part of Theorem \ref{main}]
 Let $X$ be a tropical variety such that $|X|$ is not a rational affine linear subspace of $\mathbb R^n$.
 We may assume that $|X|$ is not included in any proper rational affine linear subspace of $\mathbb R^n$ by Lemma \ref{fg reduction} and the observation after it.
 Hence, by Proposition \ref{trop unbounded}, $|X|$ is unbounded in all rational directions.
 In order to use Theorem \ref{not fg}, we check that $n \geq 2$.
 If $n =0$, $|X| = \{ \mathbf 0 \}$.
 It contradicts to the assumption that $|X|$ is not an affine linear subspace.
 If $n =1$, $|X|$ is a singleton or the whole space $\mathbb R$ (by Lemma \ref{n in n}).
 It also contradicts to the assumption.
 Thus $n \geq 2$, and hence by Theorem \ref{not fg}, $\mathbf E(X)$ is not finitely generated. \qedhere
\end{proof}

Thus, for almost all tropical variety $X$, $\mathbf E(X)$ is not finitely generated.
The only if part of this theorem is a consequence of the following more general result.

\begin{thm}
  \label{main2}
  Assume that $n \geq 2$.
  Let $Z$ be a proper closed subset of $\mathbb R^n$ that is unbounded in all rational directions.
  Then $\mathbf E(Z)$ is not finitely generated.
\end{thm}

The strategy of the proof of Theorem \ref{main2} is as follows.
First we assume that $\mathbf E(Z)$ is finitely generated and fix a generating set.
Then we construct a pair $(f,g) \in \mathbf E(Z)$ such that $f \neq g$ and the distance of any two lattice points of the Newton polytope of $f$ is sufficiently large.
The existence of such pair causes a contradiction.
See Section 3 for detail.

Note that we cannot apply Theorem \ref{main2} for showing the only if part of Theorem \ref{main1} directly because $|X|$ is not unbounded in all rational directions in general.
In particular, if $|X|$ is included in a proper rational affine linear subspace of $\mathbb R^n$, then $|X|$ is not unbounded in all rational directions.
To overcome this issue, we show that we may replace the ambient space with a lower dimensional one.
See Section 4 for detail.

Finally, we give an example of minimal generating set of $\mathbf E(L)$ for the standard tropical line $L \subset \mathbb R^2$.
See Section \ref{section min gen} for detail.

\begin{thm}
  \label{main3}
  The set
  $$\mathcal S = \left\{ \left( \overline{0 \oplus x^u y^v}, f_{\Delta(\{ (0,0), (u,v) \})} \right) \ \middle| \ \begin{aligned}
  &\text{$u$ and $v$ are integers with $\gcd(u,v) = 1$,}\\
  &\text{ $u > 0$ or $(u,v) = (0,1)$}
  \end{aligned} \right\}$$
  is a minimal generating set of $\mathbf E(L)$.
\end{thm}

This paper is organized as follows.
In Section 2, we recall some basic definitions and facts about semirings, tropical algebra, tropical varieties, and Newton polytopes.
Also we show some lemmas we need later.
In Section 3, we consider a way to prove that a given congruence on $\mathbb T[x_1^{\pm}, \ldots, x_n^{\pm}]$ is not finitely generated, and use it to show Theorem \ref{main2}.
In Section 4, we discuss about the reduction of the ambient spaces.
The result of this section is necessary for showing Theorem \ref{main1}.
In Section 5, we prove Theorem \ref{main1}.
In Section 6, we give an example of minimal generating set of $\mathbf E(L)$ for the standard tropical line $L$.
This section is almost independent of Section 3-5.
In Appendix, we give proofs of some facts about free abelian groups and polytopes, which are used in this paper.

For Theorem~\ref{main1}, the text says: “The only if part of this theorem is a consequence of the following more general result” (Theorem~\ref{main2}), but “we cannot apply Theorem~\ref{main2} for showing the only if part of Theorem~\ref{main1} directly because $|X|$ is not unbounded in all rational directions in general.” “To overcome this issue, we show that we may replace the ambient space with a lower dimensional one.”

For Theorem~\ref{main2}, the stated proof strategy is: “First we assume that $\mathbf E(Z)$ is finitely generated and fix a generating set. Then we construct a pair $(f,g) \in \mathbf E(Z)$ such that $f \neq g$ and the distance of any two lattice points of the Newton polytope of $f$ is sufficiently large. The existence of such pair causes a contradiction.”

Question:
Let $X$ be a tropical variety in $\mathbb R^n$, and let $|X|$ denote its support. For a subset $Z\subset \mathbb R^n$, define the congruence
$$\mathbf E(Z)=\{(f,g)\in \mathbb T[x_1^{\pm},\ldots,x_n^{\pm}]\mid f(\mathbf a)=g(\mathbf a)\text{ for all }\mathbf a\in Z\},$$
where $\mathbb T[x_1^{\pm},\ldots,x_n^{\pm}]$ is the tropical Laurent polynomial semiring. A rational affine linear subspace of $\mathbb R^n$ means an affine translate of a linear subspace defined over $\mathbb Q$ (equivalently, whose direction is spanned by rational vectors). Which statement holds for every such tropical variety $X$?

Choices:
B. $\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is an affine linear subspace of $\mathbb R^n$.
C. If $|X|$ is a rational affine linear subspace of $\mathbb R^n$, then $\mathbf E(|X|)$ is finitely generated.
D. $\mathbf E(|X|)$ is finitely generated whenever $|X|$ is contained in a rational affine linear subspace of $\mathbb R^n$.
E. $\mathbf E(|X|)$ is finitely generated if and only if $|X|$ is a finite union of rational affine linear subspaces of $\mathbb R^n$.

equality with a rational affine linear subspace replaced by mere containment

single rational affine linear subspace replaced by finite union


Question:
Let \(X\) be a tropical variety in \(\mathbb R^n\), and let
\[
\mathbf E(|X|)=\{(f,g)\in \mathbb T[x_1^{\pm},\ldots,x_n^{\pm}]^2\mid f|_{|X|}=g|_{|X|}\}
\]
be the congruence of Laurent tropical polynomials that agree on the support \(|X|\) of \(X\). Which of the following statements is equivalent to the condition that \(\mathbf E(|X|)\) is finitely generated?

The support \(|X|\) is a rational affine linear subspace of \(\mathbb R^n\).

Choices:
B. The support \(|X|\) is an affine linear subspace of \(\mathbb R^n\).
C. The support \(|X|\) is contained in a rational affine linear subspace of \(\mathbb R^n\).
D. The support \(|X|\) is a finite union of rational affine linear subspaces of \(\mathbb R^n\).
E. There exists a rational affine linear subspace \(W\subset \,\mathbb R^n\) such that \(|X|\subseteq W\) and \(\mathbf E(|X|)=\mathbf E(W)\).

rationality of the affine linear subspace

equality with a rational affine linear subspace weakened to mere containment

single affine linear piece replaced by finite union

replacement of the geometric condition on \(|X|\) by congruence equality with a larger rational affine linear subspace


\label{th1}
Let $p>3$.
Assume that $\Gamma$ is a simple closed smooth curve with unit length in $\Omega$, and it is also a non-degenerate critical point of the functional
\begin{equation}\label{e91}
\mathcal{K}(\Gamma)=\int_\Gamma \mathbf{\Psi}^{\frac{p+3}{2p}}dvol_{\mathfrak{g}},
\end{equation}
where $\mathfrak{g}(X,Y)=\langle A^*X,Y\rangle$ is a Riemannian metric on $\mathbb R^2$ and $A^*$ is the adjoint matrix for $A$. We also assume that the following inequality holds on $\Gamma$:
\begin{eqnarray*}
\Upsilon_0&=&-\alpha^{1-p}\left[ 2\alpha^{-1}\alpha'a_{22}+\beta^{-1}\beta'b_{11}+b_{22}-a_{33}+\frac{p+3}2\alpha^{p-2}\beta^{-2}\mathbf{q}_{tt}+\frac{p+2}{2(p+3)}\alpha^{1-p} \beta^2 (a_{32})^2\right. \\
&&\left.-\frac{p+1}{p+3}\alpha^{1-p}\beta^2 a_{32}b_{21}-\frac{2}{p+3}\alpha^{1-p}\beta^2(b_{21})^2 +a_{11}\left(\beta^{-1}\beta''+2\alpha^{-1}\beta^{-1}\alpha'\beta'-\beta^{-2}(\beta')^2\right)\right]>0,
\end{eqnarray*}
where $a_{ij}$'s and $b_{ij}$'s is defined in Lemma \ref{lm8}, $\alpha$ and $\beta$ are defined in \eqref{80}, and $\mathbf{q}$ is defined in \eqref{e88}.
Then for each  integer $N>0$, there exists a sequence of $\varepsilon$, \textit{i.e.}, $\{\varepsilon_l\}$ converging to $0$ such that \eqref{1} has a positive solutions $u_{\varepsilon_l}$ with exactly $N$ concentration layers at mutual distance $O(\varepsilon_l|\log\varepsilon_l|)$, the center of mass of $N$ concentration layers collapses to $\Gamma$ at speed $\varepsilon_l^{1+\mu}$ for small positive constant $\mu\in(0,1/2)$. More precisely $u_{\varepsilon_l}$ has the form
\begin{eqnarray*}
u_{\varepsilon_l}(y_1,y_2)\approx -\mathbf{\Psi}^{\frac1p}(y_1,y_2)+ \mathbf{\Psi}^{\frac1p}(\gamma(\theta))\sum_{k=1}^N w\left(\left[\frac{\mathbf{\Psi}^{\frac{p-1}p}(1-\langle\gamma',n\rangle)}{\langle A^*n,n\rangle}\right]^{\frac12}\frac{t-\varepsilon_l f_k}{\varepsilon_l}\right),
\end{eqnarray*}
where $\gamma$ is a natural parametrization of $\Gamma$, $n$ is the unit vector defined in \eqref{e102} and $w$ is  the unique solution of the following problem
\begin{equation}\label{29}
-w''=|1-w|^p -1, \qquad w>0 \quad \mathrm{in} \quad \mathbb R, \qquad w'(0)=w(\pm \infty).
\end{equation}
In the expression above, the functions $f_j$'s satisfy
\[\|f_j\|_{L^\infty(0,1)}\le C|\log\varepsilon_l|^2,\qquad \sum_{j=1}^N f_j =O(\frac1{|\log\varepsilon_l|^{\frac32}}),\]
\[\min_{1\le j\le N}(f_{j+1}-f_j)\approx\frac2{\sqrt{p}}|\log\varepsilon_l|\left[\frac{\langle A^*n,n\rangle}{\Psi^{\frac{p-1}p}(1-\langle\gamma',n\rangle)}\right]^{\frac12}\]
and solves the Jacobi-Toda system for $j=1,2,\cdots,N$
\begin{eqnarray*}
\nonumber&&\varepsilon^2\alpha^{1-p}\beta\left\{-a_{11}f_j''+\left[a_{22}-b_{11}-a_{11}\left(\beta^{-1}\beta'+2\alpha^{-1}\alpha'\right)\right]f_j'+\left[a_{22}(\beta^{-1}\beta'+2\alpha^{-1}\alpha') +b_{22}-a_{33}\right]f_j\right. \\
\nonumber&&+\left[\frac{p+3}2\alpha^{p-2}\beta^{-2} \mathbf{q}_{tt}+\frac{p+2}{2(p+3)}\alpha^{1-p}\beta^2(a_{32})^2-\frac{p+1}{p+3}\alpha^{1-p}\beta^2 a_{32}b_{21}-\frac2{p+3}\alpha^{1-p}\beta^2 (b_{21})^2\right]f_j \\
&&C_0 p\alpha_p\left[e^{-\sqrt{p}\beta(f_j-f_{j-1})}-e^{-\sqrt{p}\beta (f_{j+1}-f_j)}\right]\approx0.
\end{eqnarray*}

