\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


\label{thm:main}
    Let $Q=\Spec(A)$ be an affine algebraic rack with unit $e\in Q$, and let $\mathfrak{q}\coloneq \Der_k(A,k)$ be the vector space of $k$-derivations of $A$. Define a Leibniz bracket $[\cdot,\cdot]\colon\mathfrak{q}\times\mathfrak{q}\to\mathfrak{q}$ by convolution with respect to the corack operation $\nabla$ of $A$ (see Section \ref{sec:nabla}):
    \[
    [D,E]\coloneq (D\otimes E)\circ\nabla.
    \]
    Then the assignment $Q\mapsto(\mathfrak{q},[\cdot,\cdot])$ defines a covariant functor $\Rack(\Aff)\to\leib$ sending morphisms of affine algebraic racks $\varphi\colon Q\to R$ to their derivatives $(d\varphi)_e\colon \mathfrak{q}\to\mathfrak{r}$ at $e$.

    Furthermore, if $G$ is an affine algebraic group and $Q=\Conj(G)$ is its conjugation quandle scheme, then $(\mathfrak{q},[\cdot,\cdot])$ is the Lie algebra of $G$. On the other hand, if $k=\mathbb{R}$ or $\mathbb{C}$ and $Q(k)$ is a Lie rack, then $(\mathfrak{q},[\cdot,\cdot])$ is the Leibniz algebra of $Q(k)$ as a Lie rack.

\begin{rmk}\label{rmk:recover}
    If $G$ is an affine algebraic group and $Q=\Conj(G)$ is its conjugation quandle scheme, then $(\q,[\cdot,\cdot])$ is the Lie algebra of $G$. To see this, note that $Q$ and $G$ have the same underlying scheme and identity $e$, so they also have the same tangent space $\q$ at $e$. By the definition of $\Conj(G)$, the adjoint representations $\Ad$ of $Q$ and $\ad$ of $\q$ are therefore the same as those of $G$ and its Lie algebra (see, for example, \cite{milne}*{Rem.\ 10.24}). Hence, the claim follows from Proposition \ref{prop:adj}.\footnote{Alternatively, one can use Example \ref{ex:corack-conj} and the Hopf algebra axioms to directly (if tediously) verify that the Leibniz bracket $[D,E]=(D\otimes E)\circ\nabla$ agrees with the usual Lie bracket $(D\otimes E-E\otimes D)\circ\Delta$ on $\q=\operatorname{Lie}(G)$ (see \cite{waterhouse}*{Sec.\ 12.2}); we leave the details to the reader. As a hint, first deduce the identity $\eps\circ S=\eps$ from the axiom $\mu\circ (S\otimes\id)\circ\Delta=\eta\circ\eps$ by post-composing with $\eps$, and then deduce from these two equalities that $D\circ S=-D$.}

    Similarly, if $k=\R$ or $\mathbb{C}$ and $Q(k)$ is an algebraic Lie rack, then our definitions of $\Ad$ and $\ad$ coincide with the ones for Lie racks in \cite{kinyon}, so $(\q,[\cdot,\cdot])$ is the Leibniz algebra of $Q(k)$ as a Lie rack.
\end{rmk}

\begin{thm}\label{thm:main}
    Let $Q=\Spec(A)$ be an affine algebraic rack with unit $e\in Q$, and let $\q\coloneq \Der_k(A,k)$ be the vector space of $k$-derivations of $A$. Define a Leibniz bracket $[\cdot,\cdot]\colon\q\times\q\to\q$ by convolution with respect to the corack operation $\nabla$ of $A$ (see Section \ref{sec:nabla}):
    \[
    [D,E]\coloneq (D\otimes E)\circ\nabla.
    \]
    Then the assignment $Q\mapsto(\q,[\cdot,\cdot])$ defines a covariant functor $\Rack(\Aff)\to\leib$ sending morphisms of affine algebraic racks $\phi\colon Q\to R$ to their derivatives $(d\phi)_e\colon \q\to\mathfrak{r}$ at $e$.

    Furthermore, if $G$ is an affine algebraic group and $Q=\Conj(G)$ is its conjugation quandle scheme, then $(\q,[\cdot,\cdot])$ is the Lie algebra of $G$. On the other hand, if $k=\R$ or $\mathbb{C}$ and $Q(k)$ is a Lie rack, then $(\q,[\cdot,\cdot])$ is the Leibniz algebra of $Q(k)$ as a Lie rack.
\end{thm}

This paper introduces \emph{algebraic racks}, which are pointed racks internal to the category of schemes of finite type over a field $k$. We show that affine algebraic racks provide a solution to a version of the well-known \emph{coquecigrue} problem for Leibniz algebras over arbitrary fields $k$.

\subsubsection{}
Loday's \cite{loday} original version of the \emph{coquecigrue} problem from 1993 seeks algebraic structures on smooth manifolds that differentiate to Leibniz algebras in a way that recovers the Lie group--Lie algebra correspondence. \emph{Leibniz algebras} are vector spaces $\mathfrak{q}$ equipped with a bilinear mapping $[\cdot,\cdot]\colon \mathfrak{q}\times\mathfrak{q}\to\mathfrak{q}$ called the \emph{Leibniz bracket} that satisfies the derivation form of the Jacobi identity
 \[
 [X,[Y,Z]]=[[X,Y],Z]+[Y,[X,Z]].
 \]
Bloh \cite{bloh} introduced Leibniz algebras in 1965 to generalize Lie algebras, which are precisely those Leibniz algebras whose Leibniz bracket is alternating. See \cite{leibniz} for an introduction to Leibniz algebras.

As a solution to the \emph{coquecigrue} problem over $\mathbb{R}$ and $\mathbb{C}$, Kinyon \cite{kinyon} in 2007 introduced \emph{Lie racks}, which are pointed rack objects (see Section \ref{sec:pro}) in the category of smooth manifolds; see the survey \cite{ongay} for further discussion. \emph{Racks} \cite{fenn} and similar structures called \emph{quandles} \citelist{\cite{joyce}\cite{matveev}} were originally introduced as invariants of knots and 3-manifolds; see \citelist{\cite{book}\cite{quandlebook}} for introductions to the theory.

