\begin{proposition}\label{prop: rank 3 excluded minor}
    If $M$ is a simple rank-$3$ matroid with a line $L$ and distinct points $e_1, e_2, e_3$ on $L$ each on at least two long lines, then $M$ is not a positroid.
    In particular, $M$ has a minor from Figure \ref{fig: rank-3 excluded minors}.
\end{proposition}

\begin{theorem} \label{thm: main}
For all integers $r, \ell \ge 1$, if $M$ is a simple rank-$r$ positroid with no $U_{2,\ell+2}$-minor, then $|M| \le \ell(r - 1) + 1$.
Moreover, if $r \ge 2$ then equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2, \ell+1}$.
\end{theorem}

\begin{proposition}\label{prop: rank 4 excluded minor}
    If $M$ is a simple rank-$4$ matroid with a line $L$, distinct points $e_1, e_2, e_3$ on $L$, and distinct planes $P_1$, $P_2$, $P_3$ through $L$ so that for $i = 1,2,3$ the point $e_i$ is on at least two long lines in $P_i$, then $M$ is not a positroid.
    In particular, $M$ has a minor from Figure \ref{fig: rank-3 excluded minors} or Figure \ref{fig: rank-4 excluded minor}.
\end{proposition}

\label{sec: introduction}

Extremal matroid theory was formally introduced in 1993 with the following problem posed by Joseph Kung \cite{Kung-1993}.

\begin{problem} \label{prob: Kung problem}
Let $\cM$ be a class of matroids.
Determine the maximum number of elements of a simple rank-$r$ matroid in $\cM$, and characterize the matroids for which equality holds.
\end{problem}

This maximum does not exist for all classes of matroids.
For instance, there is no finite answer to Problem~\ref{prob: Kung problem} for the class of all matroids, or for any class containing all uniform matroids.
However, Kung \cite[Theorem 4.3]{Kung-1993} proved that every simple rank-$r$ matroid with no $U_{2, \ell+2}$-minor has at most $\frac{\ell^r - 1}{\ell - 1}$ elements.
If $\ell$ is a prime power, then this bound is tight and is achieved by the rank-$r$ projective geometry over the finite field of order $\ell$. It is thus natural to study Problem~\ref{prob: Kung problem} for classes of matroids without $U_{2,\ell}$-minors.

In this paper, we study the class of \textit{positroids} without $U_{2,\ell}$-minors. Positroids, which are matroids representable by a matrix over $\bR$ with all maximal subdeterminants nonnegative, occupy a central position at the intersection of combinatorics, algebraic geometry, and physics. They serve as combinatorial models for the totally nonnegative Grassmannian, which in turn underlies the construction of the amplituhedron in quantum field theory. The study of amplituhedra and related objects remains an active and influential area in theoretical physics \cite{AHT}. Through these connections, positroids emerge as unifying objects that integrate discrete combinatorial frameworks with geometric and physical theory. Combinatorially, they are remarkably rich: Postnikov~\cite{postnikov} showed that positroids are in bijection with decorated permutations, Grassmann necklaces, plabic graphs, and \reflectbox{L}-diagrams, providing multiple equivalent viewpoints.
Positroids also form a minor-closed class of matroids. It is thus natural to ask extremal questions about positroids and their minors. There has been work enumerating both positroids \cite{Williams05} and connected positroids \cite{ARW16}, but as far as the authors are aware, no other extremal results about positroids are known. Our main result tightly resolves Problem~\ref{prob: Kung problem} for the class of $U_{2,\ell+2}$-minor-free positroids.

This result lies at the intersection of three areas of research: extremal matroid theory, structural properties of positroids, and structural properties of matroids with no $M(K_4)$-minor.
We will explain how Theorem \ref{thm: main} continues lines of research in these three areas, beginning with extremal matroid theory.
The following beautiful result of Geelen and Nelson \cite[Theorem 1.2]{Geelen-Nelson-2010} generalizes the initial work of Kung and tightly solves Problem \ref{prob: Kung problem} for the class of matroids with no $U_{2,\ell+2}$-minor, for all values of $\ell$ and all sufficiently large values of $r$.

\begin{theorem}[Geelen, Nelson \cite{Geelen-Nelson-2010}] \label{thm: exlude a line}
Let $\ell\ge 2$ be an integer and let $q$ be the largest prime power at most $\ell$.
For $r$ sufficiently large, every simple rank-$r$ matroid with no $U_{2,\ell+2}$-minor has at most $\frac{q^r - 1}{q - 1}$ elements, with equality only for the rank-$r$ projective geometry over the order-$q$ finite field.
\end{theorem}

\begin{theorem}[Geelen, Nelson, Walsh \cite{Geelen-Nelson-Walsh-2024}] \label{thm: complex exclude a line}
Let $\ell\ge 2$ be an integer and let $r$ be a sufficiently large integer.
Then every simple rank-$r$ $\bC$-representable matroid with no $U_{2,\ell+2}$-minor has at most $\ell\binom{r}{2} + r$ elements, with equality only for the rank-$r$ Dowling geometry over the cyclic group of order $\ell$.
\end{theorem}

\begin{theorem} \label{thm: main result with proof}
For all integers $r, \ell \ge 1$, if $M$ is a simple rank-$r$ matroid with no minor from Figure \ref{fig: rank-3 excluded minors} or Figure \ref{fig: rank-4 excluded minor} and no $U_{2,\ell+2}$-minor, then $|M| \le \ell(r - 1) + 1$.
Moreover, equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2, \ell+1}$.
\end{theorem}
\begin{proof}
First suppose that $\ell = 1$.
If $M$ is a simple rank-$r$ matroid with more than $r$ elements, then $r \ge 2$ and $M$ has a circuit with at least three elements.
But every circuit with at least three elements has a $U_{2,3}$-minor \cite[Example 3.1.9]{Oxley-2011}, a contradiction.
So $M$ is a rank-$r$ matroid with $r$ elements, and it follows that $M$ is the parallel connection of $r - 1$ copies of $U_{2,2}$, as desired.
So we may assume that $\ell \ge 2$.
We will proceed by induction on $r$.
The statement is clearly true when $r \le 2$ so we may assume that $r \ge 3$.
If $M$ has an element $e$ that is not on any long lines, then $M/e$ is simple, $|M/e| = |M| - 1$, and $r(M/e) = r - 1$, so by induction we have
\begin{align*}
 |M| = 1 + |M/e| \le 1 + \ell(r - 2) + 1 = \ell(r - 1) + 2 - \ell < \ell(r - 1) + 1,
\end{align*}
as desired.
So we may assume that every element of $M$ is on a long line.
Let $L$ be a long line of $M$.
We will now consider the connected components of $M/L$.
Let $\cX$ be the partition of $E(M) - L$ given by the ground sets of the connected components of $M/L$.
Let $X \in \cX$, and let $N = M|(L \cup X)$.
We will first bound the number of elements of $N \del L$.

\begin{claim} \label{claim: bound the size of each component}
 At most two elements in $L$ are on a long line of $N$ other than $L$, and $|N \del L| \le \ell(r(N) - 2)$.
\end{claim}
\begin{proof}
Suppose that distinct elements $e_1$, $e_2$, and $e_3$ in $L$ are on long lines $L_1$, $L_2$, and $L_3$, respectively, of $N$ other than $L$.
Let $K = \si(N/L)$.
By deleting elements from nontrivial parallel classes of $N/L$ we may assume that $K$ is a restriction of $N/L$.
Note that the elements of $K$ are in one-to-one correspondence with the planes of $N$ containing $L$.
For $i = 1,2,3$ let $p_i$ be the element of $K$ in the plane $\cl_N(L \cup L_i)$ of $N$.
Since $K$ is connected, it follows from \cite[Theorem 4.3.1]{Oxley-2011} that $K$ has a connected minor $K_1 = K \del D/C$ on ground set $\{p_1, p_2, p_3\}$.
We may assume that $C$ is independent in $K$ and therefore in $N/L$, which implies that $r_{N/C}(L) = r_N(L) = 2$.
Since $K_1$ is connected, it is isomorphic to $U_{1,3}$ or $U_{2,3}$.
Let $N_1$ be obtained from $N$ by deleting, for each element $d$ in $D$, the elements in $\cl_N(L \cup d) - L$, and then contracting $C$.
Then $K_1 = \si(N_1/L)$ and $r(N_1) = r(K_1) + 2$.
We will show that $N_1$ has a minor from Figure \ref{fig: rank-3 excluded minors} or \ref{fig: rank-4 excluded minor}.

\begin{claim}
 Some $x \in L$ is on more than one long line of $M$.
\end{claim}
\begin{proof}
Let $e \in L$ and let $M_1 = M/e$.
Let $P = L - e$, and note that $P$ is a nontrivial parallel class of $M_1$.
Also note that $\elem(M_1) = |M| - \ell = \ell(r(M_1) - 1) + 1$,
so by induction, $\si(M_1)$ is obtained by taking parallel connections of copies of $U_{2,\ell+1}$.
This implies that $P$ is on a long line $L_1$ of $M_1$.
Let $F$ be the plane $L \cup L_1$ of $M$.
Note that $|F| \ge |L| + |L_1| - 1 \ge 2(\ell + 1) - 1 = 2\ell + 1$.
If $F - L$ has a $3$-element independent set $I$, then $M|F$ has an $M(K_4)$-restriction (if each pair of points in $I$ spans a point on $L$) or a $U_{2, \ell+2}$-minor, a contradiction.
So $F - L$ has rank $2$.
If $\cl(F - L)$ does not intersect $L$, then for every $f \in F - L$ we have $(M|F)/f \cong U_{2,\ell+2}$, a contradiction.
So $\cl(F - L)$ intersects $L$ in some element $x$.
Since $|F - L| \ge \ell \ge 2$, it follows that $\cl(F - L)$ is a long line of $M$ that contains $x$ and is not equal to $L$, as desired.
\end{proof}

Let $r, \ell$ be integers with $r \ge 2$ and $\ell \ge 3$.
Let $W(r,\ell)$ be the matroid with a basis $B = \{b_1, b_2, \dots, b_r\}$ and $\lfloor \frac{\ell - 1}{2} \rfloor$ elements freely placed (see \cite[page 270]{Oxley-2011}) in the span of $\{b_i, b_{i+1}\}$ for all $i \in [r]$, reading indices modulo $r$.
Let $W(r, \ell)^+$ be obtained from $W(r, \ell)$ by freely placing one point in $\cl(\{b_1, b_2\})$.
Then $W(2, \ell) \cong U_{2,\lfloor \frac{\ell - 1}{2} \rfloor + 2}$, and $W(r, 3)$ and $W(r, 4)$ are both isomorphic to the rank-$r$ whirl $\mathcal W^r$ when $r \ge 3$.
Since the whirl $\mathcal W^r$ is $3$-connected (see \cite[Example 8.3.4]{Oxley-2011}) and $W(r,\ell)$ has $\mathcal W^r$ as a restriction, it follows from \cite[page 295, Exercise 5]{Oxley-2011} that $W(r,\ell)$ and $W(r, \ell)^+$ are $3$-connected.
And it is straightforward to check that the natural cyclic ordering of $E(W(r, \ell))$ (first $b_1$, then all elements in $\cl(\{b_1, b_2\}) - b_2$, then $b_2$, and so on) or $W(r,\ell)^+$ satisfies the condition of Theorem \ref{thm:Bonin's condition}, so $W(r, \ell)$ and $W(r, \ell)^+$ are positroids.
Finally, one can check that if $r \ge 4$ and $e \notin \{b_1, \dots, b_r\}$, then $\si(W(r, \ell)/e) \cong W(r, \ell)$, and that $\si(W(r, \ell)/b_i) \cong W(r,\ell) \del (\cl(\{b_{i-1},b_i, b_{i+1}) - \{b_{i-1}, b_{i+1}\})$, and from these observations one can prove that $W(r,\ell)$ is $U_{2,\ell+2}$-minor-free, and when $\ell$ is even, $W(r, \ell)^+$ is $U_{2, \ell+2}$-minor-free.
Noting that $|W(r, \ell)| = r + r\lfloor \frac{\ell-1}{2} \rfloor$, we make the following conjecture.

This result lies at the intersection of three areas of research: extremal matroid theory, structural properties of positroids, and structural properties of matroids with no $M(K_4)$-minor.
We will explain how Theorem \ref{thm: main} continues lines of research in these three areas, beginning with extremal matroid theory.
The following beautiful result of Geelen and Nelson \cite[Theorem 1.2]{Geelen-Nelson-2010} generalizes the initial work of Kung and tightly solves Problem \ref{prob: Kung problem} for the class of matroids with no $U_{2,\ell+2}$-minor, for all values of $\ell$ and all sufficiently large values of $r$.

\begin{theorem}[Geelen, Nelson \cite{Geelen-Nelson-2010}] \label{thm: exlude a line}
Let $\ell\ge 2$ be an integer and let $q$ be the largest prime power at most $\ell$.
For $r$ sufficiently large, every simple rank-$r$ matroid with no $U_{2,\ell+2}$-minor has at most $\frac{q^r - 1}{q - 1}$ elements, with equality only for the rank-$r$ projective geometry over the order-$q$ finite field.
\end{theorem}

Projective geometries are not representable over fields of characteristic zero (see \cite[page 205, Exercise 8]{Oxley-2011}), inviting the following question: for a field $\bF$ with characteristic zero, what is the maximum number of elements of a simple rank-$r$ $\bF$-representable matroid with no $U_{2, \ell+2}$-minor?
This was answered by Geelen, Nelson, and Walsh \cite[Theorem 1.0.5]{Geelen-Nelson-Walsh-2024} in the case that $\bF = \bC$.

\begin{theorem}[Geelen, Nelson, Walsh \cite{Geelen-Nelson-Walsh-2024}] \label{thm: complex exclude a line}
Let $\ell\ge 2$ be an integer and let $r$ be a sufficiently large integer.
Then every simple rank-$r$ $\bC$-representable matroid with no $U_{2,\ell+2}$-minor has at most $\ell\binom{r}{2} + r$ elements, with equality only for the rank-$r$ Dowling geometry over the cyclic group of order $\ell$.
\end{theorem}

When $\bF \in \{\bR, \mathbb Q\}$, there is a lower bound of $2\binom{r}{2} + r$ and an upper bound of $2 \binom{r}{2} + f(\ell)\cdot r$ with $f$ a function of $\ell$, due to Geelen, Nelson, and Walsh \cite[Theorem 1.0.6]{Geelen-Nelson-Walsh-2024}, but no exact solution to Problem~\ref{prob: Kung problem} is known for all $\ell$.
Theorem \ref{thm: main} can be seen as a further specialization of Theorems \ref{thm: exlude a line} and \ref{thm: complex exclude a line} to the subclass of $\bR$-representable matroids consisting of positroids.