\begin{theorem}\label{th1}
Let $p>3$.
Assume that $\Gamma$ is a simple closed smooth curve with unit length in $\Omega$, and it is also a non-degenerate critical point of the functional
\begin{equation}\label{e91}
\mathcal{K}(\Gamma)=\int_\Gamma \mathbf{\Psi}^{\frac{p+3}{2p}}dvol_{\mathfrak{g}},
\end{equation}
where $\mathfrak{g}(X,Y)=\langle A^*X,Y\rangle$ is a Riemannian metric on $\R^2$ and $A^*$ is the adjoint matrix for $A$. We also assume that the following inequality holds on $\Gamma$:
\begin{eqnarray*}
\Upsilon_0&=&-\alpha^{1-p}\left[ 2\alpha^{-1}\alpha'a_{22}+\beta^{-1}\beta'b_{11}+b_{22}-a_{33}+\frac{p+3}2\alpha^{p-2}\beta^{-2}\mathbf{q}_{tt}+\frac{p+2}{2(p+3)}\alpha^{1-p} \beta^2 (a_{32})^2\right. \\
&&\left.-\frac{p+1}{p+3}\alpha^{1-p}\beta^2 a_{32}b_{21}-\frac{2}{p+3}\alpha^{1-p}\beta^2(b_{21})^2 +a_{11}\left(\beta^{-1}\beta''+2\alpha^{-1}\beta^{-1}\alpha'\beta'-\beta^{-2}(\beta')^2\right)\right]>0,
\end{eqnarray*}
where $a_{ij}$'s and $b_{ij}$'s is defined in Lemma \ref{lm8}, $\alpha$ and $\beta$ are defined in \eqref{80}, and $\mathbf{q}$ is defined in \eqref{e88}.
Then for each  integer $N>0$, there exists a sequence of $\varepsilon$, \textit{i.e.}, $\{\varepsilon_l\}$ converging to $0$ such that \eqref{1} has a positive solutions $u_{\varepsilon_l}$ with exactly $N$ concentration layers at mutual distance $O(\varepsilon_l|\log\varepsilon_l|)$, the center of mass of $N$ concentration layers collapses to $\Gamma$ at speed $\varepsilon_l^{1+\mu}$ for small positive constant $\mu\in(0,1/2)$. More precisely $u_{\varepsilon_l}$ has the form
\begin{eqnarray*}
u_{\varepsilon_l}(y_1,y_2)\approx -\mathbf{\Psi}^{\frac1p}(y_1,y_2)+ \mathbf{\Psi}^{\frac1p}(\gamma(\theta))\sum_{k=1}^N w\left(\left[\frac{\mathbf{\Psi}^{\frac{p-1}p}(1-\langle\gamma',n\rangle)}{\langle A^*n,n\rangle}\right]^{\frac12}\frac{t-\varepsilon_l f_k}{\varepsilon_l}\right),
\end{eqnarray*}
where $\gamma$ is a natural parametrization of $\Gamma$, $n$ is the unit vector defined in \eqref{e102} and $w$ is  the unique solution of the following problem
\begin{equation}\label{29}
-w''=|1-w|^p -1, \qquad w>0 \quad \mathrm{in} \quad \R, \qquad w'(0)=w(\pm \infty).
\end{equation}
In the expression above, the functions $f_j$'s satisfy
\[\|f_j\|_{L^\infty(0,1)}\le C|\log\varepsilon_l|^2,\qquad \sum_{j=1}^N f_j =O(\frac1{|\log\varepsilon_l|^{\frac32}}),\]
\[\min_{1\le j\le N}(f_{j+1}-f_j)\approx\frac2{\sqrt{p}}|\log\varepsilon_l|\left[\frac{\langle A^*n,n\rangle}{\Psi^{\frac{p-1}p}(1-\langle\gamma',n\rangle)}\right]^{\frac12}\]
and solves the Jacobi-Toda system for $j=1,2,\cdots,N$
\begin{eqnarray*}
\nonumber&&\varepsilon^2\alpha^{1-p}\beta\left\{-a_{11}f_j''+\left[a_{22}-b_{11}-a_{11}\left(\beta^{-1}\beta'+2\alpha^{-1}\alpha'\right)\right]f_j'+\left[a_{22}(\beta^{-1}\beta'+2\alpha^{-1}\alpha') +b_{22}-a_{33}\right]f_j\right. \\
\nonumber&&+\left[\frac{p+3}2\alpha^{p-2}\beta^{-2} \mathbf{q}_{tt}+\frac{p+2}{2(p+3)}\alpha^{1-p}\beta^2(a_{32})^2-\frac{p+1}{p+3}\alpha^{1-p}\beta^2 a_{32}b_{21}-\frac2{p+3}\alpha^{1-p}\beta^2 (b_{21})^2\right]f_j \\
&&C_0 p\alpha_p\left[e^{-\sqrt{p}\beta(f_j-f_{j-1})}-e^{-\sqrt{p}\beta (f_{j+1}-f_j)}\right]\approx0.
\end{eqnarray*}
\end{theorem}

\begin{equation}\label{e55}
-\varepsilon^2 \Delta u+V(y)u=u^p, \qquad \mathrm{in} \qquad \R^\mathcal{N}.
\end{equation}

The Ambrosetti-Prodi problem is of the following form:
\begin{equation}\label{e51}
\left\{\begin{array}{ll}
-\Delta u=\zeta(u)-t\mathbf{\Psi}_0 (x), &\mbox{in $\Omega$,} \\
u=0,&\mbox{on $\partial \Omega$},
\end{array}\right.
\end{equation}
where $t>0$, $\Omega\subset \mathbb R^\mathcal{N}$ is a smooth bounded domain,  $\mathbf{\Psi}_0$ is the an eigenfunction of $-\Delta$ subject to Dirichlet boundary condition corresponding to the first eigenvalue, the function $\zeta(t)$ satisfies
\[-\infty\le\mu=\lim_{t\to +\infty} \frac{\zeta(t)}{t}>\lim_{t\to +\infty} \frac{\zeta(t)}{t}=\nu\le +\infty,\]
and the interval $(\mu, \nu)$ contains some eigenvalues of $-\Delta$ subject to Dirichlet boundary condition.

Problem \eqref{e51} was first studied by Ambrosetti and Prodi \cite{Ambrosetti_Prodi1972}. It was widely discussed in 1980's( see \cite{deFigueiredo1984,deFigueiredo_Solimini1984,Lazer_McKenna1981,Lazer_McKenna1984,Lazer_McKenna1988,Ruf_Srikanth986a,Ruf_Srikanth1986,Solimini1985,Ruf_Solimini1986} for example). The main results in these literature are that if $g(t)$ grows subcritical at infinity, \eqref{e51} has at least two solutions: one is local minimizer of the Euler-Lagrange functional, the other is the mountain-pass solution. Breuer, McKenna and Plum \cite{Breuer_McKenna_Plum2003} considered the case that $\zeta(t)=t^2$ and $\Omega$ is a unit square in $\mathbb R^2$. Using a computer assisted proof, they showed  that \eqref{e51} has at least 4 solutions.  By comparing Morse index of mountain pass solution in different spaces, de Figueiredo, Srikanth and Santra \cite{deFigueiredo_Srikanth_Santra2005}  found a non-radial solution of \eqref{e51} under the condition $\Omega$ is a unit ball and $\zeta(t)=t^2$.

However, in the case of $\zeta(t)=|t|^p$, where $1<p<\frac{\mathcal{N}+2}{\mathcal{N}-2}$ for $\mathcal{N}\ge3$ and $p>1$ for $\mathcal{N}=2$, \eqref{e51} becomes the following problem
\begin{equation}\label{e52}
\left\{\begin{array}{ll}
-\Delta u=|u|^p-t\mathbf{\Psi}_0 (x), &\mbox{in $\Omega$,} \\
u=0,&\mbox{on $\partial \Omega$}.
\end{array}\right.
\end{equation}
Dancer and Yan \cite{Dancer_Yan2005} constructed arbitrary many peak solutions of \eqref{e52} for $t>0$ large enough.  It shows that Lazer-McKenna conjecture \cite{Lazer_McKenna1981} holds in this case. Dancer and Yan \cite{Dancer_Yan2005} also proved that the mountain pass solution of \eqref{e51} has a sharp peak near the boundary for $t$ large enough. As a conclusion,   \eqref{e51} has solutions concentrating at some points in $\Omega$ or the  ones on $\partial \Omega$ as $t\to \infty$. These results have been extended to different kind of nonlinearities( see \cite{Dancer_Yan2005a, Dancer_Santra2007, delPino_Munoz2006, Li_Yan_Yang2007, Li_Yan_Yang2006, Molle_Passaseo2010, Wei_Yan2007} for instance and references therein).

However, these results concern only point concentrating solutions. Based on some numerical evidence, Hollman and McKenna \cite{Hollman_McKenna2011} asked \eqref{e52} that  whether there exist other types of concentrations  for the solutions to \eqref{e52} as $t\to+\infty$. For this problem, Manna and Santra \cite{Manna_Santra2016} considered \eqref{e52} under the condition $\Omega\subset \mathbb R^2$ and $p>2$. They proved that \eqref{e52} has a family of solutions clustering along a closed curve $\Gamma\subset \Omega$, where $\Gamma$ is a non-degenerate critical point of the functional
\begin{equation}\label{e54}
\mathcal{K}_0(\Gamma)=\int_{\Gamma} \mathbf{\Psi}_0^{\frac{p+3}{2p}}(x)dvol.
\end{equation}
Later, this result was extended to high dimensional case by Khemiri, Mahmoudi and Messaoudi  \cite{Khemiri_Mahmoudi_Messaoudi2017}. Under the condition that $\mathcal{N}\ge 3$ and $\Omega$ contain a $k$-dimensional compact submanifold $\Gamma$, which is a non-degenerate critical point of the functional
\[\mathcal{K}_1(\Gamma)=\int_{\Gamma} \mathbf{\Psi}_0^{\left(1-\frac1p\right)\left(\frac{p+1}{p-1}-\frac{n-k}2\right)}(x)dvol,\]
They proved there exists a sequence $t=t_j\to \infty$, and  solutions  $u_{t_j}$ to \eqref{e52}, which have  concentration layers concentrating near $\Gamma$.