\subsection{Main theorem}
Kinyon also posed an analogue of the \emph{coquecigrue} problem over arbitrary fields $k$ that replaces Lie groups with affine algebraic groups. Save for unpublished work of Ba\v{s}i\'{c} and \v{S}koda \cite{skoda} in 2008 in the case that $\operatorname{char}(k)=0$, the author is unaware of any literature in this direction.
To address this, we prove the following. Let $\Rack(\Aff)$ and $\leib$ denote the categories of affine algebraic racks over $k$ and Leibniz algebras over $k$, respectively.

\begin{proof}
 We prove the theorem piecemeal in Propositions \ref{prop:adj}, \ref{prop:jacobi}, and \ref{prop:functor} and Remark \ref{rmk:recover}.
\end{proof}

\subsubsection{}
To make Definition \ref{def:corack} more concrete, let $A$ be a $k$-algebra with (commutative and associative) multiplication $\mu\colon A\otimes A\to A$, and let $\tau\colon A\otimes A\to A\otimes A$ denote the transposition $a\otimes b\mapsto b\otimes a$. A corack algebra structure on $A$ consists of two $k$-algebra homomorphisms $\nabla,\nabla\inv\colon A\to A\otimes A$ called the \emph{corack operations} and a $k$-algebra homomorphism $\eps\colon A\to k$ called the \emph{counit}, \emph{coidentity}, or \emph{augmentation} satisfying the following five identities:
\begin{enumerate}[(C1)]
 \item (Co-invertability) $(\mathord{\id}\otimes \mu)\circ(\tau\otimes\mu)\circ(\mathrm{\id}\otimes\nabla\inv\otimes\id)\circ(\nabla\circ\id)=(\mathord{\id}\otimes \id)=(\mu\otimes\id)\circ(\mu\otimes\tau)\circ(\mathord{\id}\otimes\nabla\otimes\id)\circ(\mathord{\id}\otimes\nabla\inv)$.
 \item\label{ax:cold} (Co-left distributivity) $(\mathord{\id}\otimes\nabla)\circ\nabla=(\mu\otimes\mathord{\id}\otimes\id)\circ(\mathord{\id}\otimes\tau\otimes\id)\circ(\nabla\otimes\nabla)\circ\nabla$.
 \item (Co-right distributivity) $(\nabla\inv\otimes\id)\circ\nabla\inv=(\mathord{\id}\otimes\mathord{\id}\otimes\mu)\circ(\mathord{\id}\otimes\tau\otimes\id)\circ(\nabla\inv\otimes\nabla\inv)\circ\nabla\inv$.
 \item\label{ax:co-fixing} (Co-fixing) $(\eps\otimes \id)\circ\nabla=\id$.
 \item\label{ax:co-fixed} (Co-fixedness) $(\mathord{\id}\otimes \eps)\circ\nabla=\eps$.
\end{enumerate}
We identify $k\otimes A\cong A\cong A\otimes k$ in identities \ref{ax:co-fixing} and \ref{ax:co-fixed}. Actually, this paper will only have occasion to use identities \ref{ax:cold} and \ref{ax:co-fixing}; the other three identities are less important for our purposes than the corresponding axioms of algebraic racks.

For example, let $Q=\Spec(A)$ be an affine algebraic rack. Then for all $k$-algebras $R$ and $R$-points $x\in Q(R)$ (thought of as $k$-algebra homomorphisms $x^\sharp\colon A\to R$), the automorphism $L_x\colon Q(R)\bij Q(R)$ from \eqref{eq:lx} induces an automorphism
\begin{align}
L_x^\sharp\colon R\otimes_k A &\bij R\otimes_k A\nonumber\\
 r\otimes f&\mapsto(r\otimes 1)\cdot ((x^\sharp\otimes\id_{A})\circ \nabla)(f).\label{eq:lx-sharp-full}
\end{align}

\begin{definition}
 The \emph{Leibniz algebra of $Q$} is the vector space
 \[
 \q\coloneq \Der_k(A,k)
 \]
 equipped with the Leibniz bracket $[\cdot,\cdot]\colon \q\times\q\to \q$ defined by
 \[
 [D,E]\coloneq (D\otimes E)\circ\nabla,
 \]
 the \emph{convolution} operation with respect to $\nabla$.
\end{definition}
Writing $\nabla f=\sum \fo\otimes\ft$, we thus have
\[
[D,E](f)=\sum D(\fo)E(\ft)
\]
for all $f\in A$. Note that $[\cdot,\cdot]$ is $k$-bilinear.

\subsubsection{}
Similarly, since $\partial D$ is a $k[\delta]$-point of $Q$ and $\Ad\colon Q\to\GL(\q)$ is a morphism of schemes, we obtain a $k[\delta]$-point of $\GL(\q)$ \[\Ad_{\partial D}\in\GL(\q)(k[\delta])=\GL(k[\delta]\otimes_k \q).\] In particular, under the identification $k[\delta]\otimes_kA \bij A[\delta]$, the definition of the derivative shows that
\begin{equation}\label{eq:adj-partial}
 \Ad_{\partial D}( 1\otimes E)= E\circ L_{\partial D}^\sharp\colon A[\delta]\to k[\delta]
\end{equation}
for all $k$-derivations $E\in\q$. We use this to show the following. \begin{prop}\label{prop:adj}
 When considered as a map $\ad\colon\q\to\mathfrak{gl}(\q)$, the adjoint representation of $\q$ is given by the left argument of the Leibniz bracket $D\mapsto [D,\cdot]$. That is, $\ad_D(E)=[D,E]$ for all $k$-derivations $D,E\in\q$. In particular, $\q$ is closed under $[\cdot,\cdot]$.
\end{prop}

\begin{rmk}\label{rmk:recover}
 If $G$ is an affine algebraic group and $Q=\Conj(G)$ is its conjugation quandle scheme, then $(\q,[\cdot,\cdot])$ is the Lie algebra of $G$. To see this, note that $Q$ and $G$ have the same underlying scheme and identity $e$, so they also have the same tangent space $\q$ at $e$. By the definition of $\Conj(G)$, the adjoint representations $\Ad$ of $Q$ and $\ad$ of $\q$ are therefore the same as those of $G$ and its Lie algebra (see, for example, \cite{milne}*{Rem.\ 10.24}). Hence, the claim follows from Proposition \ref{prop:adj}.\footnote{Alternatively, one can use Example \ref{ex:corack-conj} and the Hopf algebra axioms to directly (if tediously) verify that the Leibniz bracket $[D,E]=(D\otimes E)\circ\nabla$ agrees with the usual Lie bracket $(D\otimes E-E\otimes D)\circ\Delta$ on $\q=\operatorname{Lie}(G)$ (see \cite{waterhouse}*{Sec.\ 12.2}); we leave the details to the reader. As a hint, first deduce the identity $\eps\circ S=\eps$ from the axiom $\mu\circ (S\otimes\id)\circ\Delta=\eta\circ\eps$ by post-composing with $\eps$, and then deduce from these two equalities that $D\circ S=-D$.}