Theorem \ref{thm: main} also continues a recent line of research that studies positroids from a structural perspective.
For example, Bonin \cite{Bonin-2024} studied matroid amalagams of positroids, Quail \cite{Quail-2024} studied positroids representable over the finite fields of order $3$ and of order $4$, Quail and Rombach \cite{Quail-Rombach-2025} studied graphic positroids, and Park \cite{Park-2023} studied excluded minors for positroids.
Our proof of Theorem \ref{thm: main} relies on two new structural properties (Propositions \ref{prop: rank 3 excluded minor} and \ref{prop: rank 4 excluded minor}) for low-rank positroids.

Finally, \Cref{thm: main} contributes to a line of research studying matroids with no $M(K_4)$-minor, where $M(K_4)$ is the graphic matroid of the complete graph $K_4$.
For example, Sims \cite{Sims-1977} proved that the class of $M(K_4)$-minor-free matroids is complete, Brylawski \cite{Brylawski-1971} characterized the binary matroids with no $M(K_4)$-minor, and Oxley \cite{Oxley-1987} characterized the $3$-connected ternary matroids with no $M(K_4)$-minor.
More recently, work of Pendavingh and van der Pol \cite{Pendavingh-vanderPol-2015} and van der Pol \cite{vanderPol-2023} investigated the importance of $M(K_4)$ in matroid enumeration.
Most notably for this paper, Kung studied Problem \ref{prob: Kung problem} for the class of matroids with no $M(K_4)$-minor and no $U_{2, \ell+2}$-minor, proving that every such simple rank-$r$ matroid has at most $(6\ell^{\ell - 1} + 8\ell)\cdot r$ elements \cite[Theorem 1.1]{Kung-1988}. The graphic matroid $M(K_4)$ is not a positroid, so this also bounds from above the maximum number of elements of a simple rank-$r$ positroid with no $U_{2, \ell+2}$-minor.
To the best of our knowledge, this was in fact the previous best upper bound for this quantity.
There are several other notable $M(K_4)$-minor-free classes of matroids that contain the class of positroids, such as gammoids, strongly base-orderable matroids, and $k$-base-orderable matroids.
We will discuss these further in Section \ref{sec: future work}.

The rest of this paper is structured as follows.
In Section \ref{sec: preliminaries}, we will give more background on matroids and positroids.
In Section \ref{sec: excluded minors}, we will use a list of $9$ known excluded minors for the class of positroids to prove two structural properties for low-rank positroids.
In Section \ref{sec: the extremal examples}, we will describe the matroids for which equality holds in Theorem \ref{thm: main} and prove that they are in fact positroids and are $U_{2, \ell+2}$-minor-free.
We will prove Theorem \ref{thm: main} in Section \ref{sec: the main proof}, and conclude in Section \ref{sec: future work} by discussing several directions for future work.

Our proof of Theorem \ref{thm: main} relies on two new structural properties (Propositions \ref{prop: rank 3 excluded minor} and \ref{prop: rank 4 excluded minor}) for low-rank positroids. The paper then (i) uses a list of $9$ known excluded minors for the class of positroids to prove these two structural properties for low-rank positroids (Section \ref{sec: excluded minors}); (ii) describes the matroids for which equality holds in Theorem \ref{thm: main} and proves that they are positroids and are $U_{2, \ell+2}$-minor-free (Section \ref{sec: the extremal examples}); and (iii) proves Theorem \ref{thm: main} in Section \ref{sec: the main proof}.

Inequality or Bound
Biconditional or Equivalence

Question: Let $r,2 ell\ge 1$ be integers. Recall that a positroid is a matroid representable over $\mathbb{R}$ by a matrix whose maximal minors are all nonnegative, and that $U_{2,m}$ denotes the rank-$2$ uniform matroid on $m$ elements. If $M$ is a simple rank-$r$ positroid with no $U_{2,\ell+2}$-minor, which quantitative estimate for the number of elements $|M|$ holds, and for $r\ge 2$ when does equality occur?

The bound $|M|\le \ell(r-1)+1$ always holds. Moreover, if $r\ge 2$, then equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2,\ell+1}$.

Choices:
B. The bound $|M|\le \ell r+1$ always holds. Moreover, if $r\ge 2$, then equality holds if and only if $M\cong U_{2,\ell+2}$ or, more generally, $M$ is obtained by taking parallel connections of copies of $U_{2,\ell+2}$.
C. The bound $|M|\le \ell(r-1)+1$ always holds.
D. For each fixed $\ell\ge 1$, there exists an integer $r_0(\ell)$ such that whenever $r\ge r_0(\ell)$ and $M$ is a simple rank-$r$ positroid with no $U_{2,\ell+2}$-minor, one has $|M|\le \ell(r-1)+1$; moreover, for such sufficiently large $r$, equality holds if and only if $M$ can be obtained by taking parallel connections of copies of $U_{2,\ell+1}$.
E. The bound $|M|\le \ell(r-1)+1$ always holds. Moreover, if $r\ge 2$, then equality holds if and only if every long line of $M$ has exactly $\ell+1$ points.

all-ranks statement replaced by sufficiently-large-r

global parallel-connection structure replaced by local long-line condition


\begin{theorem}\label{main} Let $T>0$, $l_j>0,\ j=1,...,N$, $s\in [0,3]$ be given. Suppose $f\in W^{\frac{s}{3},1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^{\frac{6}{6-s}}(0,T;\mathbb{H}^\frac{s}{3}(\mathcal{T}))$ and
\begin{align*}
(u^0,g_0,g,p)\in \mathbb{H}^s(\mathcal{T})\times H^{\frac{s-1}{3}}(0,T)\times \left[H^\frac{s}{3}(0,T)\right]^{N}\times \left[H^{\frac{s+1}{3}}(0,T)\right]^N.
\end{align*}
satisfying the $s$-compatibility conditions, thus there exists $T^*\in (0,T)$ such that the IBVP \eqref{kdV} admits a unique solution $u \in \mathcal{X}_{s,T^*}$, where
$$\mathcal{X}_{s,T}=\left\{v \in \mathbb{B}^{*}_{s,T};\ \partial_x^ku_j\in L_x^\infty(0,l_j;H^\frac{s+1-k}{3}(0,T)),\ k=0,1,2,\ j=1,2,...,N\right\},$$
\begin{align*}
	\mathbb{B}_{s,T}^*:=\mathbb{B}_{s,T}\cap H^\frac{s}{3}(0,T;\mathbb{H}^1(\mathcal{T})).
\end{align*}
\end{theorem}

In mathematics and physics, a quantum graph is a network composed of vertices connected by edges, where each edge carries a differential or pseudo-differential equation. When every edge is endowed with a natural metric, the structure is referred to as a metric graph. A common illustration is a power network: the transmission lines act as edges connecting transformer stations (vertices). In such a case, the governing differential equations may describe the voltage along each line, with boundary conditions at the vertices ensuring that the total current flowing into and out of each vertex sums to zero.

The concept of quantum graphs traces back to the work of Linus Pauling in the 1930s, who used them to model free electrons in organic molecules. Since then, they have emerged in a wide range of contexts, including quantum chaos, waveguide analysis, photonic crystals, and Anderson localization—the suppression of wave propagation in disordered media. Quantum graphs also arise as limit models of systems composed of thin wires and serve as important theoretical tools in mesoscopic physics and nanotechnology. A simplified formulation of quantum graphs was later proposed by Freedman \textit{et al.} in \cite{Freedman}.

Beyond solving the differential equations in specific applications, the study of quantum graphs encompasses fundamental questions such as well-posedness, controllability, and identifiability. For example, controllability concerns determining the inputs required to steer the system toward a desired configuration, such as ensuring stable power distribution across a grid. Identifiability, on the other hand, focuses on optimal measurement placement—for instance, choosing pressure gauges in a water network to detect and locate potential leaks.

\subsection{Dispersive systems on graph structure}
In recent years, the study of nonlinear dispersive equations on metric graphs has attracted growing interest from mathematicians, physicists, chemists, and engineers (see \cite{BK, BlaExn08, BurCas01, K, Mug15} for detailed discussions and references). A fundamental setting for such investigations is the star graph $\mathcal{G}$, a metric graph consisting of $N$ half-lines $(0,\infty)$ meeting at a single common vertex $\nu = 0$. On each edge, one typically defines a nonlinear dispersive equation—most prominently, the nonlinear Schrödinger equation (see the works of Adami \textit{et al.} \cite{AdaNoj14, AdaNoj15} and Angulo and Goloshchapova \cite{AngGol17a, AngGol17b}).
The introduction of nonlinearities into dispersive models on such networks creates a rich framework for exploring soliton propagation, energy transfer, and nonlinear wave interactions. A central analytical difficulty arises at the vertex, where the graph may exhibit bifurcation or multi-bifurcation phenomena, especially in more intricate network topologies.

Other nonlinear dispersive systems on graphs have also been the subject of extensive study. For example, in the context of well-posedness theory, Cavalcante \cite{Cav} established local well-posedness for the Cauchy problem associated with the Korteweg–de Vries (KdV) equation on a metric star graph composed of one negative and two positive half-lines joined at a common vertex $\nu = 0$ (the so-called $\mathcal{Y}$-junction).

Another important example is the Benjamin–Bona–Mahony (BBM) equation: Bona and Cascaval \cite{bona} proved local well-posedness in the Sobolev space $H^1$, while Mugnolo and Rault \cite{Mugnolobbm} showed the existence of traveling-wave solutions on graphs. Using a different approach, Ammari and Crépeau \cite{AmCr1} derived results concerning well-posedness and stabilization of the BBM equation in star-shaped networks with bounded edges. Moreover, Mugnolo \textit{et al.} \cite{MugNoSe} provided a full characterization of boundary conditions under which the Airy-type evolution equation
$$u_t = \alpha u_{xxx} + \beta u_x, \quad \alpha \in \mathbb{R} \setminus \{0\}, \ \beta \in \mathbb{R},$$
generates contraction semigroups on star graphs. We also refer to the recent work on Airy dynamics on graphs with looping edges \cite{PavaMunoz24}, which provides a complementary perspective on this setting.
\medskip

Significant progress has also been achieved in the fields of control theory and inverse problems on networks. Ignat \textit{et al.} \cite{Ignat2011} examined inverse problems for the heat and Schrödinger equations on tree-like structures, while Baudouin and Yamamoto \cite{Baudouin} proposed a unified and simplified framework for coefficient identification problems. Furthermore, Ammari and Crépeau \cite{Ammari and Crepeau 2018}, Cerpa \textit{et al.} \cite{Cerpa, Cerpa1}, and Parada \textit{et al.} \cite{Parada2022a, Parada2022b} established stabilization and boundary controllability results for the KdV equation on star-shaped graphs, contributing substantially to the control-theoretic understanding of dispersive models on networks. Recently, in \cite{CapistranoParadaDaSilva25}, we studied the analytical structure of the entire functions associated with the KdV operator, addressing control problems involving mixed Dirichlet–Neumann boundary controls.

\subsection{Setting of problem and functional framework} In the context of graph structure, let us consider the following: given $N\geq 2$, consider a star-shaped network $\mathcal{T}$ with $N$ edges described by intervals $I_j=(0,l_j)$, $l_j>0$ for $j=1,...,N$. Denoting by $e_1,...,e_N$ the edges of $\mathcal{T}$ one has $\mathcal{T}=\bigcup_{j=1}^Ne_j$. Here, we study the local  well-posedness for a system composed of $N$ nonlinear KdV equations posed on $\mathcal{T}$ and coupled by the boundary, namely
		\begin{align}\label{kdV}
			\begin{cases}
				\partial_t u_j+\partial_x u_j+\partial_x^3 u_j+u_j\partial_x u_j=0,&t \in (0,T),\ x \in (0,l_j),\ j=1,...,N\\
				u_j(t,0)=u_1(t,0),&t\in (0,T),\ j=1,...,N\\
				\displaystyle\sum_{j=1}^N\partial_x^2 u_j(t,0)=-\alpha u_1(t,0)-\frac{N}{3}u_1^2(t,0)+g_0(t),&t \in (0,T)\\
				u_j(t,l_j)=p_j(t),\ \ \ \ \ \partial_x u_j(t,l_j)=g_j(t),&t\in (0,T),\ j=1,...,N\\
				u_j(0,x)=u_j^0(x),&x \in (0,l_j),\ j=1,...,N
			\end{cases}
		\end{align}
where $u=(u_1,...,u_N)$ stands the state of the system, $p_j$ and $g_j$ are the controls inputs and $\alpha>\frac{N}{2}$. We represent this situation in the figure \eqref{fig:network}.
		\begin{figure}[H]
			\centering
			\begin{tikzpicture}
				\draw(0, 0) circle (1mm);
				\draw[line width=0.8pt](0.02, 0.09) -- (0.5, 1.95);
				\draw[fill=black](0.53, 2.04) circle (1mm);
				\draw[line width=0.8pt](0.07, 0.07) -- (1.33, 1.5);
				\draw[fill=black](1.39, 1.57) circle (1mm);
				\draw[line width=0.8pt](0.1, 0.04) -- (1.85, 0.78);
				\draw[fill=black](1.93, 0.815) circle (1mm);
				\draw[line width=0.8pt](0.08, -0.04) -- (1.75, -1);
				\draw[fill=black](1.825, -1.045) circle (1mm);
				\draw[line width=0.8pt](0.05, -0.09) -- (1.02, -1.73);
				\draw[fill=black](1.07, -1.8) circle (1mm);
				\draw[fill=black](1.45, 0.2) circle (0.3mm);
				\draw[fill=black](1.45, 0.05) circle (0.3mm);
				\draw[fill=black](1.45, -0.1) circle (0.3mm);
				\draw node (no1) at (-0.3,0) {$0$};
				\draw node (no1) at (0.19, 1.7) {$e_1$};
				\draw node (no1) at (0.86, 1.35) {$e_2$};
				\draw node (no1) at (1.35, 0.8) {$e_3$};
				\draw node (no1) at (1.7, -0.6) {$e_{_{N-1}}$};
				\draw node (no1) at (1.06, -1.25) {$e_{_N}$};
				\draw(3.95,-0.5) circle (1mm);
				\draw node (no1) at (6.4,-0.5) {{\footnotesize \hspace{10mm} internal controlled node}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ };
				\draw[fill=black](3.95,-1) circle (1mm);
				\draw node (no1) at (6.4,-1) {{\footnotesize \hspace{13mm} external controlled node}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ };
			\end{tikzpicture}
			\label{fig:network}
			\caption{Network with $N$ edges}
	\end{figure}

