\begin{theorem}\label{Th13}
Let $2\le p <\infty,\;0<s<1$ and $0\le \Lambda \le 1$. Suppose $\Omega \in \mathbb{H}^n$ is a bounded open set and $u \in HW_{{\rm{loc}}}^{1,p}\left( \Omega \right) \cap{ L_{sp}^{p - 1}\left( {{\mathbb{H}^n}} \right)} $ is a weak solution of \eqref{eq0}. Then $u \in C_{loc}^\gamma \left( \Omega  \right)$ for any $0 < \gamma  < 1$. Moreover, for any $0 < \gamma  < 1$ and any Kor\'{a}nyi ball ${B_{2R}}\left( {{\xi _0}} \right) \subset  \subset \Omega $ with $0<R<1$, there exists a positive constant $c=c(Q,p,s,\gamma)$ such that
\begin{equation}\label{eq482}
  \mathop {\sup }\limits_{\xi  \ne \eta  \in {B_{\frac{R}{2}}}\left( {{\xi _0}} \right)} \frac{{\left| {u\left( \xi  \right) - u\left( \eta  \right)} \right|}}{{{{\left\| {{\eta ^{ - 1}} \circ \xi } \right\|}^\gamma _{\mathbb{H}^n}}}} \le \frac{c}{{{R^\gamma }}}\left( {{{{\left\| u \right\|}_{{L^\infty }\left( {{B_R}\left( {{\xi _0}} \right)} \right)}}} + {\rm{Tai}}{{\rm{l}}_{p - 1,sp,p}}\left( {u;{\xi _0},R} \right)} \right),
\end{equation}
where the definition of ${\rm{Tai}}{{\rm{l}}_{p - 1,sp,p}}\left( {u;{\xi _0},R} \right)$ see \eqref{eq26} below.
\end{theorem}

\label{sec-1}

The study of regularity theory for degenerate elliptic equations has long stood as a cornerstone of modern analysis, tracing its intellectual lineage to the groundbreaking contributions of De Giorgi, Nash, and Moser. In the context of the Heisenberg group $\mathbb{H}^n$, a paradigmatic example of a sub-Riemannian manifold, the analysis of quasilinear degenerate elliptic equations of $p$-sub-Laplacian type has undergone profound development. In the pioneering work \cite{CL97}, Capogna established $C^{1,\alpha}$-regularity for the case $p=2$. Subsequently, Mukherjee and Zhong\cite{MZ21} extended this result to quasilinear degenerate elliptic equations with $p$-growth conditions for all $1<p<\infty$ via De Giorgi's method. Further progress includes: fractional differentiability estimates for nonlinear non-differentiable degenerate elliptic equations obtained by Zhang and Niu\cite{ZN22}; ${C^{\alpha }}$-regularity of minimizers for generalized Orlicz functionals encompassing $p,q$-growth conditions established by Zhang and Niu\cite{ZN20}; weak differentiability of weak solutions under the constraint $1<p<4$ for quasilinear equations with $p,q$-growth conditions proved by Zhang and Li\cite{ZL23}. A sufficient condition excluding the Lavrentiev phenomenon for non-autonomous integral functionals with $p,q$-growth condition in the same setting provided by Zhang and Niu\cite{ZN231}. For additional developments and comprehensive references, we refer the reader to \cite{ZN22} and the works cited therein.

The regularity theory for purely nonlocal equations involving fractional powers of the $p$-Laplacian on the Heisenberg group has witnessed remarkable progress in recent years, spurred by pivotal advancements such as the introduction of nonlocal tailing mechanisms and the establishment of nonlocal Harnack and H\"{o}lder regularity estimates. These contributions, as reported in Roncal and Thangavelu\cite{RT16}, Frank, Lieb, and Seiringer\cite{FLS08}, Ferrari and Franchi\cite{FF15}, Ferrari, Miranda, Pallara, Pinamonti and Sire\cite{FMPPS18}, Frank, Gonzalez, Monticelli and Tan\cite{FGMT15}, Manfredini, Palatucci, Piccinini and Polidoro\cite{MPPP23}, and Palatucci and Piccinini\cite{PP22}, collectively constitute a transformative body of work reshaping the analytical foundations of the field. Moreover, Palatucci and Piccinini in 2022 (see \cite{PP22}) and 2023 (see \cite{PP23}) proposed an open problem how to determine the regularity of nonlocal double-phase equations in the Heisenberg group. Fang, Zhang and Zhang provided a solution in \cite{FZZ24}.