\begin{enumerate}
 \item Does the Leibniz algebra of a (not necessarily affine) algebraic rack have a natural Leibniz algebra structure? That is, does the functor $\Rack(\Aff)\to\leib$ from Theorem \ref{thm:main} extend to a functor $\Rack(\Sch)\to\leib$?
 \item If $G$ is an algebraic group, then its Lie algebra $\mathrm{Lie}(G)$ is isomorphic to the space of left-invariant $k$-derivations $D\colon \O(G)\to\O(G)$, that is, $k$-derivations that satisfy the identity $\Delta\circ D=(\mathord{\id}\otimes D)\circ\Delta$. (See, for example, \cite{milne}*{Sec.\ 10e}.) Do Leibniz algebras of algebraic racks have a similar interpretation?
 \item Mirroring the representation theory of algebraic groups and comodules of commutative Hopf algebras (see, for example, \cite{milne}*{Sec.\ 4e}), develop a theory of affine representations of affine algebraic racks $Q$ and $\O(Q)$-comodules. Also, study their relationships with the induced omni-representations (see \cite{omni}) of Leibniz algebras. See Comment \ref{com:rep} for further discussion.
 \item Building upon the previous question, introduce a cohomology theory for algebraic racks and their representations. Study its relationships with Leibniz algebra cohomology \cite{lp}, traditional rack and quandle cohomology \cite{covez2}, and (in the case of conjugation quandle schemes) the cohomology of algebraic groups and their representations (see \cite{bendel}).
 \item More broadly, which theorems about affine algebraic groups and their Lie algebras extend to affine algebraic racks and their Leibniz algebras? (Cf.\ Remark \ref{rmk:triv}.)
 \item Determine which Leibniz algebras are \emph{algebraic} or ``integrate to an algebraic rack.'' That is, given $\q$, determine whether there exists an algebraic rack whose Leibniz algebra is $\q$.
 \item La Rosa and Mancini \cite{larosa}*{Thm.\ 4.6} proved that if $Q$ is a Lie rack with Leibniz algebra $\q$, then $\q$ is a Lie algebra if and only if $Q$ is a quandle. Does the same hold for algebraic racks?
 \item Study non-pointed algebraic racks (that is, \emph{rack objects} in $\Sch$; cf.\ \cite{rack-roll}). In particular, generalize the results of Takahashi \citelist{\cite{takahashi}\cite{takahashi2}} from quandle varieties to rack schemes.
 \item Develop the general theory of corack algebras, perhaps in analogy to the theory of Hopf algebras. What relationships do they have with rack bialgebras (see \citelist{\cite{bardakov}\cite{alexandre}})?
 \item Racks \cite{fenn} and quandles \citelist{\cite{joyce}\cite{matveev}} were originally introduced to construct invariants of knots and 3-manifolds; see \citelist{\cite{book}\cite{quandlebook}}. In the past decade, racks (for example, \citelist{\cite{randal}\cite{shusterman}}) and quandles (for example, \citelist{\cite{fan}\cite{bianchi}\cite{bianchi2}\cite{takahashi1}}) have also enjoyed applications as invariants of various algebro-geometric objects. Use algebraic racks to enhance these invariants, and compare them to other algebro-geometric invariants.
\end{enumerate}

\begin{proof}
    We prove the theorem piecemeal in Propositions \ref{prop:adj}, \ref{prop:jacobi}, and \ref{prop:functor} and Remark \ref{rmk:recover}.
\end{proof}

\subsubsection{}
Theorem \ref{thm:main} is an algebraic analogue of a result of Kinyon's result for Lie racks in \cite{kinyon}*{Thm.\ 3.4}.
Compared to that result, two advantages of Theorem \ref{thm:main} are that it holds for all fields $k$ and gives an explicit formula for the Leibniz bracket $[\cdot,\cdot]$ that does not require computing any derivatives. That said, the Leibniz bracket agrees with the derivative of the \emph{adjoint representation} of $Q$ at the identity; see Proposition \ref{prop:adj}.

The last part of Theorem \ref{thm:main} states that our construction recovers the usual functors sending affine algebraic groups to their Lie algebras (see, for example, \cite{waterhouse}*{Sec.\ 12.2}) and affine algebraic Lie racks over $\mathbb{R}$ or $\mathbb{C}$ to their Leibniz algebras (see \cite{kinyon}). As racks and quandles have enjoyed increasing interest in both quantum algebra and algebraic geometry in the past decade, this paper lays the foundations for what we expect will be a very rich area of study; cf.\ Section \ref{sec:open}.

\subsection{Literature review}
While there does not appear to be any published work on the algebraic \emph{coquecigrue} problem over arbitrary fields $k$---see \cite{skoda} for a preliminary report of unpublished work of Ba\v{s}i\'{c} and \v{S}koda from 2008 in the case that $\operatorname{char}(k)=0$---there is a rich literature on the \emph{coquecigrue} problem for smooth manifolds over $\mathbb{R}$ and $\mathbb{C}$. See \cite{ongay} for a discussion of the literature up to 2014.

Although the first work on the \emph{coquecigrue} problem is due to Datuashvili \cite{Datuashvili} in 2004, most modern approaches to the problem employ Lie racks, which Kinyon \cite{kinyon} introduced in 2007 in response to the problem.
For example, Covez \cite{covez} in 2013 introduced \emph{local augmented Lie racks}, which address the \emph{coquecigrue} problem with the caveat that an algebraic structure is only defined in a neighborhood of the identity. By contrast, Bordemann and Wagemann \cite{bordemann} in 2016 introduced a global rack-theoretic solution with the caveat of not being functorial.

Within the last decade, various authors have classified special types of Lie racks \citelist{\cite{abchir2}\cite{benayadi}\cite{dibartolo}}. These include Lie racks corresponding to certain kinds of nilpotent Leibniz algebras \citelist{\cite{larosa}\cite{larosa2}} and symmetric Leibniz algebras \cite{abchir}, as well as all Lie racks corresponding to Lie algebras \cite{larosa}*{Thm.\ 4.6}. For further work on the \emph{coquecigrue} problem, see \citelist{\cite{alexandre}\cite{dherin}}. For non-rack-theoretic approaches, see \citelist{\cite{smith}\cite{rodriguez}\cite{uslu}}.

\subsubsection{}
Although there does not appear to be existing literature on rack schemes, we note that quandle varieties and ordinary set-theoretic racks have enjoyed increasing attention in algebraic geometry in the past decade. In particular, quandle varieties provide algebro-geometric analogues of Riemannian symmetric spaces \citelist{\cite{takahashi}\cite{takahashi2}} and applications to geometric invariant theory \cite{fan}.