Let us introduce some notation used throughout this work. From now on, we have the following:
	 \begin{itemize}
	 \item[i.] The vector $u$ is given by $u=(u_1,...,u_N) \in \mathbb{L}^2(\mathcal{T})=\displaystyle\prod_{j=1}^NL^2(0,l_j)$
	and initial data is $u^0=(u^0_1,...,u^0_N)$.
	 \item[ii.] The inner product in $\mathbb{L}^2(\mathcal{T})$ will be given by  $$\left(u,z\right)_{\mathbb{L}^2(\mathcal{T})}=\sum_{j=1}^N\int_0^{l_j}u_jz_jdx,\ \ u,z\in \mathbb{L}^2(\mathcal{T}).$$
	Moreover,  $$\mathbb{H}^s(\mathcal{T})=\displaystyle\prod_{j=1}^NH^s(0,l_j), \ \ s \in \mathbb{R}.$$
	 \end{itemize}
	\begin{itemize}
		\item[iii.] Define also the following spaces:
		$$\mathbb{H}_0^k(\mathcal{T})=\displaystyle\prod_{j=1}^{N}H_0^k(0,l_j),\ k\in \mathbb{N}$$
		and
		$$H_r^k(0,l_j)=\left\{v \in H^k(0,l_j),\ \partial_x^iv(l_j)=0,\ 0\leq i\leq k-1\right\},\ k=1,...,N,$$
		where $v^{(m)}=\frac{dv}{dx^m}$ and the index $r$ is related to the null right boundary conditions. In addition,
		$$\mathbb{H}^k_r(\mathcal{T})=\prod_{j=1}^{N}H_r^k(0,l_j)\quad \text{and} \quad \|u\|_{\mathbb{H}^k_r(\mathcal{T})}^2=\sum_{j=1}^{n}\|u_j\|_{H^k(0,l_j)}^2\ k \in \mathbb{N}.$$
		\item[iv.] Consider the following characterization:
		$$H_r^{-1}(0,l_j)=\left(H_r^{1}(0,l_j)\right)'$$
		as the dual space of $H_r(0,l_j)$ with respect to the pivot space $L^2(0,l_j)$ and $\mathbb{H}_r^{-1}$ denotes the cartesian product of $H_r^{-1}(0,l_j)$.
\item[v.] For $k\in \mathbb{N}$, the derivative $\partial_x^ku$ means the row vector
\begin{align*}
\partial_x^k u:=(\partial_x^ku_1,...,\partial_x^ku_N).
\end{align*}
The derivative $\partial_t^ku$ is understood in the same way.
\item[vi.] Under the above notation, we have
\begin{align*}
	C^k\left([0,T];\mathbb{H}^s(\mathcal{T})\right)=\prod_{j=1}^NC^k\left([0,T];H^s(0,l_j)\right),\ k=0,1,2,... \text{ \ and \ }s \in \mathbb{R}
\end{align*}
as well as
\begin{align*}
	L^p\left(0,T;\mathbb{H}^s(\mathcal{T})\right)=\prod_{j=1}^NL^p\left(0,T;H^s(0,l_j)\right),\ p\geq 1 \text{ \ and \ }s \in \mathbb{R}.
\end{align*}
Hence, for $f \in 	C^k\left([0,T];\mathbb{H}^s(\mathcal{T})\right)$ and $q \in L^p\left(0,T;\mathbb{H}^s(\mathcal{T})\right)$ we can write
\begin{align*}
f=(f_1,...,f_N)\text{ \ and \ }q=(q_1,...,q_N)
\end{align*}
with
\begin{align*}
f_j \in C^k\left([0,T];H^s(0,l_j)\right)\text{ \ and \ }q_j \in L^p\left(0,T;H^s(0,l_j)\right),\ j=1,...N.
\end{align*}
\item[vii.] We will consider the spaces
$$\mathbb{H}^k_e(\mathcal{T})=\left\{u=(u_1,...,u_N)\in \mathbb{H}^k_r(\mathcal{T});\ u_j(0)=u_k(0),\ j,k=1,...,N\right\}\ k \in \mathbb{N},$$
$$\mathbb{H}^3_c(\mathcal{T})=\left\{u=(u_1,...,u_N)\in \mathbb{H}^3(\mathcal{T});\ u_j(0)=u_k(0),\ j,k=1,...,N\right\},$$
and
$$\mathbb{B}_{s,T}=C([0,T],\mathbb{H}^s(\mathcal{T}))\cap L^2(0,T,\mathbb{H}^{s+1}(\mathcal{T})),$$
with the norm
$$\displaystyle\|u\|_{\mathbb{B}_{s,T}}:=\|u\|_{C([0,T],\mathbb{H}^s(\mathcal{T}))}+\|u\|_{L^2(0,T,\mathbb{H}^{s+1}(\mathcal{T}))}=\max_{t\in [0,T]}\|u\|_{\mathbb{H}^s(\mathcal{T})}+\left(\int_0^T\|u(t,\cdot)\|_{\mathbb{H}^{s+1}}^2dt\right)^\frac{1}{2}.$$
When $s=0$ we write $\mathbb{B}_T$ instead $\mathbb{B}_{0,T}$.
\item [viii.] Once understand the appropriate spaces in each situation, we will use the notations $g=(g_1,...,g_N)$ and $p=(p_1,...,p_N)$.
\end{itemize}

\subsection{Main results and structure of the article} With the previous framework in hand, our goal here is to prove the local and global well-posedness for the initial boundary value problem (IBVP) \eqref{kdV}. The first main result establishes the local well-posedness for this system in sharp spaces.

\subsection{Dispersive systems on graph structure}
In recent years, the study of nonlinear dispersive equations on metric graphs has attracted growing interest from mathematicians, physicists, chemists, and engineers (see \cite{BK, BlaExn08, BurCas01, K, Mug15} for detailed discussions and references). A fundamental setting for such investigations is the star graph $\mathcal{G}$, a metric graph consisting of $N$ half-lines $(0,\infty)$ meeting at a single common vertex $\nu = 0$. On each edge, one typically defines a nonlinear dispersive equation—most prominently, the nonlinear Schrödinger equation (see the works of Adami \textit{et al.} \cite{AdaNoj14, AdaNoj15} and Angulo and Goloshchapova \cite{AngGol17a, AngGol17b}).
The introduction of nonlinearities into dispersive models on such networks creates a rich framework for exploring soliton propagation, energy transfer, and nonlinear wave interactions. A central analytical difficulty arises at the vertex, where the graph may exhibit bifurcation or multi-bifurcation phenomena, especially in more intricate network topologies.

Another important example is the Benjamin–Bona–Mahony (BBM) equation: Bona and Cascaval \cite{bona} proved local well-posedness in the Sobolev space $H^1$, while Mugnolo and Rault \cite{Mugnolobbm} showed the existence of traveling-wave solutions on graphs. Using a different approach, Ammari and Crépeau \cite{AmCr1} derived results concerning well-posedness and stabilization of the BBM equation in star-shaped networks with bounded edges. Moreover, Mugnolo \textit{et al.} \cite{MugNoSe} provided a full characterization of boundary conditions under which the Airy-type evolution equation
$$u_t = \alpha u_{xxx} + \beta u_x, \quad \alpha \in \mathbb{R} \setminus \{0\}, \ \beta \in \mathbb{R},$$
generates contraction semigroups on star graphs. We also refer to the recent work on Airy dynamics on graphs with looping edges \cite{PavaMunoz24}, which provides a complementary perspective on this setting.
\medskip

Let us introduce some notation used throughout this work. From now on, we have the following:
 \begin{itemize}
 \item[i.] The vector $u$ is given by $u=(u_1,...,u_N) \in \mathbb{L}^2(\mathcal{T})=\displaystyle\prod_{j=1}^NL^2(0,l_j)$
 and initial data is $u^0=(u^0_1,...,u^0_N)$.
 \item[ii.] The inner product in $\mathbb{L}^2(\mathcal{T})$ will be given by $$\left(u,z\right)_{\mathbb{L}^2(\mathcal{T})}=\sum_{j=1}^N\int_0^{l_j}u_jz_jdx,\ \ u,z\in \mathbb{L}^2(\mathcal{T}).$$
 Moreover, $$\mathbb{H}^s(\mathcal{T})=\displaystyle\prod_{j=1}^NH^s(0,l_j), \ \ s \in \mathbb{R}.$$
 \end{itemize}
 \begin{itemize}
 \item[iii.] Define also the following spaces:
 $$\mathbb{H}_0^k(\mathcal{T})=\displaystyle\prod_{j=1}^{N}H_0^k(0,l_j),\ k\in \mathbb{N}$$
 and
 $$H_r^k(0,l_j)=\left\{v \in H^k(0,l_j),\ \partial_x^iv(l_j)=0,\ 0\leq i\leq k-1\right\},\ k=1,...,N,$$
 where $v^{(m)}=\frac{dv}{dx^m}$ and the index $r$ is related to the null right boundary conditions. In addition,
 $$\mathbb{H}^k_r(\mathcal{T})=\prod_{j=1}^{N}H_r^k(0,l_j)\quad \text{and} \quad \|u\|_{\mathbb{H}^k_r(\mathcal{T})}^2=\sum_{j=1}^{n}\|u_j\|_{H^k(0,l_j)}^2\ k \in \mathbb{N}.$$
 \item[iv.] Consider the following characterization:
 $$H_r^{-1}(0,l_j)=\left(H_r^{1}(0,l_j)\right)'$$
 as the dual space of $H_r(0,l_j)$ with respect to the pivot space $L^2(0,l_j)$ and $\mathbb{H}_r^{-1}$ denotes the cartesian product of $H_r^{-1}(0,l_j)$.
\item[v.] For $k\in \mathbb{N}$, the derivative $\partial_x^ku$ means the row vector
\begin{align*}
\partial_x^k u:=(\partial_x^ku_1,...,\partial_x^ku_N).
\end{align*}
The derivative $\partial_t^ku$ is understood in the same way.
\item[vi.] Under the above notation, we have
\begin{align*}
 C^k\left([0,T];\mathbb{H}^s(\mathcal{T})\right)=\prod_{j=1}^NC^k\left([0,T];H^s(0,l_j)\right),\ k=0,1,2,... \text{ \ and \ }s \in \mathbb{R}
\end{align*}
as well as
\begin{align*}
 L^p\left(0,T;\mathbb{H}^s(\mathcal{T})\right)=\prod_{j=1}^NL^p\left(0,T;H^s(0,l_j)\right),\ p\geq 1 \text{ \ and \ }s \in \mathbb{R}.
\end{align*}
Hence, for $f \in C^k\left([0,T];\mathbb{H}^s(\mathcal{T})\right)$ and $q \in L^p\left(0,T;\mathbb{H}^s(\mathcal{T})\right)$ we can write
\begin{align*}
f=(f_1,...,f_N)\text{ \ and \ }q=(q_1,...,q_N)
\end{align*}
with
\begin{align*}
f_j \in C^k\left([0,T];H^s(0,l_j)\right)\text{ \ and \ }q_j \in L^p\left(0,T;H^s(0,l_j)\right),\ j=1,...N.
\end{align*}
\item[vii.] We will consider the spaces
$$\mathbb{H}^k_e(\mathcal{T})=\left\{u=(u_1,...,u_N)\in \mathbb{H}^k_r(\mathcal{T});\ u_j(0)=u_k(0),\ j,k=1,...,N\right\}\ k \in \mathbb{N},$$
$$\mathbb{H}^3_c(\mathcal{T})=\left\{u=(u_1,...,u_N)\in \mathbb{H}^3(\mathcal{T});\ u_j(0)=u_k(0),\ j,k=1,...,N\right\},$$
and
$$\mathbb{B}_{s,T}=C([0,T],\mathbb{H}^s(\mathcal{T}))\cap L^2(0,T,\mathbb{H}^{s+1}(\mathcal{T})),$$
with the norm
$$\displaystyle\|u\|_{\mathbb{B}_{s,T}}:=\|u\|_{C([0,T],\mathbb{H}^s(\mathcal{T}))}+\|u\|_{L^2(0,T,\mathbb{H}^{s+1}(\mathcal{T}))}=\max_{t\in [0,T]}\|u\|_{\mathbb{H}^s(\mathcal{T})}+\left(\int_0^T\|u(t,\cdot)\|_{\mathbb{H}^{s+1}}^2dt\right)^\frac{1}{2}.$$
When $s=0$ we write $\mathbb{B}_T$ instead $\mathbb{B}_{0,T}$.
\item [viii.] Once understand the appropriate spaces in each situation, we will use the notations $g=(g_1,...,g_N)$ and $p=(p_1,...,p_N)$.
\end{itemize}

\subsection{Main results and structure of the article} With the previous framework in hand, our goal here is to prove the local and global well-posedness for the initial boundary value problem (IBVP) \eqref{kdV}. The first main result establishes the local well-posedness for this system in sharp spaces.

\subsection{Main results and structure of the article} With the previous framework in hand, our goal here is to prove the local and global well-posedness for the initial boundary value problem (IBVP) \eqref{kdV}. The first main result establishes the local well-posedness for this system in sharp spaces.