Baraket \textit{et. al.} \cite{Baraket_Khemiri_Mahmoudi_Messaoudi2018} considered the following Neumann problem:
\begin{equation}\label{e53}
\left\{\begin{array}{ll}
-\Delta u=|u|^p-t\psi (x), &\mbox{in $\Omega$,} \\
\frac{\partial u}{\partial \mathbf{n}}=0,&\mbox{on $\partial \Omega$},
\end{array}\right.
\end{equation}
where  $\Omega\subset\mathbb R^\mathcal{N}$ is a smooth bound domain and $\mathbf{n}$ is the unit outward normal vector of $\partial\Omega$.
Under the condition $\mathcal{N}=2$, they proved that \eqref{e53} has a solution $u_t$ concentrating along a curve $\Gamma\subset \bar{\Omega}$ with $t>0$ large enough. The curve $\Gamma$ intersects $\partial\Omega$ with right angle and divides $\Omega$ into two part. Moreover $\Gamma$ is a non-degenerate critical point of the functional $\mathcal{K}_2(\Gamma)=\int_{\Gamma} \psi^{\frac{p+3}{2p}}(x)dvol$.  However, under the condition that  $\psi\equiv1$ and  $\mathcal{N}\ge 2$, Bendahou, Khemiri and Mahmoudi \cite{Bendahou_Khemiri_Mahmoudi2020} constructed a family of new solutions to \eqref{e53}, which has large number of spikes concentrating along an interior  straight line in $\Omega$ for $t\to+\infty$. Using the similar method, Ao, Fu and Liu \cite{Ao_Fu_Liu2022} constructed a similar type of  solutions  which concentrate along a segment of boundary $\partial \Omega$ in the two dimensional case.

From these results, we recognize that the high dimensional concentration behaviours of Ambrosetti-Prodi type problem is similar to that of Nonlinear Schr\"{o}dinger equation:
\begin{equation}\label{e55}
-\varepsilon^2 \Delta u+V(y)u=u^p, \qquad \mathrm{in} \qquad \mathbb R^\mathcal{N}.
\end{equation}
In 2003, Ambrosetti, Malchiodi and Ni \cite{Ambrosetti_Malchiodi_Ni2003} raised the following conjecture:
\begin{conjecture}
Let $\Gamma$ be a $k$-dimensional submanifold in $\mathbb R^\mathcal{N}$ and a nondegenerate critical point of the following functional
\[\mathcal{K}(\Gamma)=\int_\Gamma V^{\frac{p+1}{p-1}-\frac12(\mathcal{N}-k)}dvol,\]
where $1<p<\frac{n+2-k}{n-2+k}$ for $\mathcal{N}\ge 3$ and $p>1$ for $\mathcal{N}=2$.
Then there exists a family of solutions to \eqref{e55} concentrating along $\Gamma$ at least for a subsequence $\varepsilon=\varepsilon_j\to 0$.
\end{conjecture}
del Pino, Kowalczyk and Wei \cite{delPino_Kowalczyk_Wei2007} first showed that this conjecture holds in the case of $\mathcal{N}=2$ and $k=1$.  Wang, Wei and Yang \cite{Wang_Wei_Yang2011} proved that this conjecture in the case of $\mathcal{N}\ge3$ and $k=\mathcal{N}-1$.  And Mahmoudi, Sanchez and Yao \cite{Mahmoudi_Sanchez_Yao2015} proved this conjecture is valid for all cases.

However, the following more general Neumann version of \eqref{e55} also has some solutions concentrating on high dimensional set:
\begin{equation}\label{e90}
\left\{\begin{array}{ll}
\varepsilon^2 \Div(\nabla_{\mathfrak{a}(y)}u)-V(y)u+u^p=0, \qquad u>0, &\mbox{in $\Omega$,}\\
\nabla_{\mathfrak{a}(y)} u\cdot \mathbf{n}=0,  &\mbox{on $\partial\Omega$,}
\end{array}\right.
\end{equation}
where $\Omega\subset \mathbb R^2$ is a smooth bounded domain, $\mathbf{n}$ is the unit outward normal of $\partial \Omega$,
\[\mathfrak{a}(y)=(\mathfrak{a}_1(y),\mathfrak{a}_2(y), \cdots, \mathfrak{a}_N(y))\]
and
\[\nabla_{\mathfrak{a}(y)}u=(\mathfrak{a}_1(y)u_{y_1}, \mathfrak{a}_2(y)u_{y_2}, \cdots, \mathfrak{a}_N(y)u_{y_N}).\]
In the case of $\mathfrak{a}\equiv 1$ and $V(y)\equiv 1$, Wei and Yang \cite{Wei_Yang2008} constructed a sequence of solutions concentrating near a segment $\Gamma_2$ in $\Omega$. The segment intersects $\partial \Omega$ with right angle and separates $\Omega$ into two parts. Unlike the solutions constructed in \cite{delPino_Kowalczyk_Wei2007,Wang_Wei_Yang2011,Mahmoudi_Sanchez_Yao2015}, the solutions constructed in \cite{Wei_Yang2008}  have multiple  concentration layers.
Wei, Xu and Yang \cite{Wei_Xu_Yang2018} considered the case that $\mathfrak{a}(y)\equiv 1$. Let $\Gamma\subset\bar{\Omega}$ be a curve intersecting $\partial \Omega$ with right angle and separating $\Omega$ into two parts. Provided $\Gamma$ is a nondegenerate critical point of $\int_\Gamma V^{\frac{p+1}{p-1}-\frac12}$, they constructed a  sequence of solutions to \eqref{e90} clustering along $\Gamma$. Recently, Wei and Yang \cite{Wei_Yang2021} extended the result in \cite{Wei_Yang2008} into the general case. They  constructed a sequence of solutions to \eqref{e90} with multiple
concentration layers which concentrate near a closed curve $\Gamma_0\subset\Omega$ or a curve $\Gamma_1$ intersecting $\partial \Omega$ with right angle and separating $\Omega$ into two parts. Moreover,  $\Gamma_0$ and $\Gamma_1$ are all the nondegenerate geodesics embedded into the Riemannian manifold $\mathbb R^2$ with the metric $V^{\frac{2(p+1)}{p-1}-1}[\mathfrak{a}_2(y)dy_1^2+\mathfrak{a}_1(y)dy_2^2]$.

Inspired by \cite{Wei_Yang2021,Wei_Yang2008}, we consider the following problem
\begin{equation}\label{2}
\left\{\begin{array}{ll}
-\Div (A(x)\nabla u)=|u|^p-t\mathbf{\Psi}(x), &\mbox{in $\Omega$,} \\
u=0, & \mbox{on $\partial \Omega$},
\end{array}
\right.
\end{equation}
where $\Omega \subset \mathbb R^2$ is a smooth bounded domain, $t>0$ is a constant, and  $\mathbf{\Psi}(x)$ is an eigenfunction corresponding to the first Dirichlet eigenvalue of the operator $\mathfrak{L}(u)=-\Div (A(x)\nabla u)$ on $\Omega$.
Moreover, $A(x)=\{A_{ij}(x)\}_{2\times 2}$ is a symmetric positive defined matrix function, satisfying
\begin{equation}\label{e98}
\lambda |\alpha|^2 \leq \langle A(x)\alpha, \alpha \rangle \leq \Lambda |\alpha|^2,\qquad \mathrm{for} \qquad x,\alpha \in \mathbb R^n.
\end{equation}
We notice $\Div(\nabla_{\mathfrak{a}(y)}u)$ is a special form of $\Div (A(x)\nabla u)$. Based on previous work, we suspect whether \eqref{2} has a similar solution to that constructed in \cite{Wei_Yang2021,Wei_Yang2008}.

Our motivation for writing this paper is twofold. First, we plan to construct solutions to \eqref{2} with multiple concentration layers. Second, we explore the influence of the matrix $A(y)$ to the high dimensional concentration behaviours of the solutions to \eqref{2}.

Let $\varepsilon^2=t^{-(p-1)/p}$. It is easy to get that $u$ is a solution of \eqref{2} if and only if $t^{-\frac1p}u$ is the solution of the following problem
\begin{equation}\label{1}
\left\{\begin{array}{ll}
-\varepsilon^2\Div (A(x)\nabla u)=|u|^p-\mathbf{\Psi}(x), &\mbox{in $\Omega$,} \\
u=0, & \mbox{on $\partial \Omega$},
\end{array}
\right.
\end{equation}
Then we get the following theorem.

Baraket \textit{et. al.} \cite{Baraket_Khemiri_Mahmoudi_Messaoudi2018} considered the following Neumann problem:
\begin{equation}\label{e53}
\left\{\begin{array}{ll}
-\Delta u=|u|^p-t\psi (x), &\mbox{in $\Omega$,} \\
\frac{\partial u}{\partial \mathbf{n}}=0,&\mbox{on $\partial \Omega$},
\end{array}\right.
\end{equation}
where $\Omega\subset\mathbb R^\mathcal{N}$ is a smooth bound domain and $\mathbf{n}$ is the unit outward normal vector of $\partial\Omega$.
Under the condition $\mathcal{N}=2$, they proved that \eqref{e53} has a solution $u_t$ concentrating along a curve $\Gamma\subset \bar{\Omega}$ with $t>0$ large enough. The curve $\Gamma$ intersects $\partial\Omega$ with right angle and divides $\Omega$ into two part. Moreover $\Gamma$ is a non-degenerate critical point of the functional $\mathcal{K}_2(\Gamma)=\int_{\Gamma} \psi^{\frac{p+3}{2p}}(x)dvol$. However, under the condition that $\psi\equiv1$ and $\mathcal{N}\ge 2$, Bendahou, Khemiri and Mahmoudi \cite{Bendahou_Khemiri_Mahmoudi2020} constructed a family of new solutions to \eqref{e53}, which has large number of spikes concentrating along an interior straight line in $\Omega$ for $t\to+\infty$. Using the similar method, Ao, Fu and Liu \cite{Ao_Fu_Liu2022} constructed a similar type of solutions which concentrate along a segment of boundary $\partial \Omega$ in the two dimensional case.