Ordinary racks and quandles provide invariants of not only knots and 3-manifolds but also various objects of study in algebraic geometry, particularly arithmetic algebraic geometry. For example, quandles provide invariants of arithmetic schemes \cite{takahashi1} and categorical constructions of Hurwitz spaces \citelist{\cite{bianchi}\cite{bianchi2}}. Racks have similar applications in the theory of Hurwitz spaces and algebraic function fields \citelist{\cite{randal}\cite{shusterman}}. The increasing attention racks and quandles have received in algebraic geometry provides another motivation for developing a modern scheme-theoretic treatment of algebraic racks.

\begin{notationC}
    We work over a field $k$ of arbitrary characteristic.
    By a \emph{scheme}, we mean a scheme of finite type over $k$. Unless we are specifically discussing Leibniz algebras or Lie algebras, by a \emph{$k$-algebra} we mean a finitely generated commutative associative $k$-algebra.
    Let $\Sch$, $\Aff$, $\Alg$, $\leib$, $\mathsf{Grp}$, $\mathsf{Grp}(\Sch)$, and $\mathsf{Set}$ denote the categories of schemes, affine schemes, $k$-algebras, Leibniz algebras over $k$, groups, algebraic groups (that is, group objects in $\Sch$), and sets, respectively. Denote the $k$-algebra of \emph{dual numbers} by $k[\delta]\coloneq k[\delta]/(\delta^2)$. Given a morphism of affine schemes $\varphi\colon\Spec(B)\to\Spec(A)$, let $\varphi^\sharp$ denote the induced homomorphism $A\to B$.
\end{notationC}

\subsection{Organization of the paper}

In Section \ref{sec:ptd-racks}, we recall the definition of the category of pointed racks $\Rack$ and introduce pointed rack objects internal to cartesian monoidal categories.

In Section \ref{sec:alg-racks}, we introduce the categories of algebraic racks $\Rack(\Sch)$ and corack algebras $\Corack(\Alg)$. We also discuss the functor of points approach for algebraic racks.

In Section \ref{sec:corack}, we specialize our attention to the category of affine algebraic racks $\Rack(\Aff)$, which is equivalent to the opposite category $(\Corack(\Alg))\op$. As examples, we compute the corack algebras corresponding to three classes of affine algebraic racks. We also show that corack algebras are almost never coassociative or cocommutative, unlike Hopf algebras of algebraic groups.

In Section \ref{sec:pre-pf}, we discuss our approach to proving Theorem \ref{thm:main}, and we recall various facts about $k$-derivations and tangent spaces.

In Section \ref{sec:pf}, we prove Theorem \ref{thm:main}. A major step in the proof is to show that the Leibniz bracket $[\cdot,\cdot]$ in Theorem \ref{thm:main} agrees with the scheme-theoretic derivative $\ad\coloneq (d\Ad)_e$ of the \emph{adjoint representation} of $Q$ at the identity $e$.

In Section \ref{sec:ol}, we illustrate Theorem \ref{thm:main} with a class of affine algebraic racks $\OL_n$ whose Leibniz algebras $\ol_n(k)$ are not generally Lie algebras.
Then we provide several results relating the structure of an affine algebraic rack to that of its Leibniz algebra, including sufficient conditions for the latter to be abelian. These results recover the analogous results for Lie algebras of affine algebraic groups.

In Section \ref{sec:open}, we propose directions for future research.

\begin{ack}
    An early draft of this paper was submitted as the final project for a course on affine algebraic groups taught by Prof.\ Bogdan Ion at the University of Pittsburgh in fall 2025.
\end{ack}

We "prove the theorem piecemeal in Propositions \ref{prop:adj}, \ref{prop:jacobi}, and \ref{prop:functor} and Remark \ref{rmk:recover}." A "major step in the proof" is "to show that the Leibniz bracket $[\cdot,\cdot]$ in Theorem \ref{thm:main} agrees with the scheme-theoretic derivative $\ad\coloneq (d\Ad)_e$ of the \emph{adjoint representation} of $Q$ at the identity $e$."

We prove the theorem piecemeal in the following results. \begin{prop}\label{prop:adj}
    When considered as a map $\ad\colon\q\to\mathfrak{gl}(\q)$, the adjoint representation of $\q$ is given by the left argument of the Leibniz bracket $D\mapsto [D,\cdot]$. That is, $\ad_D(E)=[D,E]$ for all $k$-derivations $D,E\in\q$. In particular, $\q$ is closed under $[\cdot,\cdot]$.
\end{prop} \begin{prop}\label{prop:jacobi}
    The convolution operation $[\cdot,\cdot]$ satisfies the Jacobi identity.
\end{prop} \begin{prop}\label{prop:functor}
    The assignments $Q\mapsto\q$ and $\phi\mapsto \phi'\coloneq (d\phi)_e$ define a covariant functor $\Rack(\Aff)\to\leib$.
\end{prop} We also use the following remark. \begin{rmk}\label{rmk:recover}
    If $G$ is an affine algebraic group and $Q=\Conj(G)$ is its conjugation quandle scheme, then $(\q,[\cdot,\cdot])$ is the Lie algebra of $G$. To see this, note that $Q$ and $G$ have the same underlying scheme and identity $e$, so they also have the same tangent space $\q$ at $e$. By the definition of $\Conj(G)$, the adjoint representations $\Ad$ of $Q$ and $\ad$ of $\q$ are therefore the same as those of $G$ and its Lie algebra (see, for example, J. S. Milne, \emph{Algebraic Groups, Lie Groups, and their Arithmetic Subgroups}*{Rem.\ 10.24}). Hence, the claim follows from the preceding proposition.\footnote{Alternatively, one can use Example \ref{ex:corack-conj} and the Hopf algebra axioms to directly (if tediously) verify that the Leibniz bracket $[D,E]=(D\otimes E)\circ\nabla$ agrees with the usual Lie bracket $(D\otimes E-E\otimes D)\circ\Delta$ on $\q=\operatorname{Lie}(G)$ (see W. C. Waterhouse, \emph{Introduction to Affine Group Schemes}*{Sec.\ 12.2}); we leave the details to the reader. As a hint, first deduce the identity $\eps\circ S=\eps$ from the axiom $\mu\circ (S\otimes\id)\circ\Delta=\eta\circ\eps$ by post-composing with $\eps$, and then deduce from these two equalities that $D\circ S=-D$.}

    Similarly, if $k=\R$ or $\mathbb{C}$ and $Q(k)$ is an algebraic Lie rack, then our definitions of $\Ad$ and $\ad$ coincide with the ones for Lie racks in \cite{kinyon}, so $(\q,[\cdot,\cdot])$ is the Leibniz algebra of $Q(k)$ as a Lie rack.
\end{rmk} A major step in the proof is to show that the Leibniz bracket $[\cdot,\cdot]$ in the main theorem agrees with the scheme-theoretic derivative $\ad\coloneq (d\Ad)_e$ of the \emph{adjoint representation} of $Q$ at the identity $e$.

\label{thm:main}
    Let $Q=\Spec(A)$ be an affine algebraic rack with unit $e\in Q$, and let $\mathfrak{q}\coloneq \Der_k(A,k)$ be the vector space of $k$-derivations of $A$. Define a Leibniz bracket $[\cdot,\cdot]\colon\mathfrak{q}\times\mathfrak{q}\to\mathfrak{q}$ by convolution with respect to the corack operation $\nabla$ of $A$ (see below):
    \[
    [D,E]\coloneq (D\otimes E)\circ\nabla.
    \]
    Then the assignment $Q\mapsto(\mathfrak{q},[\cdot,\cdot])$ defines a covariant functor $\Rack(\Aff)\to\leib$ sending morphisms of affine algebraic racks $\varphi\colon Q\to R$ to their derivatives $(d\varphi)_e\colon \mathfrak{q}\to\mathfrak{r}$ at $e$.