\begin{lemma}\label{lemma for v}
 Let $(u^0,g_0, g,p)\in \mathbb{X}_{3,T}$, $f\in W^{1,1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^2(0,T;\mathbb{H}^1(\mathcal{T}))$ and $v$ the solution of \eqref{v_j}. Then for $\beta=0,1,2,3$ we have:
 \begin{enumerate}
 \item[i.] The function
 \begin{align*}
 \begin{array}{rcl}
 [0,T]&\rightarrow&\mathcal{D}'(0,l_j)\\
 s&\mapsto&\partial_x^\beta v_j(s,\cdot)
 \end{array}
 \end{align*}
 is integrable in $[0,T]$.
 \item[ii.]For any $t \in [0,T]$ we have $\displaystyle\int_0^tv_j(s,\cdot)ds \in H^4(0,l_j)$ and for $\beta=0,1,2,3$ we have
 \begin{align*}
 \partial_x^\beta\int_0^tv_j(s,\cdot)ds=\int_0^t\partial_x^\beta v_j(s,\cdot)ds.
 \end{align*}
 \end{enumerate}
\end{lemma}
The previous lemma can give us the regularity for $u$.
\begin{proposition}\label{W.P._s=3}
Let $(u^0,g_0,g,p) \in \mathbb{H}^3(\mathcal{T})\times H^{\frac{3-1}{3}}(0,T)\times [H^\frac{3}{3}(0,T)]^N\times [H^{\frac{3+1}{3}}(0,T)]^N$ satisfying the linear compatibility conditions for $s=3$ and $f\in W^{1,1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^2(0,T;\mathbb{H}^1(\mathcal{T}))$ be. For $j=1,...,N$ the function $t\mapsto u_j(t,\cdot)$ given by \eqref{u_j} satisfies the following items:
\begin{enumerate}
 \item[i.] $u_j \in C^1([0,T],L^2(0,l_j))\cap C([0,T];H^3(0,l_j))$;
 \item[ii.] $u\in \mathbb{B}_{3,T}=C([0,T];\mathbb{H}^3(\mathcal{T}))\cap L^2(0,T;\mathbb{H}^4(\mathcal{T}))$ is the unique classical solution to the problem \eqref{LkdV}; and
 \begin{equation*}
 \begin{split}
 \|u\|_{\mathbb{B}_{3,T}}\leq &C\left( \|u^0\|_{\mathbb{H}^3(\mathcal{T})}+\|g_0\|_{H^\frac{3-1}{3}(0,T)}+\|g\|_{[H^\frac{3}{3}(0,T)]^N}\right.\\
 &\left.+\|p\|_{[H^\frac{3+1}{3}(0,T)]^N}+\|f\|_{W^{1,1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^2(0,T;\mathbb{H}^1(\mathcal{T}))}\right)
\end{split}
 \end{equation*}
 for some constant $C>0$.
 \item[iii.] $u \in H^1(0,T;\mathbb{H}^1(\mathcal{T}))$ and
 \begin{equation*}
 \begin{split}
 \|u\|_{H^1(0,T;\mathbb{H}^1(\mathcal{T}))}\leq& C\left(\|u^0\|_{\mathbb{H}^3(\mathcal{T})}+\|g_0\|_{H^\frac{3-1}{3}(0,T)}+\|g\|_{[H^\frac{3}{3}(0,T)]^N}\right.\\ &\left.+\|p\|_{[H^\frac{3+1}{3}(0,T)]^N}+\|f\|_{W^{1,1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^2(0,T;\mathbb{H}^1(\mathcal{T}))}\right),
 \end{split}
 \end{equation*}
 for some constant $C>0$.
 \item [iv.] $\partial_x^ku_j\in L_x^\infty\left([0,l_j];H^\frac{3+1-k}{3}(0,T)\right)$ and
 \begin{align*}
 \|\partial_x^ku_j\|_{L_x^\infty\left([0,l_j];H^\frac{3+1-k}{3}(0,T)\right)}\leq& C\left(\|u^0\|_{\mathbb{H}^3(\mathcal{T})}+\|g_0\|_{H^\frac{3-1}{3}(0,T)}+\|g\|_{[H^\frac{3}{3}(0,T)]^N}\right.\\ &\left.+\|p\|_{[H^\frac{3+1}{3}(0,T)]^N}+\|f\|_{W^{1,1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^2(0,T;\mathbb{H}^1(\mathcal{T}))}\right).
 \end{align*}
\end{enumerate}
\end{proposition}
\begin{proof}
Consider $j \in\{1,...,N\}$.
\vspace{0.2 cm}
\noindent\textbf{i.} Since $v_j\in C([0,T],L^2(0,l_j))$ it is immediate that by \eqref{u_j} that $u_j \in C^{1}([0,T],L^2(0,l_j))$.

\section{Local well-posedness of the nonlinear problem}\label{sec4}
Now, we are in a position to prove the first main result of the article, namely, the local well-posedness of the following problem
\begin{align}\label{NLKdV}
 \left\{
 \begin{array}{ll}
 \partial_t u_j+\partial_x [(a_j+1)u_j]+\partial_x^3 u_j+u_j\partial_x u_j=f_j,&t \in (0,T),\ x \in (0,l_j),\\
 u_j(t,0)=u_1(t,0),&t\in (0,T),\\
 \displaystyle\sum_{j=1}^N\partial_x^2 u_j(t,0)=-\alpha u_1(t,0)-\frac{N}{3}(u_1+a_1)^2(t,0)+g_0(t),&t \in (0,T)\\
 u_j(t,l_j)=p_j(t),\ \ \ \ \ \partial_x u_j(t,l_j)=g_j(t),&t\in (0,T),\\
 u_j(0,x)=u_j^0(x),&x \in (0,l_j),
 \end{array}
 \right.
\end{align}
where $j=1,...,N$ and $a$ is a given function. Remember that: for $s\geq 0$ and $T>0$ the space
\begin{align*}
\mathcal{X}_{s,T}=\left\{v \in \mathbb{B}_{s,T}^*;\ \partial_x^ku_j\in L_x^\infty(0,l_j;H^\frac{s+1-k}{3}(0,T)),\ k=0,1,2,\ j=1,2,...,N\right\}
\end{align*}
is a Banach space with the norm
\begin{align*}
\|v\|_{\mathcal{X}_{s,T}}=\|v\|_{\mathbb{B}_{s,T}^*}+\sum_{k=0}^2\|\partial_x^kv\|_{\prod_{j=1}^NL_x^\infty(0,l_j;H^\frac{s+1-k}{3}(0,T))}.
\end{align*}
Note that \eqref{kdV} is a particular case of \eqref{NLKdV} with $a=0$.
\begin{proof}[Proof of Theorem \ref{main}] Fix $\theta \in (0,T)$ and define $\Gamma_a:\mathcal{X}_{s,\theta}\rightarrow \mathcal{X}_{s,\theta}$ putting, for each $u \in \mathcal{X}_{s,\theta}$, $\Gamma_au$ as being the solution $v$ of
\begin{align*}
 \left\{
 \begin{array}{ll}
 \partial_t v_j+\partial_x [(a_j+1)v_j]+\partial_x^3 v_j=-u_j\partial_x u_j+f_j,&t \in (0,\theta),\ x \in (0,l_j),\ j=1,...,N,\\
 v_j(t,0)=v_1(t,0),&t\in (0,\theta),\ j=1,...,N,\\
 \displaystyle\sum_{j=1}^N\partial_x^2 v_j(t,0)=-\alpha v_1(t,0)-\frac{N}{3}(u_1+a_1)^2(t,0)+g_0(t),&t \in (0,\theta,)\\
 v_j(t,l_j)=p_j(t),\ \ \ \ \ \partial_x v_j(t,l_j)=g_j(t),&t\in (0,\theta),\ j=1,...,N,\\
 v_j(0,x)=u_j^0(x),&x \in (0,l_j),\ j=1,...,N.
 \end{array}
 \right.
\end{align*}
From Proposition \ref{W.P.LKdV-a} we have
\begin{align*}
& \|\Gamma_au\|_{\mathcal{X}_{s,\theta}}\leq C\left(\|u^0\|_{\mathbb{H}^s(\mathcal{T})}+\|g_0-\frac{N}{3}(u_1+a_1)^2(\cdot,0)\|_{H^\frac{s-1}{3}(0,\theta)}+\|g\|_{[H^\frac{s}{3}(0,\theta)]^N}\right.\\
&\left.+\|p\|_{[H^\frac{s+1}{3}(0,\theta)]^N}+\|u\partial_xu\|_{W^{\frac{s}{3},1}(0,\theta;\mathbb{L}^2(\mathcal{T}))\cap L^\frac{6}{6-s}(0,\theta;\mathbb{H}^\frac{s}{3}(\mathcal{T}))}+\|f\|_{W^{\frac{s}{3},1}(0,\theta;\mathbb{L}^2(\mathcal{T}))\cap L^\frac{6}{6-s}(0,\theta;\mathbb{H}^\frac{s}{3}(\mathcal{T}))} \right).
\end{align*}
Using Lemma \ref{lemma 2.4} we obtain
\begin{align*}
 \|\partial_xu v\|_{W^{\frac{s}{3},1}(0,\theta;\mathbb{L}^2(\mathcal{T}))\cap L^\frac{6}{6-s}(0,\theta;\mathbb{H}^\frac{s}{3}(\mathcal{T}))}\leq C(\theta^\frac{1}{2}+\theta^\frac{1}{3})\|u\|_{\mathbb{B}_{s,\theta}^*}\|u\|_{\mathbb{B}_{s,\theta}^*}.
\end{align*}
Furthermore, Lemma \ref{lemma 2.1} gives us
\begin{align*}
\left\|\frac{N}{3}(u_1+a_1)^2(\cdot,0)\right\|_{H^\frac{s-1}{3}(0,\theta)}&\leq C\theta^\alpha\frac{N}{3}\left\|(u_1+a_1)(\cdot,0)\right\|_{H^\frac{s+1}{3}(0,\theta)}^2\\
&\leq C\theta^\alpha\frac{N}{3}\|u+a\|_{\mathcal{X}_{s,\theta}}^2\\
&\leq \frac{4}{3}CN\theta^\alpha \left(\|u\|_{\mathcal{X}_{s,\theta}}^2+\|a\|_{\mathcal{X}_{s,\theta}}^2\right),
\end{align*}
where $\alpha>0$ is a constant. Thus,
\begin{align*}
 \|\Gamma_au\|_{\mathcal{X}_{s,\theta}}\leq& C\left(\|u^0\|_{\mathbb{H}^s(\mathcal{T})}+\|g_0\|_{H^\frac{s-1}{3}(0,T)}+\|g\|_{[H^\frac{s}{3}(0,T)]^N}\right.\\
&+\|p\|_{[H^\frac{s+1}{3}(0,T)]^N}+\|f\|_{W^{\frac{s}{3},1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^\frac{6}{6-s}(0,T;\mathbb{H}^\frac{s}{3}(\mathcal{T}))}\\
&\left.+C(\theta^\frac{1}{2}+\theta^\frac{1}{3})\|u\|_{\mathcal{X}_{s,\theta}}^2+\frac{4}{3}CN\theta^\alpha \|u\|_{\mathcal{X}_{s,\theta}}^2+\frac{4}{3}CN\theta^\alpha\|a\|_{\mathcal{X}_{s,\theta}}^2 \right).
\end{align*}
Defining
\begin{align*}
r=&4C\left(\|u^0\|_{\mathbb{H}^s(\mathcal{T})}+\|g_0\|_{H^\frac{s-1}{3}(0,T)}+\|g\|_{[H^\frac{s}{3}(0,T)]^N}\right.\\
&\left.+\|p\|_{[H^\frac{s+1}{3}(0,T)]^N}+\|f\|_{W^{\frac{s}{3},1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^\frac{6}{6-s}(0,T;\mathbb{H}^\frac{s}{3}(\mathcal{T}))}\right),
\end{align*}
it follows that, for $u \in B=\{v \in \mathcal{X}_{s,\theta};\ \|v\|_{\mathcal{X}_{s,\theta}}\leq r\}$,
\begin{align}\label{eq 3.2}
\|\Gamma_au\|_{\mathcal{X}_{s,\theta}}\leq \frac{r}{4}+C^2(\theta^\frac{1}{2}+\theta^\frac{1}{3})r^2+\frac{4}{3}C^2N\theta^\alpha r^2+\frac{4}{3}C^2N\theta^\alpha\|a\|_{\mathcal{X}_{s,T}}^2.
\end{align}

The proof of the previous theorem relies on several analytical tools combined with a fixed-point argument. In addition, a new notion of $s$-compatibility adapted to the graph structure is introduced (see Section \ref{sec2}), inspired by the single-edge KdV framework in \cite{BSZ 2003}. This theorem provides a sharp local well-posedness result, along with regularity properties, for the associated functions. Furthermore, we establish the existence of global solutions at the $H^s$-norm, which enables applications to control problems on star graphs and resolves several open questions that will be addressed in a forthcoming paper. The second main result of this work can be stated as follows.

\begin{theorem}\label{global_(0,3)}
Let $T>0$ and $\lambda\geq 1$ be. Consider $(u^0,g_0,g,p)\in \mathbb{X}_{s,T}^\lambda$, where $\mathbb{X}_{s,T}^\lambda$ is the set of $s$-compatible data
$$u^0 \in \mathbb{H}^{s}(\mathcal{T}), g_0 \in H^{\frac{s+\lambda-1}{3}}(0,T),\ g \in [H^\frac{s}{3}(0,T)]^N\quad \text{and}\quad p\in [H^\frac{s+\lambda+1}{3}(0,T)]^N.$$
Thus, the problem \eqref{kdV} has a unique solution $u\in \mathcal{X}_{s,T}$ which satisfies
\begin{align*}
	\|u\|_{\mathcal{X}_{s,T}}&\leq C\left(\|(u^0,g_0,g,p)\|_{0,\lambda}\right)\left(
	\begin{array}{c}
		\|u^0\|_{\mathbb{H}^s(\mathcal{T})}+\|g_0\|_{H^\frac{s+\lambda-1}{3}(0,T)}+\|g\|_{[H^\frac{s}{3}(0,T)]^N}+\|p\|_{[H^\frac{s+\lambda+1}{3}(0,T)]^N}
	\end{array}
	\right)
\end{align*}
where $C>0$ is a positive constant.
\end{theorem}
Together, these two theorems provide, for the first time, a comprehensive framework for the well-posedness of general graph structures with boundary controls. Moreover, they extend the classical result for a single KdV equation \cite{BSZ 2003} to star-shaped graphs composed of $N$ equations.
\vspace{0.2cm}

The paper is organized as follows. In Section \ref{sec2}, we introduce a general notion of $s$-compatibility for star graphs. Building on this framework, Section \ref{sec3} addresses the well-posedness of the linear system associated with \eqref{kdV}. After that, Section \ref{sec4} is devoted to proving the local well-posedness result for the system \eqref{kdV}, giving the proof of Theorem \ref{main}. The second main result, concerning global well-posedness, in the $H^s$-regime, for the system \eqref{kdV} is established in Section \ref{sec6}.

To prove Theorem~\ref{main}, the introduction states that it "relies on several analytical tools combined with a fixed-point argument." It also mentions that "a new notion of $s$-compatibility adapted to the graph structure is introduced" (Section~\ref{sec2}), "inspired by the single-edge KdV framework in \cite{BSZ 2003}." The paper’s organization further indicates the proof route: first introduce $s$-compatibility on star graphs (Section~\ref{sec2}); then treat "the well-posedness of the linear system associated with \eqref{kdV}" (Section~\ref{sec3}); and then Section~\ref{sec4} "is devoted to proving the local well-posedness result for the system \eqref{kdV}, giving the proof of Theorem~\ref{main}."

To prove the main theorem, the introduction states that it “relies on several analytical tools combined with a fixed-point argument.” It also mentions that “a new notion of $s$-compatibility adapted to the graph structure is introduced” (Section~\ref{sec2}), “inspired by the single-edge KdV framework in \cite{BSZ 2003}.” The paper’s organization further indicates the proof route: first introduce $s$-compatibility on star graphs (Section~\ref{sec2}); then treat “the well-posedness of the linear system associated with
\begin{align}\label{kdV}
			\begin{cases}
				\partial_t u_j+\partial_x u_j+\partial_x^3 u_j+u_j\partial_x u_j=0,&t \in (0,T),\ x \in (0,l_j),\ j=1,...,N\\
				u_j(t,0)=u_1(t,0),&t\in (0,T),\ j=1,...,N\\
				\displaystyle\sum_{j=1}^N\partial_x^2 u_j(t,0)=-\alpha u_1(t,0)-\frac{N}{3}u_1^2(t,0)+g_0(t),&t \in (0,T)\\
				u_j(t,l_j)=p_j(t),\ \ \ \ \ \partial_x u_j(t,l_j)=g_j(t),&t\in (0,T),\ j=1,...,N\\
				u_j(0,x)=u_j^0(x),&x \in (0,l_j),\ j=1,...,N
			\end{cases}
		\end{align}
” (Section~\ref{sec3}); and then Section~\ref{sec4} “is devoted to proving the local well-posedness result for the system above, giving the proof of the main theorem.”