However, the following more general Neumann version of \eqref{e55} also has some solutions concentrating on high dimensional set:
\begin{equation}\label{e90}
\left\{\begin{array}{ll}
\varepsilon^2 \Div(\nabla_{\mathfrak{a}(y)}u)-V(y)u+u^p=0, \qquad u>0, &\mbox{in $\Omega$,}\\
\nabla_{\mathfrak{a}(y)} u\cdot \mathbf{n}=0, &\mbox{on $\partial\Omega$,}
\end{array}\right.
\end{equation}
where $\Omega\subset \mathbb R^2$ is a smooth bounded domain, $\mathbf{n}$ is the unit outward normal of $\partial \Omega$,
\[\mathfrak{a}(y)=(\mathfrak{a}_1(y),\mathfrak{a}_2(y), \cdots, \mathfrak{a}_N(y))\]
and
\[\nabla_{\mathfrak{a}(y)}u=(\mathfrak{a}_1(y)u_{y_1}, \mathfrak{a}_2(y)u_{y_2}, \cdots, \mathfrak{a}_N(y)u_{y_N}).\]
In the case of $\mathfrak{a}\equiv 1$ and $V(y)\equiv 1$, Wei and Yang \cite{Wei_Yang2008} constructed a sequence of solutions concentrating near a segment $\Gamma_2$ in $\Omega$. The segment intersects $\partial \Omega$ with right angle and separates $\Omega$ into two parts. Unlike the solutions constructed in \cite{delPino_Kowalczyk_Wei2007,Wang_Wei_Yang2011,Mahmoudi_Sanchez_Yao2015}, the solutions constructed in \cite{Wei_Yang2008} have multiple concentration layers.
Wei, Xu and Yang \cite{Wei_Xu_Yang2018} considered the case that $\mathfrak{a}(y)\equiv 1$. Let $\Gamma\subset\bar{\Omega}$ be a curve intersecting $\partial \Omega$ with right angle and separating $\Omega$ into two parts. Provided $\Gamma$ is a nondegenerate critical point of $\int_\Gamma V^{\frac{p+1}{p-1}-\frac12}$, they constructed a sequence of solutions to \eqref{e90} clustering along $\Gamma$. Recently, Wei and Yang \cite{Wei_Yang2021} extended the result in \cite{Wei_Yang2008} into the general case. They constructed a sequence of solutions to \eqref{e90} with multiple
concentration layers which concentrate near a closed curve $\Gamma_0\subset\Omega$ or a curve $\Gamma_1$ intersecting $\partial \Omega$ with right angle and separating $\Omega$ into two parts. Moreover, $\Gamma_0$ and $\Gamma_1$ are all the nondegenerate geodesics embedded into the Riemannian manifold $\mathbb R^2$ with the metric $V^{\frac{2(p+1)}{p-1}-1}[\mathfrak{a}_2(y)dy_1^2+\mathfrak{a}_1(y)dy_2^2]$.

Inspired by \cite{Wei_Yang2021,Wei_Yang2008}, we consider the following problem
\begin{equation}\label{2}
\left\{\begin{array}{ll}
-\Div (A(x)\nabla u)=|u|^p-t\mathbf{\Psi}(x), &\mbox{in $\Omega$,} \\
u=0, & \mbox{on $\partial \Omega$},
\end{array}
\right.
\end{equation}
where $\Omega \subset \mathbb R^2$ is a smooth bounded domain, $t>0$ is a constant, and $\mathbf{\Psi}(x)$ is an eigenfunction corresponding to the first Dirichlet eigenvalue of the operator $\mathfrak{L}(u)=-\Div (A(x)\nabla u)$ on $\Omega$.
Moreover, $A(x)=\{A_{ij}(x)\}_{2\times 2}$ is a symmetric positive defined matrix function, satisfying
\begin{equation}\label{e98}
\lambda |\alpha|^2 \leq \langle A(x)\alpha, \alpha \rangle \leq \Lambda |\alpha|^2,\qquad \mathrm{for} \qquad x,\alpha \in \mathbb R^n.
\end{equation}
We notice $\Div(\nabla_{\mathfrak{a}(y)}u)$ is a special form of $\Div (A(x)\nabla u)$. Based on previous work, we suspect whether \eqref{2} has a similar solution to that constructed in \cite{Wei_Yang2021,Wei_Yang2008}.

Our motivation for writing this paper is twofold. First, we plan to construct solutions to \eqref{2} with multiple concentration layers. Second, we explore the influence of the matrix $A(y)$ to the high dimensional concentration behaviours of the solutions to \eqref{2}.

Let $\varepsilon^2=t^{-(p-1)/p}$. It is easy to get that $u$ is a solution of \eqref{2} if and only if $t^{-\frac1p}u$ is the solution of the following problem
\begin{equation}\label{1}
\left\{\begin{array}{ll}
-\varepsilon^2\Div (A(x)\nabla u)=|u|^p-\mathbf{\Psi}(x), &\mbox{in $\Omega$,} \\
u=0, & \mbox{on $\partial \Omega$},
\end{array}
\right.
\end{equation}
Then we get the following theorem.

Our motivation for writing this paper is twofold. First, we plan to construct solutions to \eqref{2} with multiple concentration layers. Second, we explore the influence of the matrix $A(y)$ to the high dimensional concentration behaviours of the solutions to \eqref{2}.

Multiplying the both sides of \eqref{e71} by $w_{ji,x}$ and $Z_{ji}$, respectively and integrating, we get
\[a_{11}\tilde{\alpha}^{1-p}\Lambda_1''=\int_\R hw_{ji,x}dx+c_{ji}\int_\R w_x^2dx\]
and
\[a_{11}\tilde{\alpha}^{1-p}\Lambda_2''+\lambda_0 \Lambda_2=\int_\R hZ_{ji}dx+d_{ji}.\]
Then we get
\[\|c_{ji}\|_{L^2(0,1/\varepsilon)}\le C\|\Lambda_1\|_{H^2(0,1/\varepsilon)}+\|h\|_{L^2(\mathfrak{S})}\]
and
\[\|d_{ji}\|_{L^2(0,1/\varepsilon)}\le C\|\Lambda_2\|_{H^2(0,1/\varepsilon)}+\|h\|_{L^2(\mathfrak{S})}.\]
From this fact, we have
\begin{eqnarray*}
&&\|c_{j2}(w_{j1,x}-w_{j2,x})+d_{j2}(Z_{j1}-Z_{j2})\|_{L^2(\mathfrak{S})} \\
&\le& \left[\|c_{ji}\|_{L^2(0,1/\varepsilon)}+\|d_{ji}\|_{L^2(0,1/\varepsilon)}\right] \|\mathbf{f}_1-\mathbf{f}_2\|_{H^2(0,1)} \\
&\le& C\|\mathbf{f}_1-\mathbf{f}_2\|_{H^2(0,1)}\left[\|h\|_{L^2(\mathfrak{S})}+\|\Lambda_1\|_{H^2(0, 1/\varepsilon)}+\|\Lambda_2\|_{H^2(0, 1/\varepsilon)}\right].
\end{eqnarray*}
Denote
\[\tilde{\Lambda}_1(z)=-\int_\R \phi_2(z,x)(w_{j1,x}-w_{j2,x})dx, \quad \mathrm{and} \quad \tilde{\Lambda}_2(z)=-\int_\R \phi_2(z,x)(Z_{j1}-Z_{j2})dx\]
From direct computation, we get
\[\|\tilde{\Lambda}_1\|_{H^2(0,1/\varepsilon)}+\|\tilde{\Lambda}_2\|_{H^2(0,1/\varepsilon)}\le C\|\phi_2\|_{H^2(\mathfrak{S})}\|\mathbf{f}_1-\mathbf{f}_2\|_{H^2(0,1)}\]
and
\[\|p[|w_{j1}-1|^{p-2}(w_{j1}-1)-|w_{j2}-1|^{p-2}(w_{j2}-1)]\phi_2\|_{L^2(\mathfrak{S})}\le C\|f_1-f_2\|_{H^2(0,1)}\|\phi_2\|_{H^2(\mathfrak{S})}.\]
From the priori estimate in \eqref{e21}, we arrive at
\[\|\phi^*\|_{H^2(S)}\le C\|\mathbf{f}_1-\mathbf{f}_2\|_{H^2(0,1)}\left[\|h\|_{L^2(\mathfrak{S})}+\|\Lambda_1\|_{H^2(0, 1/\varepsilon)}+\|\Lambda_2\|_{H^2(0, 1/\varepsilon)}\right].\]
Hence \eqref{e72} holds.
\end{proof}
\begin{proposition}\label{pre1}
Problem \eqref{48} has a unique solution $\phi=\tilde{T}(h)$, where $\tilde{T}$ is a linear operator and $\phi$ satisfies the following estimate
\begin{equation}\label{e74}
\|\phi\|_{H^2(\mathfrak{S})}\leq C\|h\|_{L^2(\mathfrak{S})}.
\end{equation}
Moreover, $\tilde{T}$ is Lipschitz continuous on $\mathbf{f}$ and $\mathbf{e}$:
\begin{equation}\label{e73}
\|\tilde{T}_{\mathbf{f}_1,\mathbf{e}_1}-\tilde{T}_{\mathbf{f}_2,\mathbf{e}_2}\|\le C\left[ \|\mathbf{f}_1-\mathbf{f}_2\|_{H^2(\mathfrak{S})}+\|\mathbf{e}_1-\mathbf{e}_2\|_*\right].
\end{equation}
\end{proposition}
\begin{proof}
According to the argument above, we only need to solve the system \eqref{50}-\eqref{52} and the problem \eqref{54}.
With Lemma \ref{lme4}, we first consider problem \eqref{50}-\eqref{52} for fixed function $\tilde{\psi}$. It is written into the following problem
\begin{eqnarray}\label{e22}
\nonumber \tilde{\phi}_j&=&T_j\left[\tilde{\chi}_j h+p\left[|w_j-1|^{p-2}(w_j-1)-|\mathcal{V}-1|^{p-2}(\mathcal{V}-1)\right]\tilde{\chi}_j \tilde{\phi}_j \right. \\ &&\left.-p\left[|\mathcal{V}-1|^{p-2}(\mathcal{V}-1)+1\right]\tilde{\chi}_j\tilde{\psi}, \Lambda_{j1}, \Lambda_{j2}\right], \qquad j=1,2,\cdots, N.
\end{eqnarray}
From tedious computation and Sobolev imbbeding theorem in \cite[Theorem 8.8]{Brezis2011}, we get
\[\|\Lambda_{j1}\|_{H^2(0,1/\varepsilon)}+\|\Lambda_{j2}\|_{H^2(0,1/\varepsilon)}\leq C\varepsilon^{1/2}\sum_{k=1}^N\|\tilde{\phi}_k\|_{H^2(\mathfrak{S})}+C\|\tilde{\psi}\|_{H^2(\mathfrak{S})},\]
\[\|p[|\mathcal{V}-1|^{p-2}(\mathcal{V}-1)+1]\tilde{\chi}_j\tilde{\psi}\|_{L^2(\mathfrak{S})}\le C\|\tilde{\psi}\|_{H^2(\mathfrak{S})}\]
and
\[\|p\left[|w_j-1|^{p-2}(w_j-1)-|\mathcal{V}-1|^{p-2}(\mathcal{V}-1)\right]\tilde{\chi}_j \tilde{\phi}_j\|_{L^2(\mathfrak{S})}\leq C\varepsilon^{1/2}\|\tilde{\phi}_j\|_{H^2(\mathfrak{S})}.\]
Using the fixed point theorem, we get \eqref{e22} has a unique solution $\tilde{\phi}_j=\tilde{\phi}_j(\tilde{\psi})$, $j=1,2,\cdots,N$. Moreover $\tilde{\phi}_j$ satisfies the following estimate
\begin{equation}\label{e24}
\|\tilde{\phi}_j\|_{H^2(\mathfrak{S})}\leq C\left[\|h\|_{L^2(\mathfrak{S})}+\|\tilde{\psi}\|_{H^2(\mathfrak{S})}\right].
\end{equation}
It is easy to get $\phi_j$ is Lipschitz dependent on $\psi$:
\begin{equation}\label{e70}
\|\tilde{\phi}_j(\tilde{\psi}_1)-\tilde{\phi}_j(\tilde{\psi}_2)\|_{H^2(\mathfrak{S})}\le C\|\tilde{\psi}_1-\tilde{\psi}_2\|_{H^2(\mathfrak{S})}.
\end{equation}