    Furthermore, if $G$ is an affine algebraic group and $Q=\Conj(G)$ is its conjugation quandle scheme, then $(\mathfrak{q},[\cdot,\cdot])$ is the Lie algebra of $G$. On the other hand, if $k=\mathbb{R}$ or $\mathbb{C}$ and $Q(k)$ is a Lie rack, then $(\mathfrak{q},[\cdot,\cdot])$ is the Leibniz algebra of $Q(k)$ as a Lie rack.,

Question:
Let $Q=\operatorname{Spec}(A)$ be an affine algebraic rack over a field $k$ with unit $e\in Q$, and let $\nabla\colon A\to A\otimes A$ be the corresponding corack operation. Define
\[
\mathfrak q:=\operatorname{Der}_k(A,k),\qquad [D,E]:=(D\otimes E)\circ \nabla\quad (D,E\in\mathfrak q).
\]
For a morphism of affine algebraic racks $\varphi\colon Q\to R$, let $(d\varphi)_e$ denote its derivative at the unit. Also, for an affine algebraic group $G$, write $\operatorname{Conj}(G)$ for the conjugation quandle scheme on $G$ (rack operation $x\triangleright y=xyx^{-1}$). If $k=\mathbb R$ or $\mathbb C$, let $Q(k)$ denote the Lie rack of $k$-points when it is a pointed smooth rack. Which statement is valid?

The bracket $[D,E]=(D\otimes E)\circ\nabla$ makes $\mathfrak q$ into a Leibniz algebra, and the assignment $Q\mapsto (\mathfrak q,[\cdot,\cdot])$ defines a covariant functor $\operatorname{Rack}(\operatorname{Aff})\to \operatorname{Leib}$ that sends each morphism $\varphi\colon Q\to R$ to its derivative $(d\varphi)_e\colon \mathfrak q\to \mathfrak r$. Moreover, if $G$ is an affine algebraic group and $Q=\operatorname{Conj}(G)$, then $(\mathfrak q,[\cdot,\cdot])$ is the Lie algebra of $G$; and if $k=\mathbb R$ or $\mathbb C$ and $Q(k)$ is a Lie rack, then $(\mathfrak q,[\cdot,\cdot])$ is the Leibniz algebra of $Q(k)$ as a Lie rack.

Choices:
B. The bracket $[D,E]=(D\otimes E)\circ\nabla$ makes $\mathfrak q$ into a Lie algebra, and the assignment $Q\mapsto (\mathfrak q,[\cdot,\cdot])$ defines a contravariant functor $\operatorname{Rack}(\operatorname{Aff})\to \operatorname{Leib}$ that sends each morphism $\varphi\colon Q\to R$ to the transpose of its derivative at the unit. Moreover, if $G$ is an affine algebraic group and $Q=\operatorname{Conj}(G)$, then $(\mathfrak q,[\cdot,\cdot])$ is the Lie algebra of $G$; and if $k=\mathbb R$ or $\mathbb C$ and $Q(k)$ is a Lie rack, then $(\mathfrak q,[\cdot,\cdot])$ is the Leibniz algebra of $Q(k)$ as a Lie rack.
C. The bracket $[D,E]=(D\otimes E)\circ\nabla$ makes $\mathfrak q$ into a Leibniz algebra, and for every morphism of affine algebraic racks $\varphi\colon Q\to R$, the derivative $(d\varphi)_e\colon \mathfrak q\to \mathfrak r$ is a homomorphism of Leibniz algebras.
D. The bracket $[D,E]=(D\otimes E)\circ\nabla$ makes $\mathfrak q$ into a Leibniz algebra, and the assignment $Q\mapsto (\mathfrak q,[\cdot,\cdot])$ defines a covariant functor $\operatorname{Rack}(\operatorname{Aff})\to \operatorname{Leib}$ sending each morphism $\varphi\colon Q\to R$ to its derivative $(d\varphi)_e\colon \mathfrak q\to \mathfrak r$. Moreover, if $G$ is an affine algebraic group and $Q=\operatorname{Conj}(G)$, then $(\mathfrak q,[\cdot,\cdot])$ is canonically isomorphic to the Leibniz algebra underlying the adjoint representation of $G$; and if $k=\mathbb R$ or $\mathbb C$, then for every affine algebraic rack $Q$ the $k$-points $Q(k)$ form a Lie rack and $(\mathfrak q,[\cdot,\cdot])$ is the Leibniz algebra of $Q(k)$ as a Lie rack.
E. The bracket $[D,E]=(D\otimes E)\circ\nabla$ makes $\mathfrak q$ into a Leibniz algebra, and the assignment $Q\mapsto (\mathfrak q,[\cdot,\cdot])$ defines a covariant functor $\operatorname{Rack}(\operatorname{Aff})\to \operatorname{Leib}$ sending each morphism $\varphi\colon Q\to R$ to its derivative $(d\varphi)_e\colon \mathfrak q\to \mathfrak r$. Moreover, if $G$ is an affine algebraic group and $Q=\operatorname{Conj}(G)$, then $(\mathfrak q,[\cdot,\cdot])$ agrees with the usual Lie algebra of $G$ only when $\operatorname{char}(k)=0$; and if $k=\mathbb R$ or $\mathbb C$ and $Q(k)$ is a Lie rack, then $(\mathfrak q,[\cdot,\cdot])$ is the Leibniz algebra of $Q(k)$ as a Lie rack.


\begin{equation}\label{eq:bound.logm}
    \log m \le \big(M+o_m(1)\big) \sqrt{st}.
\end{equation}

Notice that any graph $H$ on $m$ vertices provides a red/blue edge-coloring of $K_m$ implying that $m < R(\omega(H)+1, \alpha(H)+1)$. Thus, by \eqref{eq:bound.logm}, for every subgraph $G'$ of $G$ on $m$ vertices, we have
\begin{equation*}
     \omega(G') \cdot \alpha(G') \ge
\Big( \frac{1}{M^2} + o_m(1) \Big) (\log m)^2.
\end{equation*}
Since $\omega(G') \le \omega(G)$, we obtain that \begin{equation*}
    \alpha(G') \ge
\Big( \frac{1}{M^2} + o_m(1) \Big) \frac{(\log m)^2}{\omega(G)}.
\end{equation*}
Hence, together with the fact that $\log m = \big(1+o_n(1)\big) \log n$, \Cref{lem:erdos} yields
\begin{equation*}
    \chi(G) \le \big(M^2 +o_n(1)\big)\frac{n\cdot\omega(G)}{(\log n)^2} + \frac{n}{(\log n)^3}.
\end{equation*}
Finally, we conclude that
\begin{equation*}
    \frac{\chi(G)}{\omega(G)} \le \big(M^2 +o_n(1)\big)\frac{n}{(\log n)^2}.\qedhere
\end{equation*}
\end{proof}

\section{Proof of Theorem~\ref{thm:g_upper}} \label{subsec:numerics}
In this section, we prove Theorem~\ref{thm:g_upper}. The currently best known upper bound for $R(k,k)$ is
\begin{equation}\label{eq:diagonal_upper_bound}
    R(k,k) \le \left( 4e^{-0.14e^{-1}} \right)^{k+o(k)}.
\end{equation}

\label{sec:intro}

Ramsey Theory, often described as the investigation of when order arises from chaos, is one of the most influential areas in Graph Theory. The \emph{Ramsey number} $R(s,t)$ is the minimum $n$ such that every red/blue edge-coloring of the complete graph on $n$ vertices contains either a clique on $s$ vertices colored red or a clique on $t$ vertices colored blue.