\label{main} Let $T>0$, $l_j>0,\ j=1,...,N$, $s\in [0,3]$ be given. Suppose $f\in W^{\frac{s}{3},1}(0,T;\mathbb{L}^2(\mathcal{T}))\cap L^{\frac{6}{6-s}}(0,T;\mathbb{H}^\frac{s}{3}(\mathcal{T}))$ and
\begin{align*}
(u^0,g_0,g,p)\in \mathbb{H}^s(\mathcal{T})\times H^{\frac{s-1}{3}}(0,T)\times \left[H^\frac{s}{3}(0,T)\right]^{N}\times \left[H^{\frac{s+1}{3}}(0,T)\right]^N.
\end{align*}
satisfying the $s$-compatibility conditions, thus there exists $T^*\in (0,T)$ such that the IBVP
\begin{align}\label{kdV}
			\begin{cases}
				\partial_t u_j+\partial_x u_j+\partial_x^3 u_j+u_j\partial_x u_j=0,&t \in (0,T),\ x \in (0,l_j),\ j=1,...,N\\
				u_j(t,0)=u_1(t,0),&t\in (0,T),\ j=1,...,N\\
				\displaystyle\sum_{j=1}^N\partial_x^2 u_j(t,0)=-\alpha u_1(t,0)-\frac{N}{3}u_1^2(t,0)+g_0(t),&t \in (0,T)\\
				u_j(t,l_j)=p_j(t),\ \ \ \ \ \partial_x u_j(t,l_j)=g_j(t),&t\in (0,T),\ j=1,...,N\\
				u_j(0,x)=u_j^0(x),&x \in (0,l_j),\ j=1,...,N
			\end{cases}
		\end{align}
admits a unique solution $u \in \mathcal{X}_{s,T^*}$, where
$$\mathcal{X}_{s,T}=\left\{v \in \mathbb{B}^{*}_{s,T};\ \partial_x^ku_j\in L_x^\infty(0,l_j;H^\frac{s+1-k}{3}(0,T)),\ k=0,1,2,\ j=1,2,...,N\right\},$$
\begin{align*}
	\mathbb{B}_{s,T}^*:=\mathbb{B}_{s,T}\cap H^\frac{s}{3}(0,T;\mathbb{H}^1(\mathcal{T})).
\end{align*}

Question:
Let \(T>0\), \(l_j>0\) for \(j=1,\dots,N\), and \(s\in[0,3]\). Let \(\mathcal T\) be the star-shaped network with edges \((0,l_j)\), and write \(\mathbb L^2(\mathcal T)=\prod_{j=1}^N L^2(0,l_j)\) and \(\mathbb H^s(\mathcal T)=\prod_{j=1}^N H^s(0,l_j)\). Assume
\[
f\in W^{\frac{s}{3},1}(0,T;\mathbb L^2(\mathcal T))\cap L^{\frac{6}{6-s}}(0,T;\mathbb H^{\frac{s}{3}}(\mathcal T)),
\]
and
\[
(u^0,g_0,g,p)\in \mathbb H^s(\mathcal T)\times H^{\frac{s-1}{3}}(0,T)\times [H^{\frac{s}{3}}(0,T)]^N\times [H^{\frac{s+1}{3}}(0,T)]^N
\]
satisfy the \(s\)-compatibility conditions. Consider, for \(j=1,\dots,N\), the initial-boundary value problem
\[
\begin{cases}
\partial_t u_j+\partial_x u_j+\partial_x^3 u_j+u_j\partial_x u_j=0,& t\in(0,T),\ x\in(0,l_j),\\
u_j(t,0)=u_1(t,0),& t\in(0,T),\\
\displaystyle\sum_{j=1}^N \partial_x^2 u_j(t,0)=-\alpha u_1(t,0)-\frac{N}{3}u_1^2(t,0)+g_0(t),& t\in(0,T),\\
u_j(t,l_j)=p_j(t),\quad \partial_x u_j(t,l_j)=g_j(t),& t\in(0,T),\\
u_j(0,x)=u_j^0(x),& x\in(0,l_j).
\end{cases}
\]
Also set
\[
\mathbb B_{s,\tau}=C([0,\tau],\mathbb H^s(\mathcal T))\cap L^2(0,\tau;\mathbb H^{s+1}(\mathcal T)),
\]
\[
\mathbb B^*_{s,\tau}=\mathbb B_{s,\tau}\cap H^{\frac{s}{3}}(0,\tau;\mathbb H^1(\mathcal T)),
\]
and
\[
\mathcal X_{s,\tau}=\left\{v\in \mathbb B^*_{s,\tau}:\ \partial_x^k v_j\in L_x^{\infty}(0,l_j;H^{\frac{s+1-k}{3}}(0,\tau)),\ k=0,1,2,\ j=1,\dots,N\right\}.
\]
Which statement about existence and uniqueness is valid for this problem?

There exists \(T^*\in(0,T)\) such that the above initial-boundary value problem admits a unique solution \(u\in \mathcal X_{s,T^*}\).

Choices:
B. There exists \(T^*\in(0,T)\), depending only on \(T,s,N\) and the lengths \(l_j\), such that for every admissible \((u^0,g_0,g,p,f)\) satisfying the \(s\)-compatibility conditions, the above initial-boundary value problem admits a unique solution \(u\in \mathcal X_{s,T^*}\).
C. There exists \(T^*\in(0,T)\) such that the above initial-boundary value problem admits at most one solution in \(\mathcal X_{s,T^*}\).
D. For every \(T>0\), the above initial-boundary value problem admits a unique solution \(u\in \mathcal X_{s,T}\).
E. There exists \(T^*\in(0,T)\) such that the above initial-boundary value problem admits a unique solution \(u\in \mathbb B^*_{s,T^*}\).


\begin{thm}\label{thm:1.1}
Given $\phi\in C(\partial\Omega)$, there exist a unique harmonic function $u_\phi$ on $\Omega$ and an extension $\Phi\in C(\clos{\Omega})$ of $\phi$ such that $\Phi-u_\phi\in H^1_0(\Omega)$.
\end{thm}

Throughout we suppose that $\Omega$ is a connected bounded open subset of $\mathbb{R}^d$, $d\ge 2$, with boundary $\partial\Omega$.
Given a function $\varphi\in C(\partial\Omega)$, the \textbf{\boldmath Dirichlet problem\unboldmath} asks to
\begin{equation}
 \tag*{$D(\varphi)$}
 \label{eq:D}
 \left\{
 \begin{aligned}
 &\text{find $u\in C(\overline{\Omega})$ which is harmonic on $\Omega$} \\
 &\text{such that $\restrict{u}{\partial\Omega}=\varphi$.}
 \end{aligned}
 \right.
\end{equation}
We call such a function $u$ a \textbf{\boldmath classical solution\unboldmath} of~\ref{eq:D}.
However, a classical solution does not always exist.

In this paper we consider a generalised solution of the Dirichlet problem which is described as follows.

\begin{equation}
  \tag*{$D(\varphi)$}
  \label{eq:D}
  \left\{
    \begin{aligned}
      &\text{find $u\in C(\clos{\Omega})$ which is harmonic on $\Omega$} \\
      &\text{such that $\restrict{u}{\partial\Omega}=\phi$.}
    \end{aligned}
  \right.
\end{equation}

Throughout we suppose that $\Omega$ is a connected bounded open subset of $\RR^d$, $d\ge 2$, with boundary $\partial\Omega$.
Given a function $\phi\in C(\partial\Omega)$, the \emphdef{Dirichlet problem} asks to
\begin{equation}
 \tag*{$D(\varphi)$}
 \label{eq:D}
 \left\{
 \begin{aligned}
 &\text{find $u\in C(\clos{\Omega})$ which is harmonic on $\Omega$} \\
 &\text{such that $\restrict{u}{\partial\Omega}=\phi$.}
 \end{aligned}
 \right.
\end{equation}
We call such a function $u$ a \emphdef{classical solution} of~\ref{eq:D}.
However, a classical solution does not always exist.

In this section we develop variational solutions of the Dirichlet problem on the basis of the following result~\cite[Corollary~4.3]{AtES24}.
\begin{thm}[Extension result]\label{thm:2.1}
Let $\phi\in C(\partial\Omega)$. Then there exist an extension $\Phi\in C(\clos{\Omega})$ of $\phi$ and $M \ge 0$ such that
\begin{equation}\label{eq:2.1}
 \abs*{\int_\Omega\Phi\Delta w\dx} \le M\norm{w}_{H^1(\Omega)}
\end{equation}
for all $w\in C^\infty_\cpt(\Omega)$. This means that $\Delta\Phi\in H^{-1}(\Omega)$, the dual space of $H^1_0(\Omega)$.
\end{thm}
Note that for $u\in H^1(\Omega)$ one has
\[
 \abs*{\int_\Omega u\Delta w\dx} = \abs*{-\int_\Omega\nabla u\nabla w\dx}\le\norm{u}_{H^1(\Omega)}\norm{w}_{H^1(\Omega)}
\]
for all $w\in C^\infty_\cpt(\Omega)$ and therefore $\Delta u\in H^{-1}(\Omega)$.

Let $\phi\in C(\partial\Omega)$ and $\Phi\in C(\clos{\Omega})$ be an extension of $\phi$ satisfying~\eqref{eq:2.1}.
By the Riesz--Fréchet representation theorem there exists a unique $v\in H^1_0(\Omega)$ such that
\[
 \int_\Omega \Phi\Delta w\dx = -\int_\Omega\nabla v\nabla w\dx = \int_\Omega v\Delta w\dx
\]
for all $w\in C^\infty_\cpt(\Omega)$. This means that $u:=\Phi-v$ satisfies $\Delta u=0$ weakly. Hence $u\in C^\infty(\Omega)$ by Weyl's theorem and $\Delta u=0$ in the classical sense;
that is, $u$ is a harmonic function.
We call $u$ the \emphdef{variational solution} of~\ref{eq:D} and set $u_\phi := u$.
To do so, we have to show that the function does not depend on the extension $\Phi$ of $\phi$. This follows from the next lemma.

Next we talk about the Dirichlet principle.
We start by mentioning that the variational solution $u_\phi$ is obtained by a minimising process.
In fact, let $\phi\in C(\partial\Omega)$ and $\Phi\in C(\clos{\Omega})$ be an extension of $\phi$ satisfying the estimate~\eqref{eq:2.1}, i.e.\ $\Delta\Phi\in H^{-1}(\Omega)$.
The solution $v\in H^1_0(\Omega)$ of $\Delta v=\Delta\Phi$ is the unique function $v\in H^1_0(\Omega)$ such that
\begin{equation}\label{eq:2.5}
 \frac{1}{2}\int_\Omega\abs{\nabla v}^2\dx + \langle\Delta\Phi,v\rangle = \min\Bigl\{\frac{1}{2}\int_\Omega\abs{\nabla w}^2\dx + \langle\Delta\Phi,w\rangle \setcolon w\in H^1_0(\Omega)\Bigr\}
\end{equation}
(see e.g.~\cite[Theorem~4.24]{AU23}).
Now we can characterise when $\int_\Omega\abs{\nabla u_\phi}^2\dx<\infty$; i.e.\ when $u_\phi$ has finite energy.
Since $u_\phi\in C^\infty(\Omega)\cap C_\bdd(\Omega)$, this is equivalent to saying that $u_\phi\in H^1(\Omega)$.

\begin{thm}\label{thm:2.7}
Let $\phi\in C(\partial\Omega)$. The following statements are equivalent.
\begin{romanenum}
\item\label{en:t2.7-i}
The variational solution $u_\phi$ of~\ref{eq:D} is an element of $H^1(\Omega)$.
\item\label{en:t2.7-ii}
There exists $\Phi\in C(\clos{\Omega})\cap H^1(\Omega)$ with $\restrict{\Phi}{\partial\Omega}=\phi$.
\end{romanenum}
In that case
\[
 \int_\Omega\abs{\nabla u_\phi}^2\dx = \min\Bigl\{\int_\Omega\abs{\nabla w}^2\dx \setcolon w\in H^1(\Omega),\ \Phi-w\in H^1_0(\Omega)\Bigr\}.
\]
\end{thm}
\begin{proof}
\begin{parenum}
\item[\ref{en:t2.7-ii}$\Rightarrow$\ref{en:t2.7-i}.]
Since $\Phi\in H^1(\Omega)$, $u_\phi=\Phi-v\in H^1(\Omega)$, where $v\in H^1_0(\Omega)$ is such that $\Delta v=\Delta\Phi$.
Moreover, by~\eqref{eq:2.5} and considering $W:=\{w\in H^1(\Omega)\setcolon \Phi-w\in H^1_0(\Omega)\}$,
\begin{align*}
 \frac{1}{2}[u_\phi,u_\phi] -\frac{1}{2}[\Phi,\Phi] &= \frac{1}{2}[v,v]-[\Phi,v] \\
 &= \min\Bigl\{\frac{1}{2}[\Phi-w,\Phi-w] - [\Phi,\Phi-w] \setcolon w\in W\Bigr\} \\
 &= \min\Bigl\{\frac{1}{2}[w,w] - \frac{1}{2}[\Phi,\Phi] \setcolon w\in W\Bigr\} \\
 &= \frac{1}{2} \min\Bigl\{[w,w] \setcolon w\in W\Bigr\}-\frac{1}{2}[\Phi,\Phi].
\end{align*}
Thus $[u_\phi,u_\phi] = \min\{[w,w]\setcolon w\in W\}$ and this property characterises $u_\phi$.
\item[\ref{en:t2.7-i}$\Rightarrow$\ref{en:t2.7-ii}.]
Let $\Phi\in C(\clos{\Omega})$ be such that $\restrict{\Phi}{\partial\Omega}=\phi$ and $\Delta\Phi\in H^{-1}(\Omega)$.
Let $v\in H^1_0(\Omega)$ be such that $\Delta v=\Delta\Phi$.
Then $u_\phi=\Phi-v$.
Since $u_\phi\in H^1(\Omega)$ by hypothesis, it follows that $\Phi\in H^1(\Omega)$.\qedhere
\end{parenum}
\end{proof}

If $\Omega$ is more regular, the following holds.
\begin{thm}\label{thm:2.9}
Assume that $\Omega$ has Lipschitz boundary and let $\phi\in C(\partial\Omega)$. The following statements are equivalent.
\begin{romanenum}
\item $u_\phi\in H^1(\Omega)$
\item $\phi\in H^{1/2}(\partial\Omega)$
\end{romanenum}
\end{thm}
Here, since $\Omega$ has Lipschitz boundary, $H^{1/2}(\partial\Omega)=\{\operatorname{tr}\Phi \setcolon \Phi\in H^1(\Omega)\}$, where $\operatorname{tr}\colon H^1(\Omega)\to L^2(\partial\Omega,\mathcal{H}^{d-1})$ is the trace operator and $\mathcal{H}^{d-1}$ denotes the $(d-1)$-dimensional Hausdorff measure.
If $\phi\in C(\partial\Omega)\cap H^{1/2}(\partial\Omega)$, there even exists a $\Phi\in C(\clos{\Omega})\cap H^1(\Omega)$ such that $\restrict{\Phi}{\partial\Omega}=\phi$.
We refer to~\cite[Theorem~1.2]{AtE19:dp} for this and a proof of Theorem~\ref{thm:2.9}.
A more general result on very rough domains is~\cite[Theorem~5.8]{AtES24}.