For the other terms in \eqref{77}, we define
\[\Lambda_1=\int_\R p\eta_\delta^\varepsilon \left[|\mathcal{V}-\tilde{\alpha}^{-1}\bar{\mathbf{q}}|^{p-2}(\mathcal{V}-\tilde{\alpha}^{-1}\bar{\mathbf{q}})+|\tilde{\alpha}^{-1}\bar{\mathbf{q}}|^{p-2}(\tilde{\alpha}^{-1}\bar{\mathbf{q}}) \right]\hat{\psi}( \hat{\varphi})w_{j,x}dx\]
and
\[\Lambda_2=\int_\R \eta_\delta^\varepsilon \hat{N}(\hat{\varphi}+\hat{\psi}(\hat{\varphi}))w_{j,x}dx.\]
From direct computation, we get
\[\|\Lambda_1\|_{L^2(0,1)}\leq C\|\psi(\hat{\varphi})\|_{L^\infty }\le C(\varepsilon^{3/2-\mu})^{\min\{p,2\}}\le C\varepsilon^{\frac52+\mu_1}\]
and
\[\|\Lambda_2\|_{L^2(0,1)}\leq C\varepsilon^{1/2}(\varepsilon^{3/2-\mu})^{\min\{p,2\}}+(\varepsilon^{3/2-\mu})^{2\min\{p,2\}}\le C\varepsilon^{\frac52+\mu_1}.\]\
From \eqref{e94}, we rewrite \eqref{77} into the following form
\begin{eqnarray}\label{73}
\nonumber&&\varepsilon^2\alpha^{1-p}\beta\left\{-a_{11}f_j''+\left[a_{22}-b_{11}-a_{11}\left(\beta^{-1}\beta'+2\alpha^{-1}\alpha'\right)\right]f_j'+\left[a_{22}(\beta^{-1}\beta'+2\alpha^{-1}\alpha') +b_{22}-a_{33}\right]f_j\right. \\
\nonumber&&+\left[\frac{p+3}2\alpha^{p-2}\beta^{-2} \mathbf{q}_{tt}+\frac{p+2}{2(p+3)}\alpha^{1-p}\beta^2(a_{32})^2-\frac{p+1}{p+3}\alpha^{1-p}\beta^2 a_{32}b_{21}-\frac2{p+3}\alpha^{1-p}\beta^2 (b_{21})^2\right]f_j \\
\nonumber&&\left.+\left[-2\varepsilon a_{11}e_j' f_j'+2\varepsilon a_{22}e_j' f_j\right]\left(\int_\R w_x^2 dx\right)^{-1}\int_\R Z_x w_xdx\right\} \\
&&C_0 p\alpha_p\left[e^{-\sqrt{p}\beta(f_j-f_{j-1})}-e^{-\sqrt{p}\beta (f_{j+1}-f_j)}\right]=\varepsilon^2 M_{1j\varepsilon}.
\end{eqnarray}
Using the same procedure and \eqref{e95}, we get \eqref{82} is written into the following form
\begin{equation}\label{e80}
\varepsilon(\varepsilon^2a_{11}\alpha^{1-p}e_j''+\lambda_0 e_j) +p\alpha_p C_1\left[e^{-\sqrt{p}\beta(f_j-f_{j-1})}+e^{-\sqrt{p}\beta(f_{j+1}-f_j)}\right]=\varepsilon^2 M_{2j\varepsilon}.
\end{equation}
In the expression \eqref{73} and \eqref{e80}, $M_{ij\varepsilon}=A_{ij\varepsilon}+K_{ij\varepsilon}$, $i=1,2$, where $K_{ij\varepsilon}$ is a compact operator and $A_{ij\varepsilon}$ is a Lipschitz operator. It satisfies the following estimates
\[\|A_{ij\varepsilon}(\mathbf{f}_1,\mathbf{e}_1)-A_{ij\varepsilon}(\mathbf{f}_2,\mathbf{e}_2)\|_{L^2(0,1)}\leq C\varepsilon^{\frac12+\mu_1}\left[\|\mathbf{f}_1-\mathbf{f}_2\|_{H^2(0,1)}+\|\mathbf{e}_1-\mathbf{e}_2\|_*\right]\]
and
\begin{equation}\label{e49}
\|A_{ij\varepsilon}\|_{L^2(0,1)}\leq C\varepsilon^{\frac12+\mu_1}, \qquad \|K_{ij\varepsilon}\|_{L^2(0,1)}\leq C\varepsilon^{\frac12+\mu_1}.
\end{equation}
\section{Proof of Theorem}\label{sect8}
To prove our main theorem, we only need to find a solution to \eqref{73}-\eqref{e80}. To do job, we need some priori estimate.

\begin{remark}

The condition that  $\Gamma$ is a non-degenerate critical point of the functional $\mathcal{K}(\Gamma)$ is equivalent to the condition that $\Gamma$ is a non-degenerate geodesic embedded into the Riemannian manifold $(\mathbb R^2,\tilde{\mathfrak{g}})$, where $\tilde{\mathfrak{g}}(y)=\mathbf{\Psi}^{\frac{p+3}p}[A_{22}dy_1^2-2A_{12}(y)dy_1dy_2+A_{11}dy_2^2]$.
\end{remark}
We will use the infinite dimensional reduction method developed in \cite{delPino_Kowalczyk_Wei2007,delPino_Kowalczyk_Wei2008,Wei_Yang2021} to prove Theorem \ref{th1}. Infinite dimensional reduction method is used to find solutions to elliptic partial differential equations concentrating on high dimensional sets. To construct a solution to \eqref{1}, we first investigate the negative solution of \eqref{1} and its property in Section \ref{sectA}. However, we need to overcome the difficulty that $-\Div (A(x)\nabla u)$ is not a symmetric operator( see Lemma \ref{lm1} for detail).  To get a local approximate solution to \eqref{1}, we need to expand the operator $\Div(A(\varepsilon y)\nabla v)$ in a proper way. Wei and Yang \cite{Wei_Yang2021} developed a method to expand $\Div(\nabla_{\mathfrak{a}}(y)u)$. However we find a easier way to expand this operator. Similar to \cite{Wei_Yang2021}, to construct local approximate solution with $N$ concentration layers, we need a fine asymptotic estimate of the function $w$. But $w$ does not have the explicit expression(comparing \cite[(3.1)]{Wei_Yang2021}). To overcome this difficulty, we use the method in \cite{Gidas_Ni_Nirenberg1981} to get the asymptotic estimate of the function $w$ in the Section \ref{sectB}.

This paper is organized as follows. In Section \ref{sect1}, we set up a new modified Fermi coordinate in some neighborhood of $\Gamma$ and write down a local form of the operator $\Div (A(\varepsilon y)\nabla u)$ in the stretched modified Fermi coordinate. In Section \ref{sect2}, we find a local approximate solution to \eqref{1}. Then we conduct the gluing procedure in Section \ref{sect5} and the outer problem is solved. To solve the inner problem, we construct linear and nonlinear theory in Section \ref{sect6} and Section \ref{sect9}, respectively. From some involved calculation in Section \ref{sect4}, we reduce the problem into some partial differential equations in Section \ref{sect7}.  At last, Theorem \ref{th1} is proven in Section \ref{sect8}.

To prove Theorem~\ref{th1}, the authors "use the infinite dimensional reduction method" from \cite{delPino_Kowalczyk_Wei2007,delPino_Kowalczyk_Wei2008,Wei_Yang2021} to construct solutions of \eqref{1} concentrating on (high-dimensional) sets.

Key ingredients/steps described in the introduction:
\begin{itemize}
\item "First investigate the negative solution of \eqref{1} and its property" (Section~\ref{sectA}), while overcoming that $-\Div(A(x)\nabla u)$ "is not a symmetric operator" (Lemma~\ref{lm1}).
\item To build a local approximation, one must "expand the operator $\Div(A(\varepsilon y)\nabla v)$ in a proper way"; although Wei--Yang \cite{Wei_Yang2021} expand $\Div(\nabla_{\mathfrak{a}}(y)u)$, the authors state they "find a easier way to expand this operator."
\item As in \cite{Wei_Yang2021}, constructing a local approximate solution with $N$ layers requires "a fine asymptotic estimate of the function $w$"; since $w$ has no explicit formula, they "use the method in \cite{Gidas_Ni_Nirenberg1981} to get the asymptotic estimate of the function $w$" (Section~\ref{sectB}).
\item Paper roadmap implementing the reduction/gluing scheme:
  \begin{itemize}
  \item Section~\ref{sect1}: set up a "new modified Fermi coordinate" near $\Gamma$ and derive a local form of $\Div(A(\varepsilon y)\nabla u)$ in stretched coordinates.
  \item Section~\ref{sect2}: construct a local approximate solution to \eqref{1}.
  \item Section~\ref{sect5}: perform the "gluing procedure" and solve the outer problem.
  \item Sections~\ref{sect6} and \ref{sect9}: develop linear and nonlinear theory to solve the inner problem.
  \item Using "involved calculation" in Section~\ref{sect4}, "reduce the problem into some partial differential equations" in Section~\ref{sect7}.
  \item "At last, Theorem~\ref{th1} is proven in Section~\ref{sect8}."
  \end{itemize}
\end{itemize}

To prove the main theorem, the authors “use the infinite dimensional reduction method” from \cite{delPino_Kowalczyk_Wei2007,delPino_Kowalczyk_Wei2008,Wei_Yang2021} to construct solutions of
\begin{equation}\label{1}
\left\{\begin{array}{ll}
-\varepsilon^2\Div (A(x)\nabla u)=|u|^p-\mathbf{\Psi}(x), &\mbox{in $\Omega$,} \\
u=0, & \mbox{on $\partial \Omega$},
\end{array}
\right.
\end{equation}
concentrating on (high-dimensional) sets.