The first explicit upper bound for Ramsey numbers was obtained by Erd\H{o}s and Szekeres~\cite{E-Sz} in 1935, who showed that $R(s,t) \leqslant \binom{s+t-2}{s-1}$. Note that in the \emph{diagonal} case, this shows that $R(k,k)\leqslant 4^{k+o(k)}$.
On the other hand, Erd\H{o}s~\cite{Ramsey.lower} established the lower bound $R(k,k)\geqslant 2^{k/2+o(k)}$ in 1947 in one of the first uses of the probabilistic method. Subsequent improvements on the lower bound have only refined it by a constant multiplicative factor (see, e.g., Spencer’s bound obtained via the Lov\'asz Local Lemma~\cite{Spencer.LLL}). Since then, one of the major problems in modern combinatorics has been to determine the correct asymptotic behavior of the diagonal Ramsey number $R(k,k)$. More specifically, one could ask whether the limit
\begin{equation}\label{limit1}
    \lim_{k \to \infty} \frac{\log(R(k,k))}{k}
\end{equation}
 exists and, if so, what is its value. This question appears as Problem 1 in~\cite{chung}, as Problem 77 in~\url{https://www.erdosproblems.com/77}, and on the webpage~\url{https://mathweb.ucsd.edu/~erdosproblems/}.

In 1967, Erd\H{o}s~\cite{erdos} asked a seemingly unrelated question. Define $f(n)$ to be the maximum, over all graphs $G$ on $n$ vertices, of the ratio $\frac{\chi(G)}{\omega(G)}$, where $\chi(G)$ denotes the chromatic number of $G$ and $\omega(G)$ the clique number of $G$. Formally, we have
\begin{equation*}
    f(n)\coloneq \max \Big\{ \ \frac{\chi(G)}{\omega(G)} \ :
    \ G \text{ is a graph on } n \text{ vertices } \Big\}.
\end{equation*}

Erd\H{o}s showed that $f(n) = \Theta(n/(\log n)^2)$ and asked whether the following limit exists:
\begin{equation}\label{limit2}
    \lim_{n \to \infty} \frac{f(n)}{n/(\log n)^2}.
\end{equation}
This question also appears in~\cite{erdos2}, as Problem 627 in~\url{https://www.erdosproblems.com/627}, as Problem 53 in~\cite{chung}, and on the webpage~\url{https://mathweb.ucsd.edu/~erdosproblems/}. In this note, we establish a connection between these two problems. In particular, we show that if the following conjecture is true, a solution to the former problem results in a solution to the latter problem.

\begin{conjecture}\label{conj:weak.mult.RDC}
    For every $s, t, k \in \N$ such that $st \leqslant k^2$ we have that
    \begin{equation*}
        R(s,t) \leqslant R(k, k).
    \end{equation*}
\end{conjecture}

We remark that Conjecture~\ref{conj:weak.mult.RDC} does not seem to appear in any existing work, but it is closely related to the Ramsey Diagonal Conjecture. In Appendix~\ref{sec:DiagonalConjecture}, we briefly discuss these conjectures. The following theorem establishes a connection between the limits in \eqref{limit1} and \eqref{limit2}, should they exist, under the assumption of Conjecture~\ref{conj:weak.mult.RDC}. Throughout this note, all logarithms are in base 2.

\label{sec:intro}

The first explicit upper bound for Ramsey numbers was obtained by Erd\H{o}s and Szekeres~\cite{E-Sz} in 1935, who showed that $R(s,t) \leqslant \binom{s+t-2}{s-1}$. Note that in the \emph{diagonal} case, this shows that $R(k,k)\leqslant 4^{k+o(k)}$.
On the other hand, Erd\H{o}s~\cite{Ramsey.lower} established the lower bound $R(k,k)\geqslant 2^{k/2+o(k)}$ in 1947 in one of the first uses of the probabilistic method. Subsequent improvements on the lower bound have only refined it by a constant multiplicative factor (see, e.g., Spencer’s bound obtained via the Lov\'asz Local Lemma~\cite{Spencer.LLL}). Since then, one of the major problems in modern combinatorics has been to determine the correct asymptotic behavior of the diagonal Ramsey number $R(k,k)$. More specifically, one could ask whether the limit
\begin{equation}\label{limit1}
 \lim_{k \to \infty} \frac{\log(R(k,k))}{k}
\end{equation}
 exists and, if so, what is its value. This question appears as Problem 1 in~\cite{chung}, as Problem 77 in~\url{https://www.erdosproblems.com/77}, and on the webpage~\url{https://mathweb.ucsd.edu/~erdosproblems/}.

In 1967, Erd\H{o}s~\cite{erdos} asked a seemingly unrelated question. Define $f(n)$ to be the maximum, over all graphs $G$ on $n$ vertices, of the ratio $\frac{\chi(G)}{\omega(G)}$, where $\chi(G)$ denotes the chromatic number of $G$ and $\omega(G)$ the clique number of $G$. Formally, we have
\begin{equation*}
 f(n)\coloneq \max \Big\{ \ \frac{\chi(G)}{\omega(G)} \ :
 \ G \text{ is a graph on } n \text{ vertices } \Big\}.
\end{equation*}

Erd\H{o}s showed that $f(n) = \Theta(n/(\log n)^2)$ and asked whether the following limit exists:
\begin{equation}\label{limit2}
 \lim_{n \to \infty} \frac{f(n)}{n/(\log n)^2}.
\end{equation}
This question also appears in~\cite{erdos2}, as Problem 627 in~\url{https://www.erdosproblems.com/627}, as Problem 53 in~\cite{chung}, and on the webpage~\url{https://mathweb.ucsd.edu/~erdosproblems/}. In this note, we establish a connection between these two problems. In particular, we show that if the following conjecture is true, a solution to the former problem results in a solution to the latter problem.