Theorem~\ref{thm:5.1} is contained in Landkof~\cite[p.\,243, lines 1--5]{La72}.
We will give a very different proof based on the following result. We assume that $d\ge 3$.
\begin{thm}\label{thm:5.2}
Let $z_0\in \partial\Omega$ be a singular point of $\Omega$.
Then there exists an open set $\Omega_0\subset\RR^d$ such that $\clos{\Omega_0}\subset\Omega\cup\{z_0\}$, $z_0\in\partial\Omega_0$ and $z_0$ is a singular point of $\Omega_0$.
\end{thm}
We postpone the proof of Theorem~\ref{thm:5.2} but note that in the case of Lebesgue's domain it readily follows from its construction.
More precisely, given the potential $V\colon D\to(0,\infty)$ from Lemma~\ref{lem:lebesgue-potential-basics}, we had $\Omega=\{x\in D\setcolon \frac{1}{2}<V(x)<2\}$. We can choose e.g.~$\Omega_0=\{x\in D\setcolon \frac{1}{c}<V(r,z)<c\}$ for any $c\in (1,2)$.
Now we can easily prove Theorem~\ref{thm:5.1} using the following criterion for a boundary point to be regular, see~\cite[Theorem~6.74]{AU23}.
\begin{prop}\label{prop:5.4}
Let $z_0\in \partial\Omega$. Then $z_0$ is regular if there exists a \emphdef{global $H^1$-barrier} $b$ at $z_0$; i.e.\ a function $b\in C(\clos{\Omega})\cap H^1(\Omega)$ such that $b(x)>0$ for all $x\in\clos{\Omega}\setminus\{z_0\}$, $b(z_0)=0$ and $-\Delta b\ge 0$ weakly on $\Omega$.
\end{prop}
\begin{proof}[Proof of Theorem~\ref{thm:5.1}]
Let $z_0\in \partial\Omega$ be a singular point for $\Omega$.
By Theorem~\ref{thm:5.2} there exists an open set $\Omega_0\subset\RR^d$ such that $\clos{\Omega_0}\subset\Omega\cup\{z_0\}$ and $z_0\in\partial\Omega_0$ is a singular point with respect to $\Omega_0$.
Let $0\le\phi\in C(\partial\Omega)$ be such that $\phi(z_0)=0$ and $\phi(w)>0$ at a regular point $w\in \partial\Omega$.
Then there exists $0\le\Phi_0\in C^\infty_\cpt(\RR^d)$ such that $\phi_0:=\restrict{\Phi_0}{\partial\Omega}$ satisfies $\phi_0\le\phi$ and $\phi_0(w)>0$.
Let $u_\phi$ and $u_{\phi_0}$ be the variational (equivalently Perron) solutions with respect to $\Omega$.
Then $u_\phi\ge u_{\phi_0}\ge 0$ on $\Omega$ by the maximum principle, and $u_{\phi_0}\ne 0$.
Moreover, $u_{\phi_0}\in H^1(\Omega)$ by Theorem~\ref{thm:2.7}.

We call $u_\varphi$ the \textbf{\boldmath variational solution\unboldmath} of~\ref{eq:D}.
It attains the given boundary value $\varphi$ in the Sobolev sense.
The existence of such a variational solution is difficult to prove and was an open problem for general $\varphi\in C(\partial\Omega)$ until recently.

In the first part of this paper we start with a simple proof of Theorem~\ref{thm:1.1}, making use of a nontrivial extension result that we shall take as given.
We then establish diverse properties of the variational solution defined as above.
We show that it is well-defined; i.e.\ $u_\varphi$ does not depend on the extension $\Phi$ of $\varphi$ and it coincides with the classical solution of~\ref{eq:D} whenever a classical solution exists.
Moreover, we show that the weak maximum principle is valid for variational solutions.
In Section~\ref{sec:3} we will see that $u_\varphi$ coincides with the Perron solution.
This notion was defined by Perron in 1923~\cite{Per1923} and has been a subject of intense research.
One subject is the way in which $u_\varphi(x)$ attains the value $\varphi(z)$ as $x\to z$.
So Theorem~\ref{thm:1.1} can be interpreted as a new description of the behaviour of the Perron solution at the boundary.

In the second part of the paper, from Section~\ref{sec:4} on, we investigate the behaviour of $u_\varphi(x)$ at a fixed singular point $z_0\in\partial\Omega$, i.e.\ a point where $\lim_{x\to z_0} u_\varphi(x)=\varphi(z_0)$ is violated for some $\varphi\in C(\partial\Omega)$.

In a single page paper Lebesgue~\cite{Leb1913} considered a potential in $\mathbb{R}^3$ of a thin rod modelled by a simple mass distribution concentrated along a line segment.
He chooses two suitable level sets of that potential and considers the bounded, open, simply connected region enclosed by these level sets.
We call this region \textbf{\boldmath Lebesgue's domain\unboldmath}.

Lebesgue's domain has a unique singular point $z_0$ which is the tip of an inward pointing cusp, the famous \textbf{\boldmath Lebesgue cusp\unboldmath}.
Taking the intersection of Lebesgue's domain with a small ball, one obtains a simply connected domain with connected boundary and a singular point.
This is of great interest since in $\mathbb{R}^2$ each simply connected domain is \textbf{\boldmath Dirichlet regular\unboldmath}; i.e.\ $u_\varphi$ is a classical solution for all $\varphi\in C(\partial\Omega)$.
Our point is to consider the original Lebesgue domain, whose boundary has two connected components.
It demonstrates two striking features of the generalised solution $u_\varphi$.

\begin{enumerate}[1.]
\item It is impossible to deduce from the regularity properties of $\varphi$ whether or not $u_\varphi$ is a classical solution.
The boundary of Lebesgue's domain is the disjoint union of two closed sets $\Gamma_2$ and $\Gamma_{\frac{1}{2}}$ (which are level sets of the considered potential).
If $\varphi$ is constant on each of these two closed sets with two different constants, $u_\varphi$ is not a classical solution.
To prove this, the variational definition of $u_\varphi$ will be most useful.

\item The irregular behaviour of $u_\varphi(x)$ as $x\to z_0$ is a non-local and unstable phenomenon.
For $\phi_0\equiv 0$ one has $u_{\phi_0}\equiv 0$, but if we perturb $\phi_0$ by a small compactly supported positive smooth bump in the neighbourhood of some point far away from $z_0$, then $u_\varphi$ is no longer a classical solution.
This can be seen easily at Lebesgue's domain, but we prove it for a singular point of an arbitrary domain in Section~\ref{sec:5}.
\end{enumerate}

As explained above, there seems to be no reasonable condition on the boundary function $\varphi$ to ensure that $u_\varphi$ is a classical solution of~\ref{eq:D} -- even though there are always many classical solutions.
The situation is different if we instead consider the question of when $u_\varphi$ has finite energy $\int_\Omega\abs{\nabla u_\varphi}^2$.
This is the case if and only if $\varphi$ has an extension $\Phi\in C(\overline{\Omega})\cap H^1(\Omega)$, and then $u_\varphi$ may be determined by the Dirichlet principle as the unique minimiser of the Dirichlet energy.
As an easy consequence, if $\varphi$ is Lipschitz continuous, then $u_\varphi$ has finite energy.
These results are proved in Section~\ref{sec:2} along with other properties of the variational solution.

In the first part of this paper we start with a simple proof of Theorem~\ref{thm:1.1}, making use of a \emph{nontrivial extension result} that we shall take as given.

\label{thm:1.1}
Given $\varphi\in C(\partial\Omega)$, there exist a unique harmonic function $u_\varphi$ on $\Omega$ and an extension $\Phi\in C(\overline{\Omega})$ of $\varphi$ such that $\Phi-u_\varphi\in H^1_0(\Omega)$.,

Question: Let \(\Omega\subset \mathbb{R}^d\) be a connected bounded open set with \(d\ge 2\) and boundary \(\partial\Omega\). Given a boundary function \(\varphi\in C(\partial\Omega)\), which existence-and-uniqueness statement holds? (Here \(H^1_0(\Omega)\) denotes the Sobolev space of \(H^1\)-functions with zero trace on \(\partial\Omega\).)

There exists exactly one harmonic function \(u_\varphi\) on \(\Omega\) such that there exists at least one extension \(\Phi\in C(\overline{\Omega})\) of \(\varphi\) with \(\Phi|_{\partial\Omega}=\varphi\) and \(\Phi-u_\varphi\in H^1_0(\Omega)\).

Choices:
B. There exists exactly one harmonic function \(u_\varphi\) on \(\Omega\) such that for every extension \(\Phi\in C(\overline{\Omega})\) of \(\varphi\) one has \(\Phi-u_\varphi\in H^1_0(\Omega)\).
C. There exists at least one harmonic function \(u\) on \(\Omega\) and at least one extension \(\Phi\in C(\overline{\Omega})\) of \(\varphi\) such that \(\Phi|_{\partial\Omega}=\varphi\) and \(\Phi-u\in H^1_0(\Omega)\).
D. There exists exactly one function \(u_\varphi\in C(\overline{\Omega})\) that is harmonic on \(\Omega\) and satisfies \(u_\varphi|_{\partial\Omega}=\varphi\); equivalently, one may choose \(\Phi=u_\varphi\), so that \(\Phi-u_\varphi\in H^1_0(\Omega)\).
E. For every extension \(\Phi\in C(\overline{\Omega})\) of \(\varphi\), there exists exactly one harmonic function \(u_\varphi\) on \(\Omega\) such that \(\Phi-u_\varphi\in H^1(\Omega)\).


Question: Let \(\Omega\subset \mathbb{R}^d\) be a connected bounded open set with \(d\ge 2\) and boundary \(\partial\Omega\). For \(\varphi\in C(\partial\Omega)\), an extension of \(\varphi\) means a function \(\Phi\in C(\overline{\Omega})\) such that \(\Phi|_{\partial\Omega}=\varphi\), and \(H^1_0(\Omega)\) denotes the Sobolev space of \(H^1\)-functions with zero trace on \(\partial\Omega\). Under these assumptions, which existence statement holds?

For every \(\varphi\in C(\partial\Omega)\), there exists a unique harmonic function \(u_\varphi\) on \(\Omega\), and there exists an extension \(\Phi\in C(\overline{\Omega})\) of \(\varphi\) such that \(\Phi-u_\varphi\in H^1_0(\Omega)\).

Choices:
B. For every \(\varphi\in C(\partial\Omega)\), there exists a unique harmonic function \(u_\varphi\in C(\overline{\Omega})\) on \(\Omega\) such that \(u_\varphi|_{\partial\Omega}=\varphi\); equivalently, one may take the extension \(\Phi=u_\varphi\), so that \(\Phi-u_\varphi\in H^1_0(\Omega)\).
C. For every \(\varphi\in C(\partial\Omega)\), there exists a harmonic function \(u\) on \(\Omega\) and an extension \(\Phi\in C(\overline{\Omega})\) of \(\varphi\) such that \(\Phi-u\in H^1_0(\Omega)\).
D. For every extension \(\Phi\in C(\overline{\Omega})\) of every \(\varphi\in C(\partial\Omega)\), there exists a unique harmonic function \(u_\varphi\) on \(\Omega\) such that \(\Phi-u_\varphi\in H^1_0(\Omega)\).
E. For every \(\varphi\in C(\partial\Omega)\), there exists a unique function \(u_\varphi\in H^1(\Omega)\) and an extension \(\Phi\in C(\overline{\Omega})\) of \(\varphi\) such that \(\Phi-u_\varphi\in H^1_0(\Omega)\) and \(u_\varphi\) is harmonic on \(\Omega\) in the weak sense.

upgrades existence to classical boundary continuity and boundary equality

dropped uniqueness of the harmonic function

existential choice of extension replaced by all extensions

replaces harmonic-on-\(\Omega\) uniqueness claim by unique weak-\(H^1\) solution, forcing unnecessary finite-energy regularity


\begin{lemma} \label{lem1}
For $k\geq 2$ we have that
$$f_k = (p_{k}-1) f_{k-1}.$$
\end{lemma}

\begin{theorem} \label{thm:main}
Let $p_1, \ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\prod_{i=1}^k p_i$ be their product.
For a given choice of residue classes, i.e., one residue class $r_i$ for each prime $p_i$, the number of available elements in $[1,P_k]$ is $a_k$ and the number of free elements is $f_k$.
In other words, for every choice of residue classes, there are $a_k$ integers $n$ with $1\leq n \leq P_k$ such that $\gamma(n)\leq 1$ and $f_k$ integers with $\gamma(n)=0$.
\end{theorem}

The sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \cite{oeis}, is defined as $(a_k)_{k \geq 1}$ with $a_k$ being the determinant of the $k \times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones, i.e.,
$$ a_k = \det
 \begin{bmatrix}
2 & 1 & \ldots & 1 \\
1 & 3 & \ldots & 1 \\
\vdots & \vdots & \ddots & \vdots \\
1 & 1 & \ldots & p_k
\end{bmatrix}. $$
The first five elements, for $k=1, \ldots, 5$, of this sequence are
$$2, 5, 22, 140, 1448, \ldots,$$
see \cite{oeis} for a table of the first 300 elements of this sequence.
We relate this sequence to a concrete counting problem.
Let $p_1, \ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\prod_{i=1}^k p_i$ be their product.
We choose an arbitrary residue class $r_i$ for each prime $p_i$ and consider $k$ segments of arithmetic progressions restricted to the interval $[1,P_k]$, i.e., for $p_i$ and $r_i$ we consider
$$s_i = \{n : n= h\cdot p_i + r_i, h \in \NN_0 \} \cap [1,P_k]$$

For a given choice $r_1, \ldots, r_k$ of residue classes, let
$$\gamma(n)=\# \{p_i: n\equiv r_i \pmod{p_i} \}$$
for $1 \leq n \leq P_k$ be the number of residue classes in which $n$ is contained.
We introduce some notation:
\begin{itemize}
\item An integer $n$ is \emph{covered by a prime $p_i$} if it is contained in the residue class $r_i$.
\item An integer $n$ is \emph{free} if it is not contained in any residue class, i.e., $\gamma(n)=0$.
\item An integer $n$ is \emph{available} if it is contained in at most one residue class, i.e., $\gamma(n)\leq 1$.
\item An integer $n$ is \emph{occupied} if it is contained in at least two residue class, i.e., $\gamma(n)\geq 2$.
\end{itemize}
By definition, a free integer is also available.
Moreover, we define a second sequence $(f_k)_{k\geq 1}$ as follows:
$$ f_k = (-1)^{k} \det
 \begin{bmatrix}
 1 & 1 & \ldots & 1 & 1\\
2 & 1 & \ldots & 1& 1 \\
1 & 3 & \ldots & 1 & 1\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
1 & 1 & \ldots & p_{k} & 1
\end{bmatrix}. $$
The first five elements of this sequence for $k=1, \ldots, 5$ are
$$1, 2, 8, 48, 480 \ldots,$$
We have the following main result:

\begin{abstract}
The sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \cite{oeis}, is defined as $(a_k)_{k \geq 1}$ with $a_k$ being the determinant of the $k \times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones.
We relate this sequence to a concrete counting problem.
Choose an arbitrary residue class $r_i$ for each prime $p_i$ with $1 \leq i \leq k$ and set $P_k = \prod_{i=1}^k p_i$. We show that $a_k$ is the number of integers in $[1, P_k]$ that are contained in \emph{at most} one of the $k$ chosen residue classes.
Interestingly, we show that this sequence is closely related to the better known sequence $A005867$ for which we derive a novel characterisation in terms of determinants and which gives the number of integers in $[1, P_k]$ that are not contained in any of the $k$ residue classes.