Key ingredients/steps described in the introduction:
\begin{itemize}
\item First they “investigate the negative solution” of the boundary value problem
\begin{equation}\label{1}
\left\{\begin{array}{ll}
-\varepsilon^2\Div (A(x)\nabla u)=|u|^p-\mathbf{\Psi}(x), &\mbox{in $\Omega$,} \\
u=0, & \mbox{on $\partial \Omega$},
\end{array}
\right.
\end{equation}
and its properties (later in the paper), while overcoming that $-\Div(A(x)\nabla u)$ “is not a symmetric operator.” In particular, they use the following auxiliary lemma.

\begin{lemma}\label{lm1}
If $a\ge c$ and $u_\varepsilon$ is the minimizer of problem \eqref{6},  $u_\varepsilon(x)\to a$ on compact sets in $B_1(0)$, as $\varepsilon \to 0$.
\end{lemma}

\item To build a local approximation, one must “expand the operator $\Div(A(\varepsilon y)\nabla v)$ in a proper way”; although Wei--Yang \cite{Wei_Yang2021} expand $\Div(\nabla_{\mathfrak{a}}(y)u)$, the authors state they “find a easier way to expand this operator.”

\item As in \cite{Wei_Yang2021}, constructing a local approximate solution with $N$ layers requires “a fine asymptotic estimate of the function $w$”; since $w$ has no explicit formula, they “use the method in \cite{Gidas_Ni_Nirenberg1981} to get the asymptotic estimate of the function $w$” (later in the paper).

\item Paper roadmap implementing the reduction/gluing scheme:
  \begin{itemize}
  \item Next they set up a “new modified Fermi coordinate” near $\Gamma$ and derive a local form of $\Div(A(\varepsilon y)\nabla u)$ in stretched coordinates.
  \item Next they construct a local approximate solution to the equation above.
  \item Next they perform the “gluing procedure” and solve the outer problem.
  \item Later they develop linear and nonlinear theory to solve the inner problem.
  \item Using “involved calculation” earlier, they “reduce the problem into some partial differential equations” later.
  \item This completes the proof of the main theorem.
  \end{itemize}
\end{itemize}

\label{th1}
Let $p>3$.
Assume that $\Gamma$ is a simple closed smooth curve with unit length in $\Omega$, and it is also a non-degenerate critical point of the functional
\begin{equation}\label{e91}
\mathcal{K}(\Gamma)=\int_\Gamma \mathbf{\Psi}^{\frac{p+3}{2p}}dvol_{\mathfrak{g}},
\end{equation}
where $\mathfrak{g}(X,Y)=\langle A^*X,Y\rangle$ is a Riemannian metric on $\mathbb R^2$ and $A^*$ is the adjoint matrix for $A$. We also assume that the following inequality holds on $\Gamma$:
\begin{eqnarray*}
\Upsilon_0&=&-\alpha^{1-p}\left[ 2\alpha^{-1}\alpha'a_{22}+\beta^{-1}\beta'b_{11}+b_{22}-a_{33}+\frac{p+3}2\alpha^{p-2}\beta^{-2}\mathbf{q}_{tt}+\frac{p+2}{2(p+3)}\alpha^{1-p} \beta^2 (a_{32})^2\right. \\
&&\left.-\frac{p+1}{p+3}\alpha^{1-p}\beta^2 a_{32}b_{21}-\frac{2}{p+3}\alpha^{1-p}\beta^2(b_{21})^2 +a_{11}\left(\beta^{-1}\beta''+2\alpha^{-1}\beta^{-1}\alpha'\beta'-\beta^{-2}(\beta')^2\right)\right]>0,
\end{eqnarray*}
where the coefficients $a_{ij}$ and $b_{ij}$ are given by the following lemma.

\begin{lemma}\label{lm8}
Under the stretched modified Fermi coordinate defined by $\Phi_\varepsilon(z,s)$, we get
\begin{eqnarray*}
\nonumber\Div(A(\varepsilon y)\nabla v)&=& a_{11}(\varepsilon z)v_{zz}+2\varepsilon s a_{22}(\varepsilon z)v_{sz} +\left(a_{31}(\varepsilon z)+\varepsilon s a_{32}(\varepsilon z) +\varepsilon^2 s^2 a_{33}(\varepsilon z)\right)v_{ss} \\
&& +\varepsilon b_{11}(\varepsilon z)v_z +\left(\varepsilon b_{21}(\varepsilon z)+\varepsilon^2 s b_{22}(\varepsilon z)\right) v_s +B_0(v),
\end{eqnarray*}
where
\[a_{11}(\theta)=\frac{\langle{A^*}n,n\rangle}{1-\langle\gamma',n\rangle^2},\qquad a_{22}(\theta)=-\frac{\langle{A^*}n,n'\rangle}{1-\langle\gamma',n\rangle^2}-\frac{\langle{A^*_t}n,\gamma'\rangle}{1-\langle\gamma',n\rangle^2}, \]
\[a_{31}(\theta)=\frac{\langle{A^*}\gamma',\gamma'\rangle}{1-\langle\gamma',n\rangle^2}, \qquad a_{32}=\frac{\langle{A^*_t}\gamma',\gamma'\rangle}{1-\langle\gamma',n\rangle^2}+ \frac{2\langle{A^*}\gamma',n'\rangle}{1-\langle\gamma',n\rangle^2}-\frac{2\langle\gamma',n'\rangle\langle{A^*}\gamma',\gamma'\rangle}{\left(1-\langle\gamma',n\rangle^2\right)^2}, \]
\begin{eqnarray*}
a_{33}(\theta)&=& \frac1{1-\langle\gamma',n\rangle^2}\left(\langle{A^*}n',n'\rangle +2\langle{A^*_t}\gamma',n'\rangle+\frac12\langle{A^*_{tt}}\gamma',\gamma'\rangle\right)  \\
&&-\frac{2\langle\gamma',n'\rangle}{\left(1-\langle\gamma',n\rangle^2\right)^2}\left(\langle{A^*_t} \gamma',\gamma'\rangle +2\langle{A^*}\gamma',n'\rangle\right)  \\
&&+\left(\frac{4\langle\gamma',n'\rangle^2}{\left(1-\langle\gamma',n\rangle^2\right)^3}- \frac{\langle n',n'\rangle}{\left(1-\langle\gamma',n\rangle^2\right)^2}\right)\langle{A^*}\gamma',\gamma'\rangle,
\end{eqnarray*}
\[b_{11}(\theta)=\left[\frac{\langle{A^*}n,n\rangle}{1-\langle\gamma',n\rangle^2}\right]_{\theta}-\frac12\left(\frac1{1-\langle\gamma',n\rangle^2}\right)_\theta \langle{A^*}n,n\rangle -\frac{\langle{A^*}n,n'\rangle}{1-\langle\gamma',n\rangle^2}-\frac{\langle{A^*_t}n,\gamma'\rangle}{1-\langle\gamma',n\rangle^2},\]
\[b_{21}(\theta)=\frac{\langle\gamma',n'\rangle}{1-\langle\gamma',n\rangle^2}a_{31}+a_{32},\]
\[b_{22}(\theta)=\frac12\frac{(1-\langle\gamma',n\rangle^2)_\theta}{1-\langle\gamma',n\rangle^2}a_{22}+\left[\frac{\langle n',n'\rangle}{1-\langle\gamma',n\rangle^2}-\frac{2\langle\gamma',n'\rangle^2}{(1-\langle\gamma',n\rangle^2)^2}\right]a_{31}+\frac{\langle\gamma',n'\rangle}{1-\langle\gamma', n\rangle^2}a_{32} +\partial_\theta a_{22}+2a_{33}\]
and
\[B_0(v)=\varepsilon a_{12}(\varepsilon z,s)v_{zz}+\varepsilon^2 a_{23}(\varepsilon z.s)v_{zs}+\varepsilon^3a_{33}(\varepsilon z,s)v_{ss} +\varepsilon^2 b_{12}(\varepsilon z,s) v_z +\varepsilon^3 b_{23}(\varepsilon z,s)v_s.\]
In the expression above, funtions $a_{12}$, $a_{23}$, $a_{33}$, $b_{12}$ and $b_{23}$ are smooth functions satisfying the following estimate
\[|a_{12}(\varepsilon z, s)|\le C(1+|s|), \quad |a_{23}(\varepsilon z, s)|\le C(1+|s|^2), \quad |a_{33}(\varepsilon z, s)|\le C(1+|s|^3),\]
and
\[|b_{12}(\varepsilon z, s)|\le C(1+|s|), \quad |b_{23}(\varepsilon z, s)|\le C(1+|s|^2).\]
\end{lemma}

Moreover,
\begin{equation}\label{80}
\alpha(\theta)=\mathbf{q}(\theta,0), \qquad \beta(\theta)=\frac{\left[\mathbf{q}(\theta,0)\right]^{\frac{p-1}2}}{\left[a_{31}(\theta)\right]^{\frac12}}.
\end{equation}
and
\begin{equation}\label{e88}
\mathbf{q}(\theta, t)=\mathbf{\Psi}^{\frac1p}(\gamma(\theta)+tn(\theta)).
\end{equation}