\begin{conjecture}\label{conj:weak.mult.RDC}
 For every $s, t, k \in \N$ such that $st \leqslant k^2$ we have that
 \begin{equation*}
 R(s,t) \leqslant R(k, k).
 \end{equation*}
\end{conjecture}

We remark that Conjecture~\ref{conj:weak.mult.RDC} does not seem to appear in any existing work, but it is closely related to the Ramsey Diagonal Conjecture. In Appendix~\ref{sec:DiagonalConjecture}, we briefly discuss these conjectures. The following theorem establishes a connection between the limits in \eqref{limit1} and \eqref{limit2}, should they exist, under the assumption of Conjecture~\ref{conj:weak.mult.RDC}. Throughout this note, all logarithms are in base 2.

\begin{conjecture}\label{conj:weak.mult.RDC}
    For every $s, t, k \in \N$ such that $st \le k^2$ we have that
    \begin{equation*}
        R(s,t) \le R(k, k).
    \end{equation*}
\end{conjecture}

\begin{equation}\label{limit1}
    \lim_{k \to \infty} \frac{\log(R(k,k))}{k}
\end{equation}

\begin{equation}\label{limit2}
    \lim_{n \to \infty} \frac{f(n)}{n/(\log n)^2}.
\end{equation}

Erd\H{o}s showed that $f(n) = \Theta(n/(\log n)^2)$ and asked whether the following limit exists:
\begin{equation}\label{limit2}
 \lim_{n \to \infty} \frac{f(n)}{n/(\log n)^2}.
\end{equation}
This question also appears in~\cite{erdos2}, as Problem 627 in~\url{https://www.erdosproblems.com/627}, as Problem 53 in~\cite{chung}, and on the webpage~\url{https://mathweb.ucsd.edu/~erdosproblems/}. In this note, we establish a connection between these two problems. In particular, we show that if the following conjecture is true, a solution to the former problem results in a solution to the latter problem.

We remark that Conjecture~\ref{conj:weak.mult.RDC} does not seem to appear in any existing work, but it is closely related to the Ramsey Diagonal Conjecture. In Appendix~\ref{sec:DiagonalConjecture}, we briefly discuss these conjectures. The following theorem establishes a connection between the limits in \eqref{limit1} and \eqref{limit2}, should they exist, under the assumption of Conjecture~\ref{conj:weak.mult.RDC}. Throughout this note, all logarithms are in base 2.

In~\cite{erdos}, Erd\H{o}s mentioned that by their method, it would be easy to prove that
\begin{equation*}
 \big(c+o(1)\big) \frac{n}{(\log n)^2} \le f(n)
 \le \big(C+o(1) \big) \frac{n}{(\log n)^2},
\end{equation*}
for constants $c=\frac{1}{4}$ and $C=1$. However, this appears to be a typographical error, as a careful application of their method yields the constants $c=\frac{1}{4}$ and $C=4$. Aided by recent breakthrough developments in Ramsey theory, we can prove the following.

\begin{theorem}\label{thm:g_upper}
 The function $f(n)$ satisfies
\begin{equation}\label{eq:thm.bound.f.1}
 f(n) \le \big( 3.71943+o(1) \big) \frac{n}{(\log n)^2}.
\end{equation}
Moreover, if Conjecture~\ref{conj:weak.mult.RDC} holds, then
 \begin{equation}\label{eq:thm.bound.f.2}
 f(n) \le \big( 3.70831 + o(1) \big) \frac{n}{(\log n)^2}.
\end{equation}
\end{theorem}

\section{Proof of Theorem~\ref{thm:diagonal_to_ratio}}
\label{sec:proof_1.2}
In this section, we establish upper and lower bounds sufficient to prove Theorem~\ref{thm:diagonal_to_ratio}. We write $h = o_n(t)$ if the functions $t,h\colon \N \to \RR$ satisfy $\lim\limits_{n \to \infty} \frac{h(n)}{t(n)} = 0$. When the variable $n$ is clear from the context, we simply write $h=o(t)$. Let us define
\begin{equation*}
g(n) \coloneqq \frac{(\log n)^2}{n} f(n),\quad
L \coloneqq \liminf\limits_{k \to \infty} \; \frac{\log (R(k,k))}{k}
\quad \text{ and } \quad
M \coloneq \limsup_{t \to \infty } \; \max_{s\le t} \ \frac{\log(R(s,t))}{\sqrt{st \ }}
\end{equation*}

\begin{fact}\label{fact} Under the assumption of Conjecture~\ref{conj:weak.mult.RDC}, equality holds in \eqref{ineq:limsups}.
\end{fact}
\begin{proof}
For $0 < s\le t$, let $k$ be the positive integer such that $(k-1)^2 < st \le k^2$. Let
\[D\coloneq\limsup_{k \to \infty} \;\frac{\log (R(k,k))}{k}.\]
Conjecture~\ref{conj:weak.mult.RDC} implies that
\begin{equation*}
 R(s,t) \le R(k,k) \le 2^{(D+o_k(1))k}
\end{equation*}
 and, therefore, we have
\begin{equation*}
 \frac{\log(R(s,t))}{\sqrt{st \ }} \le \frac{\log(R(k,k))}{k-1} \le D +o_k(1) = D +o_t(1). \qedhere
\end{equation*}
\end{proof}

Let $G$ be a graph on $n$ vertices with no clique or independent set of size $k_n+1$. In other words, $\alpha(G) \le k_n$ and $\omega(G) \le k_n$. As $\chi(G) \ge n/\alpha(G)$, we conclude that
\begin{equation}\label{eq:lower.bound.inf.g}
 g(n)=\frac{f(n)}{n/(\log n)^2} \ge \frac{\chi(G)}{\omega(G)}\cdot\frac{(\log n)^2}{n} \ge
\frac{(\log n)^2}{\alpha(G)\omega(G)}\ge
\frac{(\log n)^2}{k_n^2} \ge
L^2 + o_n(1),
\end{equation}
where the last inequality is true by \eqref{eq:bound.logn}. Hence, $\liminf\limits_{n \to \infty} g(n) \ge L^2$.