\section{Introduction}
The sequence $A067549$ of The On-Line Encyclopedia of Integer Sequences \cite{oeis}, is defined as $(a_k)_{k \geq 1}$ with $a_k$ being the determinant of the $k \times k$ matrix whose diagonal contains the first $k$ prime numbers and all other elements are ones, i.e.,
$$ a_k = \det
 \begin{bmatrix}
2 & 1 & \ldots & 1 \\
1 & 3 & \ldots & 1 \\
\vdots & \vdots & \ddots & \vdots \\
1 & 1 & \ldots & p_k
\end{bmatrix}. $$
The first five elements, for $k=1, \ldots, 5$, of this sequence are
$$2, 5, 22, 140, 1448, \ldots,$$
see \cite{oeis} for a table of the first 300 elements of this sequence.
We relate this sequence to a concrete counting problem.
Let $p_1, \ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\prod_{i=1}^k p_i$ be their product.
We choose an arbitrary residue class $r_i$ for each prime $p_i$ and consider $k$ segments of arithmetic progressions restricted to the interval $[1,P_k]$, i.e., for $p_i$ and $r_i$ we consider
$$s_i = \{n : n= h\cdot p_i + r_i, h \in \NN_0 \} \cap [1,P_k]$$

For a given choice $r_1, \ldots, r_k$ of residue classes, let
$$\gamma(n)=\# \{p_i: n\equiv r_i \pmod{p_i} \}$$
for $1 \leq n \leq P_k$ be the number of residue classes in which $n$ is contained.
We introduce some notation:
\begin{itemize}
\item An integer $n$ is \emph{covered by a prime $p_i$} if it is contained in the residue class $r_i$.
\item An integer $n$ is \emph{free} if it is not contained in any residue class, i.e., $\gamma(n)=0$.
\item An integer $n$ is \emph{available} if it is contained in at most one residue class, i.e., $\gamma(n)\leq 1$.
\item An integer $n$ is \emph{occupied} if it is contained in at least two residue class, i.e., $\gamma(n)\geq 2$.
\end{itemize}
By definition, a free integer is also available.
Moreover, we define a second sequence $(f_k)_{k\geq 1}$ as follows:
$$ f_k = (-1)^{k} \det
 \begin{bmatrix}
 1 & 1 & \ldots & 1 & 1\\
2 & 1 & \ldots & 1& 1 \\
1 & 3 & \ldots & 1 & 1\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
1 & 1 & \ldots & p_{k} & 1
\end{bmatrix}. $$
The first five elements of this sequence for $k=1, \ldots, 5$ are
$$1, 2, 8, 48, 480 \ldots,$$
We have the following main result:

Moreover, our method can be generalised to counting free and available integers in arbitrary sets of $k$ arithmetic progressions generated by $k$ different primes as shown in Section \ref{sec5}.

\begin{remark}
The problem of determining the number of free integers in the interval $[1,P_k]$ with residue classes $r_i=0$ for all $i$ is already mentioned by Dickson \cite[p 439]{dickson} in the first volume of his History of the Theory of Numbers. He refers to results by A. de Polignac, J. Deschamps and H.J.S Smith for which Martin \cite{martin} provides proofs in his manuscript.
Determining the number of available integers and the connection of the two counting problems to the determinants $a_k$ and $f_k$ are new to the best of our knowledge.
\end{remark}

We aim to understand the systematic intersections in the interval $[1,P_k]$ of $k$ arithmetic progressions generated by the first $k$ prime numbers $p_1, \ldots, p_k$ and arbitrary choice of residue classes $r_1, \ldots, r_k$ with $P_k = \prod_{i=1}^k p_i$.
In particular, we want to understand whether an integer $1\leq n \leq P_k$ is not covered by any residue class, is contained in at most one or in at least two residue classes.

Assume we have generated all $k$ arithmetic progression and intersected them with the interval $[1,P_k]$ obtaining the sets
$s_i$, $1\leq i \leq k$.
For each integer $1\leq n \leq P_k$ we can now ask in how many sets $s_i$ it is contained, i.e., compute $\gamma(n)$.
We denote the number of all available integers with $av(k)$, the number of free integers with $free(k)$ and the number of integers that are contained in two or more sets with $occ(k)$, i.e.
\begin{align*}
av(k) &:= \# \{n : \gamma(n) \leq 1 \} \\
free(k) &:=\# \{n : \gamma(n) = 0 \} \\
occ(k) &:=\# \{n : \gamma(n) > 1 \}
\end{align*}
To illustrate this, we consider the cases $k=1$ and $k=2$ with $p_1=2$, $p_2=3$ as well as $P_1=2$ $P_2=6$.
It is easy to see that $occ(1)=0$, $free(1)=1$ and $av(1)=2$, because the arithmetic progression for $p_1=2$ is the first progression we consider and it contains one element in $[1,2]$.

\begin{theorem} \label{thm2}
Let $p_{j_1}, \ldots, p_{j_k}$ be $k$ distinct prime numbers and let $P_k:=\prod_{i=1}^k p_{j_i}$ be their product.
For $k\geq 2$ and a given choice of residue classes, i.e., one residue class $r_i$ for each prime $p_i$, the number of available elements in $[1,P_k]$ is $A_k(p_{j_1}, \ldots, p_{j_k})$ and the number of free elements is $F_k(p_{j_1}, \ldots, p_{j_k})$.
In other words, for every choice of residue classes, there are $A_k(p_{j_1}, \ldots, p_{j_k})$ integers $n$ with $1\leq n \leq P_k$ such that $\gamma(n)\leq 1$ and $F_k(p_{j_1}, \ldots, p_{j_k})$ integers with $\gamma(n)=0$.
\end{theorem}

Moreover, our method can be generalised to counting free and available integers in arbitrary sets of $k$ arithmetic progressions generated by $k$ different primes as shown in Section \ref{sec5}.

\begin{remark}
The problem of determining the number of free integers in the interval $[1,P_k]$ with residue classes $r_i=0$ for all $i$ is already mentioned by Dickson \cite[p 439]{dickson} in the first volume of his History of the Theory of Numbers. He refers to results by A. de Polignac, J. Deschamps and H.J.S Smith for which Martin \cite{martin} provides proofs in his manuscript.
Determining the number of available integers and the connection of the two counting problems to the determinants $a_k$ and $f_k$ are new to the best of our knowledge.
\end{remark}

\begin{remark}
Lemma \ref{lem1} below gives a new characterisation of sequence $A005867$ from the The On-Line Encyclopedia of Integer Sequences \cite{oeis2} in terms of determinants.
\end{remark}

\begin{remark}
As a side product, the proof of Theorem \ref{thm:main} provides a quick way of calculating determinants of the form $a_k$ and $f_k$.
\end{remark}

The paper is organized as follows. In Section \ref{sec2} we derive structural results about $a_k$ and $f_k$. In Section \ref{sec3} we analyse the counting problem and obtain an equation that relates the available numbers after considering $k$ progressions to the available numbers after considering $k+1$ progressions. Theorem \ref{thm:main} is proven in Section \ref{sec4} and generalised in Section \ref{sec5}.

The post-theorem introduction does not give an explicit proof outline for Theorem~\ref{thm:main}, but it indicates the structure used to prove it: Section~\ref{sec2} "derive[s] structural results about $a_k$ and $f_k$"; Section~\ref{sec3} "analyse[s] the counting problem" and "obtain[s] an equation that relates the available numbers after considering $k$ progressions to the available numbers after considering $k+1$ progressions"; then "Theorem~\ref{thm:main} is proven in Section~\ref{sec4}" (and "generalised in Section~\ref{sec5}").

The post-theorem introduction does not give an explicit proof outline for the main theorem, but it indicates the structure used to prove it: next it derives structural results about $a_k$ and $f_k$; then it analyses the counting problem and obtains an equation that relates the available numbers after considering $k$ progressions to the available numbers after considering $k+1$ progressions; then it proves the main theorem (and later generalises it).

\label{thm:main}
Let $p_1, \ldots, p_k$ be the first $k$ prime numbers and let $P_k:=\prod_{i=1}^k p_i$ be their product.
For a given choice of residue classes, i.e., one residue class $r_i$ for each prime $p_i$, the number of available elements in $[1,P_k]$ is $a_k$ and the number of free elements is $f_k$.
In other words, for every choice of residue classes, there are $a_k$ integers $n$ with $1\leq n \leq P_k$ such that $\gamma(n)\leq 1$ and $f_k$ integers with $\gamma(n)=0$.,

Question:
Let $p_1,\dots,p_k$ be the first $k$ prime numbers, and let $P_k=\prod_{i=1}^k p_i$. For each $i$, choose a residue class $r_i\pmod{p_i}$. For $1\le n\le P_k$, define
$$\gamma(n)=\#\{i: n\equiv r_i\pmod{p_i}\}.$$
Call $n$ available if $\gamma(n)\le 1$ and free if $\gamma(n)=0$. Also define
$$a_k=\det\begin{bmatrix}
2&1&\cdots&1\\
1&3&\cdots&1\\
\vdots&\vdots&\ddots&\vdots\\
1&1&\cdots&p_k
\end{bmatrix}$$
and
$$f_k=(-1)^k\det\begin{bmatrix}
1&1&\cdots&1&1\\
2&1&\cdots&1&1\\
1&3&\cdots&1&1\\
\vdots&\vdots&\ddots&\vdots&\vdots\\
1&1&\cdots&p_k&1
\end{bmatrix}.$$
Which statement holds for the integers in $[1,P_k]$, uniformly over all choices of the residue classes $r_1,\dots,r_k$?

For every choice of residue classes $r_i\pmod{p_i}$, the number of integers $n$ with $1\le n\le P_k$ and $\gamma(n)\le 1$ is exactly $a_k$, and the number of integers $n$ with $\gamma(n)=0$ is exactly $f_k$.

Choices:
B. For every choice of residue classes $r_i\pmod{p_i}$, the number of integers $n$ with $1\le n\le P_k$ and $\gamma(n)\le 1$ is exactly $a_k$, and the number of integers $n$ with $\gamma(n)=0$ is exactly $a_k-f_k$.
C. For every choice of residue classes $r_i\pmod{p_i}$, the number of integers $n$ with $1\le n\le P_k$ and $\gamma(n)=0$ is exactly $f_k$.
D. There exists a choice of residue classes $r_i\pmod{p_i}$ such that the number of integers $n$ with $1\le n\le P_k$ and $\gamma(n)\le 1$ is exactly $a_k$, and the number of integers $n$ with $\gamma(n)=0$ is exactly $f_k$.
E. For every choice of residue classes $r_i\pmod{p_i}$, the number of integers $n$ with $1\le n\le P_k$ and $\gamma(n)\le 1$ is exactly $a_k$, and the number of integers $n$ with $\gamma(n)=1$ is exactly $f_k$.

relation between available and free counts

dropped the available-count conclusion involving $a_k$

uniformity over all residue-class choices

confusion between free integers and singly covered integers


\begin{example}\label{example}
	Let $M^n$ be a closed aspherical spin manifold with $n\ge 5$, e.g., the $n$-torus. Suppose that $\pi_1(M)$ is represented by finitely many disjoint embedded circles $S^1$ in $M$, whose tubular neighborhoods are diffeomorphic to $S^1 \times B^{n-1}$. Remove these tubular neighborhoods and attach $B^2 \times S^{n-2}$ to obtain a closed manifold $M_1$. Then $M_1$ is simply connected.

	Finally, set $Z \coloneqq M_1 \# M_1$, which is a spin surgery of $M$. Note that $Z$ is simply connected, and the Dirac operator on $Z$ has vanishing $KO$-index as $Z$ is a double. Hence, $Z$ admits a metric with positive scalar curvature \cite{Stolz}.
\end{example}

The study of scalar curvature on Riemannian manifolds has played a central role in differential geometry and geometric topology for several decades. A landmark result due to Gromov and Lawson \cite{GromovLawson,GromovLawsonAnnals}, using index theory, and independently to Schoen and Yau \cite{SchoenYau3,SchoenYau}, using minimal hypersurface methods, asserts that the $n$-torus does not admit any Riemannian metric of positive scalar curvature. More generally, Gromov--Lawson--Rosenberg conjectured that no aspherical manifold admits a metric of positive scalar curvature. This conjecture is known to hold for aspherical manifolds whose fundamental groups satisfy the rational strong Novikov conjecture \cite{Rosenberg,Yu}, as well as for aspherical manifolds of dimension at most five \cite{ChodoshLi,Gromov5fold}.

The non-existence of positive scalar metric holds for more generally enlargeable manifolds, introduced by Gromov--Lawson \cite{GromovLawson,GromovLawsonAnnals}, including connected sums of torus with a closed spin manifold. Using index-theoretic techniques, Su--Zhang \cite{SuZhang} established a quantitative version of this obstruction for connected sums of enlargeable manifolds with non-spin closed manifolds. Subsequently, Wang--Zhang \cite{WangZhang} and Su \cite{SuRemark} showed that the connected sum of an enlargeable manifold with any spin manifold does not admit a metric of positive scalar curvature. Related non-existence results for positive scalar curvature on non-spin connected sums, obtained via minimal hypersurface methods, can be found in \cite{ChodoshLi,ChenChuZhu,MR4291609}.