Then for each  integer $N>0$, there exists a sequence of $\varepsilon$, \textit{i.e.}, $\{\varepsilon_l\}$ converging to $0$ such that
\begin{equation}\label{1}
\left\{\begin{array}{ll}
-\varepsilon^2\Div (A(x)\nabla u)=|u|^p-\mathbf{\Psi}(x), &\mbox{in $\Omega$,} \\
u=0, & \mbox{on $\partial \Omega$},
\end{array}
\right.
\end{equation}
has a positive solutions $u_{\varepsilon_l}$ with exactly $N$ concentration layers at mutual distance $O(\varepsilon_l|\log\varepsilon_l|)$, the center of mass of $N$ concentration layers collapses to $\Gamma$ at speed $\varepsilon_l^{1+\mu}$ for small positive constant $\mu\in(0,1/2)$. More precisely $u_{\varepsilon_l}$ has the form
\begin{eqnarray*}
u_{\varepsilon_l}(y_1,y_2)\approx -\mathbf{\Psi}^{\frac1p}(y_1,y_2)+ \mathbf{\Psi}^{\frac1p}(\gamma(\theta))\sum_{k=1}^N w\left(\left[\frac{\mathbf{\Psi}^{\frac{p-1}p}(1-\langle\gamma',n\rangle)}{\langle A^*n,n\rangle}\right]^{\frac12}\frac{t-\varepsilon_l f_k}{\varepsilon_l}\right),
\end{eqnarray*}
where $\gamma$ is a natural parametrization of $\Gamma$, $n$ is the unit vector defined by
\begin{equation}\label{e102}
n(\theta)=\frac{A(\gamma(\theta))\nu(\theta)}{|A(\gamma(\theta))\nu(\theta)|}.
\end{equation}
and $w$ is  the unique solution of the following problem
\begin{equation}\label{29}
-w''=|1-w|^p -1, \qquad w>0 \quad \mathrm{in} \quad \mathbb R, \qquad w'(0)=w(\pm \infty).
\end{equation}
In the expression above, the functions $f_j$'s satisfy
\[\|f_j\|_{L^\infty(0,1)}\le C|\log\varepsilon_l|^2,\qquad \sum_{j=1}^N f_j =O(\frac1{|\log\varepsilon_l|^{\frac32}}),\]
\[\min_{1\le j\le N}(f_{j+1}-f_j)\approx\frac2{\sqrt{p}}|\log\varepsilon_l|\left[\frac{\langle A^*n,n\rangle}{\Psi^{\frac{p-1}p}(1-\langle\gamma',n\rangle)}\right]^{\frac12}\]
and solves the Jacobi-Toda system for $j=1,2,\cdots,N$
\begin{eqnarray*}
\nonumber&&\varepsilon^2\alpha^{1-p}\beta\left\{-a_{11}f_j''+\left[a_{22}-b_{11}-a_{11}\left(\beta^{-1}\beta'+2\alpha^{-1}\alpha'\right)\right]f_j'+\left[a_{22}(\beta^{-1}\beta'+2\alpha^{-1}\alpha') +b_{22}-a_{33}\right]f_j\right. \\
\nonumber&&+\left[\frac{p+3}2\alpha^{p-2}\beta^{-2} \mathbf{q}_{tt}+\frac{p+2}{2(p+3)}\alpha^{1-p}\beta^2(a_{32})^2-\frac{p+1}{p+3}\alpha^{1-p}\beta^2 a_{32}b_{21}-\frac2{p+3}\alpha^{1-p}\beta^2 (b_{21})^2\right]f_j \\
&&C_0 p\alpha_p\left[e^{-\sqrt{p}\beta(f_j-f_{j-1})}-e^{-\sqrt{p}\beta (f_{j+1}-f_j)}\right]\approx0.
\end{eqnarray*}

Question: Let Ω⊂R2 be a smooth bounded domain, let A(x) be a smooth symmetric positive-definite 2×2 matrix field, and let Ψ be the first Dirichlet eigenfunction of L(u)=-Div(A(x)∇u) on Ω. Let Γ⊂Ω be a simple closed smooth curve of unit length with natural parametrization γ(θ), let ν(θ) be its Euclidean unit normal, and define n(θ)=A(γ(θ))ν(θ)/|A(γ(θ))ν(θ)|, q(θ,t)=Ψ(γ(θ)+tn(θ))^(1/p), α(θ)=q(θ,0), β(θ)=q(θ,0)^((p-1)/2)/a31(θ)^(1/2), and the Riemannian metric g(X,Y)=⟨A*X,Y⟩. Assume p>3, Γ is a nondegenerate critical point of K(Γ)=∫_Γ Ψ^((p+3)/(2p)) dvol_g, and on Γ one has Υ0>0, where Υ0=-α^(1-p)[2α^(-1)α'a22+β^(-1)β'b11+b22-a33+((p+3)/2)α^(p-2)β^(-2)qtt+((p+2)/(2(p+3)))α^(1-p)β^2(a32)^2-((p+1)/(p+3))α^(1-p)β^2 a32 b21-(2/(p+3))α^(1-p)β^2(b21)^2+a11(β^(-1)β''+2α^(-1)β^(-1)α'β'-β^(-2)(β')^2)], and a11,a22,a31,a32,a33,b11,b21,b22 are the coefficients arising from the stretched modified Fermi-coordinate expansion of Div(A(εy)∇v) along Γ. Under these assumptions, which statement holds for the Dirichlet problem -ε2Div(A(x)∇u)=|u|^p-Ψ(x) in Ω, u=0 on ∂Ω?

For each integer N>0, there exists a sequence {ε_l} with ε_l→0 such that the problem -ε_l2Div(A(x)∇u)=|u|^p-Ψ(x) in Ω, u=0 on ∂Ω, has a positive solution u_{ε_l} with exactly N concentration layers at mutual distance O(ε_l|log ε_l|), and the center of mass of the N layers collapses to Γ at speed ε_l^(1+μ) for some small constant μ∈(0,1/2). More precisely, in the normal coordinates y=γ(θ)+tn(θ), one has u_{ε_l}(y1,y2)≈-Ψ(y1,y2)^(1/p)+Ψ(γ(θ))^(1/p)∑_{k=1}^N w(( [Ψ(γ(θ))^((p-1)/p)(1-⟨γ',n⟩)/⟨A*n,n⟩]^(1/2) (t-ε_l f_k)/ε_l )), where w is the unique solution of -w''=|1-w|^p-1, w>0 in R, w'(0)=w(±∞). In addition, the functions f_j satisfy ||f_j||_{L^∞(0,1)}≤C|log ε_l|^2, ∑_{j=1}^N f_j=O(1/|log ε_l|^(3/2)), and min_{1≤j≤N}(f_{j+1}-f_j)≈(2/√p)|log ε_l|[⟨A*n,n⟩/(Ψ^((p-1)/p)(1-⟨γ',n⟩))]^(1/2); moreover, for j=1,2,...,N they solve the Jacobi-Toda system ε_l2α^(1-p)β{ -a11 f_j'' + [a22-b11-a11(β^(-1)β'+2α^(-1)α')]f_j' + [a22(β^(-1)β'+2α^(-1)α')+b22-a33]f_j + [((p+3)/2)α^(p-2)β^(-2)qtt + ((p+2)/(2(p+3)))α^(1-p)β^2(a32)^2 - ((p+1)/(p+3))α^(1-p)β^2 a32 b21 - (2/(p+3))α^(1-p)β^2(b21)^2]f_j } + C0 p α_p [e^(-√p β(f_j-f_{j-1})) - e^(-√p β(f_{j+1}-f_j))]≈0.

Choices:
B.
There exists varepsilon_0>0 such that for every integer N>0 and every varepsilon
a0,varepsilon_0), the problem -varepsilon^2Div(A(x)u)=|u|^p-si(x) in 9, u=0 on 99, has a positive solution u_varepsilon with exactly N concentration layers at mutual distance O(varepsilon|log varepsilon|), whose center of mass collapses to at speed varepsilon^(1+) for some small constant (0,1/2). More precisely, in the normal coordinates y=()+tn(), one has u_varepsilon(y_1,y_2)- si(y_1,y_2)^(1/p)+si(())^(1/p)_{k=1}^N w(([ si(())^((p-1)/p)(1-',n)/A*n,n]^(1/2)(t-varepsilon f_k)/varepsilon)), and the functions f_j satisfy the same bounds and JacobiToda system as in option A.
C. For each integer N>0, there exists a sequence {\varepsilon_l} with \varepsilon_l\to0 such that the problem -\varepsilon_l^2\Div(A(x)\nabla u)=|u|^p-\Psi(x) in \Omega, u=0 on \partial\Omega, has a positive solution u_{\varepsilon_l} with exactly N concentration layers whose center of mass collapses to \Gamma. More precisely, in the normal coordinates y=\gamma(\theta)+tn(\theta), one has u_{\varepsilon_l}(y_1,y_2)\approx-\Psi(y_1,y_2)^{1/p}+\Psi(\gamma(\theta))^{1/p}\sum_{k=1}^N w\!\left(\left[\frac{\Psi(\gamma(\theta))^{(p-1)/p}(1-\langle\gamma',n\rangle)}{\langle A^*n,n\rangle}\right]^{1/2}\frac{t-\varepsilon_l f_k}{\varepsilon_l}\right),
D. For each integer N>0, there exists a sequence {\varepsilon_l} with \varepsilon_l\to0 such that the problem -\varepsilon_l^2\Div(A(x)\nabla u)=|u|^p-\Psi(x) in \Omega, u=0 on \partial\Omega, has a positive solution u_{\varepsilon_l} with exactly N concentration layers at mutual distance O(\varepsilon_l), and the center of mass of the N layers collapses to \Gamma at speed \varepsilon_l^{1+\mu} for some small constant \mu\in(0,1/2). More precisely, in the normal coordinates y=\gamma(\theta)+tn(\theta), one has the same leading-order profile as in option A, while the functions f_j satisfy \|f_j\|_{L^\infty(0,1)}\le C|\log\varepsilon_l|^2, \sum_{j=1}^N f_j=O(1/|\log\varepsilon_l|^{3/2}), and \min_{1\le j\le N}(f_{j+1}-f_j)\approx \frac{2}{\sqrt p}\left[\frac{\langle A^*n,n\rangle}{\Psi^{(p-1)/p}(1-\langle\gamma',n\rangle)}\right]^{1/2}; moreover, for j=1,2,\dots,N they solve the same Jacobi-Toda system as in option A.
E. For each integer N>0, there exists a sequence {\varepsilon_l} with \varepsilon_l\to0 such that the problem -\varepsilon_l^2\Div(A(x)\nabla u)=|u|^p-\Psi(x) in \Omega, u=0 on \partial\Omega, has a positive solution u_{\varepsilon_l} with exactly N concentration layers at mutual distance O(\varepsilon_l|\log \varepsilon_l|), and the center of mass of the N layers collapses to \Gamma at speed \varepsilon_l^{1+\mu} for some small constant \mu\in(0,1/2). More precisely, in the normal coordinates y=\gamma(\theta)+tn(\theta), one has u_{\varepsilon_l}(y_1,y_2)\approx-\Psi(y_1,y_2)^{1/p}+\Psi(\gamma(\theta))^{1/p}\sum_{k=1}^N w\!\left(\left[\frac{\Psi(\gamma(\theta))^{(p-1)/p}(1-\langle\gamma',n\rangle)}{\langle A^*n,n\rangle}\right]^{1/2}\frac{t-\varepsilon_l f_k}{\varepsilon_l}\right), where w is the unique solution of -w''=|1-w|^p-1, w>0 in \mathbb R, w'(0)=w(\pm\infty). In addition, the functions f_j satisfy the same bounds as in option A, and for j=1,2,\dots,N they solve the Jacobi-Toda system obtained from option A after replacing the interaction term \;C_0 p\alpha_p\big[e^{-\sqrt p\beta(f_j-f_{j-1})}-e^{-\sqrt p\beta(f_{j+1}-f_j)}\big]\; by \;C_0 p\alpha_p\big[e^{-\sqrt p\beta(f_j-f_{j-1})}+e^{-\sqrt p\beta(f_{j+1}-f_j)}\big]\approx0.

dropped quantitative layer-spacing speed and Jacobi-Toda details


\label{thm:main}
For a skew-symmetrizable matrix $B$, the following are equivalent:
\begin{itemize}
\item[$(1)$] $B$ is of finite type.
\item[$(2)$] The $g$-fan $\mathcal{F}(B)$ is complete, that is, its support $|\mathcal{F}(B)|=\mathbb{R}^n$.
\item[$(3)$] The support $|\mathcal{F}(B)|$ contains all lattice points in $\mathbb{Z}^n$.
\end{itemize}

Cluster algebras \cite{FZ02} have been studied from many points of view. One of the basic objects in this theory is the notion of $g$-vectors \cite{FZ07}.