\begin{proof}[Proof of Theorem~\ref{thm:g_upper}]
If Conjecture~\ref{conj:weak.mult.RDC} holds, then Theorem~\ref{thm:upper}, Fact~\ref{fact}, and \eqref{eq:diagonal_upper_bound} imply~\eqref{eq:thm.bound.f.2}. Indeed,
\begin{equation*}
 f(n) \le \big( ( \log 4e^{-0.14e^{-1}} )^2 + o(1) \big) \frac{n}{(\log n)^2} < \big( 3.70831 + o(1) \big) \frac{n}{(\log n)^2}.
\end{equation*}

\begin{conjecture}\label{conj:weak.mult.RDC}
    For every $s, t, k \in \N$ such that $st \le k^2$ we have that
    \begin{equation*}
        R(s,t) \le R(k, k).
    \end{equation*}
\end{conjecture}

\begin{equation}\label{limit1}
    \lim_{k \to \infty} \frac{\log(R(k,k))}{k}
\end{equation}

\begin{equation}\label{limit2}
    \lim_{n \to \infty} \frac{f(n)}{n/(\log n)^2}.
\end{equation}

\begin{theorem}\label{thm:diagonal_to_ratio}
    If Conjecture~\ref{conj:weak.mult.RDC} holds, and $\lim\limits_{k \to \infty} \frac{\log(R(k,k))}{k}$ exists and is equal to $\mathcal{L}$, then
    \begin{equation*}
        f(n)=\big(\mathcal{L}^2+o(1)\big)\frac{n}{(\log n)^2} .
    \end{equation*}
\end{theorem}

\begin{theorem}\label{thm:g_upper}
    The function $f(n)$ satisfies
\begin{equation}\label{eq:thm.bound.f.1}
 f(n) \le \big( 3.71943+o(1) \big) \frac{n}{(\log n)^2}.
\end{equation}
Moreover, if Conjecture~\ref{conj:weak.mult.RDC} holds, then
 \begin{equation}\label{eq:thm.bound.f.2}
    f(n) \le \big( 3.70831 + o(1) \big) \frac{n}{(\log n)^2}.
\end{equation}
\end{theorem}

\begin{theorem}\label{thm:upper}
    The function $g(n)$ satisfies
    \begin{equation*}
         \limsup_{n \to \infty} g(n) \le M^2 .
    \end{equation*}
\end{theorem}

In~\cite{erdos}, Erd\H{o}s mentioned that by their method, it would be easy to prove that
\begin{equation*}
    \big(c+o(1)\big) \frac{n}{(\log n)^2} \leqslant f(n)
    \leqslant \big(C+o(1) \big)  \frac{n}{(\log n)^2},
\end{equation*}
for constants $c=\frac{1}{4}$ and $C=1$. However, this appears to be a typographical error, as a careful application of their method yields the constants $c=\frac{1}{4}$ and $C=4$. Aided by recent breakthrough developments in Ramsey theory, we can prove the following.

\begin{theorem}\label{thm:g_upper}
    The function $f(n)$ satisfies
\begin{equation}\label{eq:thm.bound.f.1}
 f(n) \leqslant \big( 3.71943+o(1) \big) \frac{n}{(\log n)^2}.
\end{equation}
Moreover, if Conjecture~\ref{conj:weak.mult.RDC} holds, then
 \begin{equation}\label{eq:thm.bound.f.2}
    f(n) \leqslant \big( 3.70831 + o(1) \big) \frac{n}{(\log n)^2}.
\end{equation}
\end{theorem}

This upper bound on $f(n)$ is the first improvement in the asymptotics of $f(n)$ since 1967.

\subsection*{Organization of the paper} In Section~\ref{sec:proof_1.2}, we introduce some notation used throughout the paper and state bounds on $f(n)$, namely Theorems~\ref{thm:lower} and~\ref{thm:upper}, establishing a connection between the asymptotic behavior of $f(n)$ and the Ramsey numbers $R(s,t)$. Theorem~\ref{thm:diagonal_to_ratio} is an immediate consequence of these results, whose proofs appear at the end of Section~\ref{sec:proof_1.2}. In Section~\ref{subsec:numerics}, we establish the numerical bounds of Theorem~\ref{thm:g_upper}. In Appendix~\ref{sec:DiagonalConjecture}, we discuss versions of the Ramsey Diagonal Conjecture and how they relate to Conjecture~\ref{conj:weak.mult.RDC}.

The post-theorem introduction says that in Section~\ref{sec:proof_1.2} the authors "state bounds on $f(n)$, namely Theorems~\ref{thm:lower} and~\ref{thm:upper}, establishing a connection between the asymptotic behavior of $f(n)$ and the Ramsey numbers $R(s,t)$." It then states that "Theorem~\ref{thm:diagonal_to_ratio} is an immediate consequence of these results, whose proofs appear at the end of Section~\ref{sec:proof_1.2}."

The post-theorem introduction says that next the authors state bounds on $f(n)$, namely the following results.

\begin{theorem}\label{thm:lower}
    The function $g(n)$ satisfies
    \begin{equation*}
         \liminf_{n \to \infty} g(n) \ge L^2
    \quad \text{ and } \quad \limsup_{n \to \infty} g(n) \ge M^2.
    \end{equation*}
\end{theorem}

\begin{theorem}\label{thm:upper}
    The function $g(n)$ satisfies
    \begin{equation*}
         \limsup_{n \to \infty} g(n) \le M^2 .
    \end{equation*}
\end{theorem}

These establish a connection between the asymptotic behavior of $f(n)$ and the Ramsey numbers $R(s,t)$. It then states that, in establishing the main theorem, it is an immediate consequence of these results, whose proofs appear later.

\label{thm:diagonal_to_ratio}
If the following conjecture holds:
\begin{conjecture}\label{conj:weak.mult.RDC}
    For every $s, t, k \in \N$ such that $st \le k^2$ we have that
    \begin{equation*}
        R(s,t) \le R(k, k).
    \end{equation*}
\end{conjecture}
and $\lim\limits_{k \to \infty} \frac{\log(R(k,k))}{k}$ exists and is equal to $\mathcal{L}$, then
\begin{equation*}
    f(n)=\big(\mathcal{L}^2+o(1)\big)\frac{n}{(\log n)^2} .
\end{equation*}

Question:
Let $R(s,t)$ denote the Ramsey number, i.e. the least integer $N$ such that every graph on $N$ vertices contains either a clique of size $s$ or an independent set of size $t$. For each $n$, define
\[
f(n)=\max\left\{\frac{\chi(G)}{\omega(G)}: G \text{ is a graph on } n \text{ vertices}\right\},
\]
where $\chi(G)$ is the chromatic number and $\omega(G)$ is the clique number of $G$. Assume that all logarithms are in base $2$. Suppose the following conjecture holds:
\[
\text{for every } s,t,k\in\mathbb N \text{ with } st\le k^2, \quad R(s,t)\le R(k,k).
\]
Also assume that the limit
\[
\lim_{k\to\infty}\frac{\log R(k,k)}{k}
\]
exists and is equal to $\mathcal L$. As $n\to\infty$, which asymptotic statement holds for $f(n)$?

\[
f(n)=\big(\mathcal L^2+o(1)\big)\frac{n}{(\log n)^2}.
\]

Choices:
B.
\[
f(n)=\big(\mathcal L+o(1)\big)\frac{n}{(\log n)^2}.
\]
C.
\[
f(n)=O\!\left(\frac{n}{(\log n)^2}\right).
\]
D.
\[
\limsup_{n\to\infty}\frac{f(n)}{n/(\log n)^2}=\mathcal L^2.
\]
E.
\[
f(n)=\big(\mathcal L^2+o(1)\big)\frac{n}{\log n}.
\]