In this paper, we extend the above non-existence of positive scalar metrics to a significantly broader class of manifolds.

\begin{abstract}
 Let $M$ be a closed aspherical manifold. Assume that the rational strong Novikov conjecture holds for $\pi_1(M)$. We show that on any spin surgery of $M$ along a region whose induced homomorphism on the fundamental group is trivial, every complete metric with non-negative scalar curvature is Ricci-flat. In particular, on the connected sum of $M$ with a spin manifold, any complete metric with non-negative scalar curvature is Ricci-flat.
\end{abstract}

The study of scalar curvature on Riemannian manifolds has played a central role in differential geometry and geometric topology for several decades. A landmark result due to Gromov and Lawson \cite{GromovLawson,GromovLawsonAnnals}, using index theory, and independently to Schoen and Yau \cite{SchoenYau3,SchoenYau}, using minimal hypersurface methods, asserts that the $n$-torus does not admit any Riemannian metric of positive scalar curvature. More generally, Gromov--Lawson--Rosenberg conjectured that no aspherical manifold admits a metric of positive scalar curvature. This conjecture is known to hold for aspherical manifolds whose fundamental groups satisfy the rational strong Novikov conjecture \cite{Rosenberg,Yu}, as well as for aspherical manifolds of dimension at most five \cite{ChodoshLi,Gromov5fold}.

The non-existence of positive scalar metric holds for more generally enlargeable manifolds, introduced by Gromov--Lawson \cite{GromovLawson,GromovLawsonAnnals}, including connected sums of torus with a closed spin manifold. Using index-theoretic techniques, Su--Zhang \cite{SuZhang} established a quantitative version of this obstruction for connected sums of enlargeable manifolds with non-spin closed manifolds. Subsequently, Wang--Zhang \cite{WangZhang} and Su \cite{SuRemark} showed that the connected sum of an enlargeable manifold with any spin manifold does not admit a metric of positive scalar curvature. Related non-existence results for positive scalar curvature on non-spin connected sums, obtained via minimal hypersurface methods, can be found in \cite{ChodoshLi,ChenChuZhu,MR4291609}.

We remark that in Theorem \ref{thm:connectedSum}, the manifold $N$ is allowed to be non-compact, and the metric on $M\#N$ is not required to have bounded geometry. The rational strong Novikov conjecture for $\pi_1(M)$ \cite[Section~2]{Rosenberg} asserts that the assembly map from the $K$-homology of $M$ to the $K$-theory of the group $C^*$-algebra of $\pi_1(M)$ is rationally injective; see Conjecture \ref{conj:Novikov} for a precise formulation. The reader may consult \cite{GongWuYu} for recent progress and a survey of classes of groups known to satisfy this conjecture. To date, no counterexample to the rational strong Novikov conjecture is known.

\begin{theorem}\label{thm:main}
 Let $M^n$ be a closed aspherical manifold and $\Omega\subset M$ a smooth region. Assume that
 $$\pi_1(\Omega)\to \pi_1(M)\textup{ is the trivial map}.$$
 Let $N^n$ be a spin manifold with boundary, and $i\colon \partial N\to \partial \Omega$ a diffeomorphism preserving the spin structure\footnote{Although $M$ itself need not be spin, the triviality of $\pi_1(\Omega)\to\pi_1(M)$ implies that $\Omega$ and $\partial\Omega$ inherit a canonical spin structure from the spin universal cover $\widetilde M$.}. Set $Z\coloneqq (M\backslash\Omega)\cup_{\partial\Omega} N$. If the rational strong Novikov conjecture holds for $\pi_1(M)$, then $Z$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $Z$ is Ricci-flat.
\end{theorem}

\begin{corollary}\label{coro:remove}
 Let $M^n$ be a closed aspherical manifold and $\Omega\subset M$ a smooth region. Assume that
 $$\pi_1(\Omega)\to \pi_1(M)\textup{ is the trivial map},$$
 and the rational strong Novikov conjecture holds for $\pi_1(M)$. Then $M\backslash\Omega$ does not carry any complete metric with positive scalar curvature.
\end{corollary}

\section{Non-existence of uniformly positive scalar curvature}\label{sec:UPSC}
In this section, we prove a simple version of Theorem \ref{thm:main} to illustrate the key ideas. This version follows from classical index-theoretic tools, while the proof of the full version requires additional new ingredients and will be presented in the next section.
\begin{proposition}\label{prop:noPSC}
 Let $M^n$ be a closed aspherical manifold and $\Omega\subset M$ a smooth region. Assume that
 $$\pi_1(\Omega)\to \pi_1(M)\textup{ is the trivial map}.$$
 Let $N^n$ be a spin manifold with boundary, and $i\colon \partial N\to \partial \Omega$ a diffeomorphism preserving the spin structure. Set $Z\coloneqq (M\backslash\Omega)\cup_{\partial\Omega} N$. If the rational strong Novikov conjecture holds for $\pi_1(M)$, then $Z$ does not carry any complete metric with uniform positive scalar curvature, i.e., scalar curvature bounded below by some positive constant.
\end{proposition}

@article{Zeidler:2019aa,
 abstract = {Let $M$ be a closed connected spin manifold such that its spinor Dirac operator has non-vanishing (Rosenberg) index. We prove that for any Riemannian metric on $V = M \times [-1,1]$ with scalar curvature bounded below by $σ> 0$, the distance between the boundary components of $V$ is at most $C_n/\sqrtσ$, where $C_n = \sqrt{(n-1)/{n}} \cdot C$ with $C < 8(1+\sqrt{2})$ being a universal constant. This verifies a conjecture of Gromov for such manifolds. In particular, our result applies to all high-dimensional closed simply connected manifolds $M$ which do not admit a metric of positive scalar curvature. We also establish a quadratic decay estimate for the scalar curvature of complete metrics on manifolds, such as $M \times \mathbb{R}^2$, which contain $M$ as a codimension two submanifold in a suitable way. Furthermore, we introduce the "$\mathcal{KO}$-width" of a closed manifold and deduce that infinite $\mathcal{KO}$-width is an obstruction to positive scalar curvature.},
 author = {Rudolf Zeidler},
 date-added = {2021-11-10 09:44:42 -0600},
 date-modified = {2021-11-10 09:44:42 -0600},
 eprint = {1905.08520},
 journal = {Journal of Differential Geometry},
 title = {Band width estimates via the {D}irac operator},
 url = {https://arxiv.org/pdf/1905.08520.pdf},
 year = {2020},
 bdsk-url-1 = {https://arxiv.org/pdf/1905.08520.pdf}}

\begin{conjecture}[Rational strong Novikov conjecture for fundamental groups of aspherical manifolds]\label{conj:Novikov}
	Let $M$ be a closed aspherical manifold and $\Gamma=\pi_1(M)$. Then $\Gamma$ satisfies the rational strong Novikov conjecture if
	\[
	\mu\colon K_*(M)\otimes\Q \to K_*(C^*_r(\Gamma))\otimes\Q
	\]
	is injective.
\end{conjecture}

\begin{theorem}\label{thm:connectedSum}
	Let $M^n$ be a closed aspherical manifold and $N$ a spin manifold. Assume that the rational strong Novikov conjecture holds for $\pi_1(M)$.
	Then the connected sum $M\# N$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $M\# N$ is Ricci-flat.
\end{theorem}

\begin{theorem}\label{thm:main}
	Let $M^n$ be a closed aspherical manifold and $\Omega\subset M$ a smooth region. Assume that
	$$\pi_1(\Omega)\to \pi_1(M)\textup{ is the trivial map}.$$
	Let $N^n$ be a spin manifold with boundary, and $i\colon \partial N\to \partial \Omega$ a diffeomorphism preserving the spin structure\footnote{Although $M$ itself need not be spin, the triviality of $\pi_1(\Omega)\to\pi_1(M)$ implies that $\Omega$ and $\partial\Omega$ inherit a canonical spin structure from the spin universal cover $\widetilde M$.}. Set $Z\coloneqq (M\backslash\Omega)\cup_{\partial\Omega} N$. If the rational strong Novikov conjecture holds for $\pi_1(M)$, then $Z$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $Z$ is Ricci-flat.
\end{theorem}

We remark that in Theorem \ref{thm:connectedSum}, the manifold $N$ is allowed to be non-compact, and the metric on $M\#N$ is not required to have bounded geometry. The rational strong Novikov conjecture for $\pi_1(M)$ \cite[Section~2]{Rosenberg} asserts that the assembly map from the $K$-homology of $M$ to the $K$-theory of the group $C^*$-algebra of $\pi_1(M)$ is rationally injective; see Conjecture \ref{conj:Novikov} for a precise formulation. The reader may consult \cite{GongWuYu} for recent progress and a survey of classes of groups known to satisfy this conjecture. To date, no counterexample to the rational strong Novikov conjecture is known.

We further generalize Theorem \ref{thm:connectedSum} to spin surgeries on aspherical manifolds.

\begin{theorem}\label{thm:main}
	Let $M^n$ be a closed aspherical manifold and $\Omega\subset M$ a smooth region. Assume that
	$$\pi_1(\Omega)\to \pi_1(M)\textup{ is the trivial map}.$$
	Let $N^n$ be a spin manifold with boundary, and $i\colon \partial N\to \partial \Omega$ a diffeomorphism preserving the spin structure\footnote{Although $M$ itself need not be spin, the triviality of $\pi_1(\Omega)\to\pi_1(M)$ implies that $\Omega$ and $\partial\Omega$ inherit a canonical spin structure from the spin universal cover $\widetilde M$.}. Set $Z\coloneqq (M\backslash\Omega)\cup_{\partial\Omega} N$. If the rational strong Novikov conjecture holds for $\pi_1(M)$, then $Z$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $Z$ is Ricci-flat.
\end{theorem}

Theorem \ref{thm:connectedSum} is a special case of Theorem \ref{thm:main} by choosing $\Omega$ to be a small ball in $M$. The assumption that $\pi_1(\Omega)$ maps trivially into $\pi_1(M)$ is essential; in Example \ref{example} we construct a spin surgery on a closed aspherical manifold that admits a metric of positive scalar curvature when this condition fails.

Our proofs rely on Dirac operator methods. A fundamental ingredient is the relative index theorem for higher indices due to Xie and Yu \cite[Theorem~A]{XieYu14}. Inspired by the approaches in \cite{SuZhang,WangXieLinf}, we construct an explicit representative of the relative higher index and compute it in the $K$-theory of the group $C^*$-algebra. We also provide a Dirac-operator-based proof of the Ricci-flatness statement, following ideas similar to those in \cite{WangZhu}.

As an immediate application, by taking $N=\partial\Omega\times[0,1)$ in Theorem \ref{thm:main}, we obtain the following corollary.

\begin{corollary}\label{coro:remove}
	Let $M^n$ be a closed aspherical manifold and $\Omega\subset M$ a smooth region. Assume that
	$$\pi_1(\Omega)\to \pi_1(M)\textup{ is the trivial map},$$
	and the rational strong Novikov conjecture holds for $\pi_1(M)$. Then $M\backslash\Omega$ does not carry any complete metric with positive scalar curvature.
\end{corollary}

The paper is organized as follows. In Section \ref{sec:preliminary}, we review the higher indices and the strong Novikov conjecture. In Section \ref{sec:UPSC}, we prove a simplified version of Theorem \ref{thm:main}, establishing the non-existence of uniformly positive scalar curvature metrics on spin surgeries. The full proof of Theorem \ref{thm:main} is given in Section \ref{sec:nnsc}.

\medskip
\noindent\textbf{Acknowledgments.}
The author would like to thank Zhizhang Xie for helpful discussions.

Our proofs rely on Dirac operator methods. A fundamental ingredient is the relative index theorem for higher indices due to Xie and Yu \cite[Theorem~A]{XieYu14}. Inspired by the approaches in \cite{SuZhang,WangXieLinf}, we construct an explicit representative of the relative higher index and compute it in the $K$-theory of the group $C^*$-algebra. We also provide a Dirac-operator-based proof of the Ricci-flatness statement, following ideas similar to those in \cite{WangZhu}. (Theorem~\ref{thm:connectedSum} is treated as a special case of Theorem~\ref{thm:main} by choosing $\Omega$ to be a small ball in $M$.)

Our proofs rely on Dirac operator methods. A fundamental ingredient is the relative index theorem for higher indices due to Xie and Yu (Theorem~A in Xie--Yu, \emph{[title not provided in ref\_dict]} (2014)). Inspired by the approaches in Su--Zhang, \emph{[title not provided in ref\_dict]}, and Wang--Xie--Linf, \emph{[title not provided in ref\_dict]}, we construct an explicit representative of the relative higher index and compute it in the $K$-theory of the group $C^*$-algebra. We also provide a Dirac-operator-based proof of the Ricci-flatness statement, following ideas similar to those in Wang--Zhu, \emph{[title not provided in ref\_dict]}. (In establishing the main theorem, we treat it as a special case of
\begin{theorem}\label{thm:main}
	Let $M^n$ be a closed aspherical manifold and $\Omega\subset M$ a smooth region. Assume that
	$$\pi_1(\Omega)\to \pi_1(M)\textup{ is the trivial map}.$$
	Let $N^n$ be a spin manifold with boundary, and $i\colon \partial N\to \partial \Omega$ a diffeomorphism preserving the spin structure\footnote{Although $M$ itself need not be spin, the triviality of $\pi_1(\Omega)\to\pi_1(M)$ implies that $\Omega$ and $\partial\Omega$ inherit a canonical spin structure from the spin universal cover $\widetilde M$.}. Set $Z\coloneqq (M\backslash\Omega)\cup_{\partial\Omega} N$. If the rational strong Novikov conjecture holds for $\pi_1(M)$, then $Z$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $Z$ is Ricci-flat.
\end{theorem}
by choosing $\Omega$ to be a small ball in $M$.)

Question:
Let $M^n$ be a closed aspherical manifold, let $N$ be a spin manifold of the same dimension $n$, and write $M\#N$ for their connected sum. Assume that the rational strong Novikov conjecture holds for $\pi_1(M)$, meaning that for $\Gamma=\pi_1(M)$ the assembly map
\[
\mu\colon K_*(M)\otimes\mathbb{Q}\to K_*(C_r^*(\Gamma))\otimes\mathbb{Q}
\]
is injective. Which of the following conclusions about complete Riemannian metrics on $M\#N$ is valid under these hypotheses?

The manifold $M\#N$ admits no complete Riemannian metric with positive scalar curvature. Moreover, if a complete Riemannian metric on $M\#N$ has non-negative scalar curvature everywhere, then that metric is Ricci-flat.

Choices:
B. The manifold $M\#N$ admits no complete Riemannian metric with non-negative scalar curvature. In particular, every complete metric on $M\#N$ must have scalar curvature negative somewhere.
C. The manifold $M\#N$ admits no complete Riemannian metric with positive scalar curvature.
D. The manifold $M\#N$ admits no complete Riemannian metric with positive scalar curvature. Moreover, if a complete Riemannian metric on $M\#N$ has non-negative scalar curvature everywhere, then that metric is flat.
E. If the manifold $M\#N$ admits no complete Riemannian metric with positive scalar curvature, then the rational strong Novikov conjecture must hold for $\pi_1(M)$. Moreover, any complete Riemannian metric on $M\#N$ with non-negative scalar curvature is Ricci-flat.