Let $B$ be a skew-symmetrizable matrix. The $g$-vectors associated with $B$ span simplicial cones in $\mathbb{R}^n$. These cones form a simplicial fan, which is called the $g$-fan of $B$ (see \cite{GHKK18,Re14}). The geometry of this fan encodes the structure of the associated cluster algebra.

A skew-symmetrizable matrix $B$ is said to be of \emph{finite type} if it has only finitely many $g$-vectors, or equivalently, finitely many cluster variables. Finite type matrices are well understood and play an important role in cluster theory.

It is then natural to ask how the finite type property of $B$ is reflected in the geometry of its $g$-fan. In particular, one may ask whether the $g$-fan covers the whole space $\mathbb{R}^n$, or whether it can be characterized by the property of containing all lattice points in $\mathbb{Z}^n$. This question is also motivated by \cite[Question~3.49]{D17} in representation theory. In this paper, we give a complete answer to this question in the framework of $g$-fans arising from skew-symmetrizable matrices. More precisely, we prove the following theorem.

\begin{abstract}
We study the $g$-fan associated with a skew-symmetrizable matrix in the sense of cluster algebras. We show that a skew-symmetrizable matrix is of finite type if and only if its $g$-fan is complete; equivalently (as we show), its support contains all lattice points.
\end{abstract}

It is then natural to ask how the finite type property of $B$ is reflected in the geometry of its $g$-fan. In particular, one may ask whether the $g$-fan covers the whole space $\mathbb{R}^n$, or whether it can be characterized by the property of containing all lattice points in $\mathbb{Z}^n$. This question is also motivated by \cite[Question~3.49]{D17} in representation theory. In this paper, we give a complete answer to this question in the framework of $g$-fans arising from skew-symmetrizable matrices. More precisely, we prove the following theorem.

The implication $(1)\Rightarrow(2)$ is known (Theorem~\ref{thm:Reading}). For skew-symmetric matrices, the implication $(2)\Rightarrow(1)$ was proved in \cite{HY25} via categorification using Jacobian algebras. In \cite{Y23}, we suggested that this implication might be proved by methods similar to those in \cite[Proposition~4.9]{A21} in general, but no complete proof was available. In this paper, we give a complete proof using scattering diagrams and their pull-back constructions.

An $n\times n$ skew-symmetrizable matrix $B$ is called
\begin{itemize}
\item \emph{mutation equivalent} to a matrix $B'$ if it is obtained from $B'$ by a finite sequence of mutations;
\item \emph{$2$-finite} if for any matrix $B'=(b'_{ij})$ mutation equivalent to $B$, $|b'_{ij}b'_{ji}|\le 3$ for all $i,j$.
\end{itemize}

\begin{definition}
A \emph{$g$-vector seed} is a pair $(C,(\mathbf{g}_1,\ldots,\mathbf{g}_n))$ consisting of the following data:
\begin{itemize}
\item[$(1)$] $C$ is a $2n\times n$ integer matrix whose upper part is skew-symmetrizable.
\item[$(2)$] $\mathbf{g}_1,\ldots,\mathbf{g}_n\in\mathbb Z^n$.
\end{itemize}
The tuple $(\mathbf{g}_1,\ldots,\mathbf{g}_n)$ is called the \emph{$g$-vector tuple}.
\end{definition}

\begin{example}\label{exam:rank 2 g-fan}
Any nonzero $2 \times 2$ skew-symmetrizable matrix is given by
\[
B_{b,c}:=\begin{bmatrix}0 & c \\ -b & 0\end{bmatrix},
\]
where $b$ and $c$ are integers and $bc > 0$. The $g$-fan $\mathcal{F}(B_{b,c})$ is well-known (see e.g. \cite[Example~1.15]{GHKK18} or \cite[Section~2]{N24}). In fact, for $bc \ge 4$, $\mathcal{F}(B_{b,c})$ contains infinitely many rays converging to the rays $r_{\pm}$ of slope $(-bc \pm \sqrt{bc(bc-4)})/2c$ (see Figure~\ref{fig:rank 2 g-vector fans}). In this case, the ray of slope $-b/2$ is contained in $(\mathbb{R}^2\setminus|\mathcal{F}(B_{b,c})|)\cup\{0\}$. In particular, the lattice point $(-2,b)$ is contained in $\mathbb{Z}^n\setminus|\mathcal{F}(B_{b,c})|$.
\end{example}

\begin{theorem}[{\cite[Theorem~10.6]{Re14}}]\label{thm:Reading}
If $B$ is of finite type, then $|\mathcal{F}(B)|=\mathbb{R}^n$.
\end{theorem}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
The implication $(1)\Rightarrow(2)$ follows from Theorem~\ref{thm:Reading}, and $(2)\Rightarrow(3)$ clearly holds. Finally, we prove $(3)\Rightarrow(1)$. By Theorem~\ref{thm:fin 2-fin}, if $B$ is not of finite type, it is mutation equivalent to $B'$ such that $B'_I=B_{b,c}$ with $bc\ge 4$ for some subset $I \subset \{1,\ldots,n\}$. By Proposition~\ref{prop:rank 2 no lattice pt}, $\mathbb{Z}^n\setminus|\mathcal{F}(B')|\neq\emptyset$, which implies that $\mathbb{Z}^n\setminus|\mathcal{F}(B)|\neq\emptyset$ by Corollary~\ref{cor:preserve lattice pt}. This completes the proof.
\end{proof}

The post-theorem text indicates: $(1)\Rightarrow(2)$ is already known (Theorem~\ref{thm:Reading}). For skew-symmetric matrices, $(2)\Rightarrow(1)$ was proved in \cite{HY25} "via categorification using Jacobian algebras." A possible approach "by methods similar to those in \cite[Proposition~4.9]{A21}" had been suggested in \cite{Y23} but lacked a complete proof. The paper claims to provide a complete proof "using scattering diagrams and their pull-back constructions."

The post-theorem text indicates: $(1)\Rightarrow(2)$ is already known. \begin{theorem}[{\cite[Theorem~10.6]{Re14}}]\label{thm:Reading}
If $B$ is of finite type, then $|\mathcal{F}(B)|=\mathbb{R}^n$.
\end{theorem}
For skew-symmetric matrices, $(2)\Rightarrow(1)$ was proved in Assem and Yurikusa, “via categorification using Jacobian algebras.” A possible approach “by methods similar to those in \cite[Proposition~4.9]{A21}” had been suggested in Yurikusa, but lacked a complete proof. The paper claims to provide a complete proof “using scattering diagrams and their pull-back constructions.”

Biconditional or Equivalence
Classification or Bijection

Question: Let $B$ be a skew-symmetrizable $n\times n$ matrix. Its associated $g$-vectors span simplicial cones in $\mathbb{R}^n$, forming the $g$-fan $\mathcal{F}(B)$, and $|\mathcal{F}(B)|$ denotes the support of this fan (the union of all its cones). A matrix $B$ is called of finite type if it has only finitely many $g$-vectors. Which statement gives the exact characterization of when $B$ is of finite type in terms of its $g$-fan?

$B$ is of finite type if and only if the $g$-fan $\mathcal{F}(B)$ is complete, i.e. $|\mathcal{F}(B)|=\mathbb{R}^n$; equivalently, if and only if $|\mathcal{F}(B)|$ contains every lattice point of $\mathbb{Z}^n$.

Choices:
B. $B$ is of finite type if and only if the support $|\mathcal{F}(B)|$ contains every nonzero lattice point of $\mathbb{Z}^n$; equivalently, the only lattice point that may fail to lie in $|\mathcal{F}(B)|$ is the origin.
C. If $B$ is of finite type, then the $g$-fan $\mathcal{F}(B)$ is complete, i.e. $|\mathcal{F}(B)|=\mathbb{R}^n$.
D. $B$ is of finite type if and only if for every lattice point $\mathbf{m}\in\mathbb{Z}^n$ there exists a matrix $B'$ mutation equivalent to $B$ such that $\mathbf{m}\in |\mathcal{F}(B')|$.
E. $B$ is of finite type if and only if the support $|\mathcal{F}(B)|$ has nonempty interior in $\mathbb{R}^n$; equivalently, if and only if $|\mathcal{F}(B)|$ contains infinitely many lattice points of $\mathbb{Z}^n$.


Question: Let $B$ be a skew-symmetrizable matrix. Its $g$-vectors span simplicial cones in $\mathbb{R}^n$, and these cones form the $g$-fan $\mathcal{F}(B)$; write $|\mathcal{F}(B)|$ for the support, i.e. the union of all cones in the fan. A matrix $B$ is said to be of finite type if it has only finitely many $g$-vectors (equivalently, finitely many cluster variables). Which of the following statements is equivalent to both the assertion that $B$ is of finite type and the assertion that the $g$-fan $\mathcal{F}(B)$ is complete, meaning $|\mathcal{F}(B)|=\mathbb{R}^n$?

The support $|\mathcal{F}(B)|$ contains all lattice points in $\mathbb{Z}^n$.

Choices:
B. The support $|\mathcal{F}(B)|$ contains all but finitely many lattice points of $\mathbb{Z}^n$.
C. The support $|\mathcal{F}(B)|$ contains every nonzero lattice point in $\mathbb{Z}^n$.
D. For every skew-symmetrizable matrix $B'$ mutation equivalent to $B$, the support $|\mathcal{F}(B')|$ contains all lattice points in $\mathbb{Z}^n$.
E. The support $|\mathcal{F}(B)|$ is dense in $\mathbb{R}^n$, that is, every nonempty open subset of $\mathbb{R}^n$ meets $|\mathcal{F}(B)|$.