However, the analysis of equations combining local and nonlocal operators-referred to as "mixed" equations-presents a formidable challenge due to the intricate interplay between classical diffusion processes and long-range interactions. While the Euclidean setting has witnessed rapid progress, with results ranging from local boundedness (see Garain and Kinnunen\cite{GK22}) to higher-order H\"{o}lder estimates (see Garain and Lindgren\cite{GL23}), extending such theories to the non-commutative geometry of the Heisenberg group demands overcoming profound technical obstacles, including the absence of embedding theorems and the inherent anisotropy of the Carnot-Carath\'{e}odory metric.

Recently, Zhang, Niu and Wu in \cite{ZNW25} proved that both the derivative in the vertical direction and the derivative in the horizontal direction of weak solutions $u \in H{W^{1,p}}\left( \Omega  \right)$ to mixed sub-Laplace and fractional sub-Laplace equations and $f \in {L^2}\left( \Omega  \right)$ belonging to the first-order local Sobolev space $HW_{loc}^{1,p}\left( \Omega  \right)$. Furthermore, under the assumption $f \in H{W^{m,p}}\left( \Omega  \right)$, it follows that $u \in HW_{loc}^{m + 2,p}\left( \Omega  \right)$. Subsequently, Zhang and Niu\cite{ZN26} achieved a significant breakthrough by establishing fundamental regularity properties for weak solutions to mixed local and nonlocal degenerate elliptic equations in $\mathbb{H}^n$. Specifically, they demonstrated the local boundedness of weak subsolutions, the H\"{o}lder continuity of weak solutions, and critically, both the Harnack inequality and the weak Harnack inequality. These results affirm that the classical difference and De Giorgi-Nash-Moser theory can be fruitfully extended to this hybrid setting, laying a robust foundation for the qualitative analysis of such equations. While H\"{o}lder continuity of the solution $u$ has been established, a central open question remains: does the horizontal gradient ${{\nabla _H}u}$ exhibit H\"{o}lder continuity, i.e., do solutions belong to the $C^{1,\alpha}$ class? This problem is exceptionally intricate in the context of degenerate ($p>2$) equations coupled with mixed operators, as standard linearization techniques prove inadequate, and the interplay between the local $p$-Laplacian structure and the nonlocal fractional term obfuscates the differentiation of the equation.

In this paper, we bridge this gap by investigating the $C^{1,\alpha}$-regularity of weak solutions to mixed local and nonlocal degenerate elliptic equations in the Heisenberg group. Building on the Harnack estimates and tail control developed in \cite{ZN26}, we aim to prove that under suitably structured assumptions, weak solutions not only exhibit H\"{o}lder continuity but also possess higher smoothness properties with their horizontal derivatives satisfying a H\"{o}lder condition. This result will extend the classical $C^{1,\alpha}$ estimates for $p$-sub-Laplace equations in $\mathbb{H}^n$ to the more intricate landscape of mixed local-nonlocal diffusion.

In this paper we focus on the analysis of the mixed local and nonlocal degenerate elliptic equations
\begin{equation}\label{eq0}
 - \Delta _{{{\mathbb{H}},p}}u + \Lambda{\left( { - \Delta _{{{\mathbb{H}},p}}} \right)^s}u = 0\;{\rm{in}}\;\Omega ,
\end{equation}
where $2\le p< \infty,\;0<s<1,\;0\le\Lambda \le 1 $, and $\Omega$ is a bounded open subset in the Heisenberg group  ${{\mathbb{H}}^n},\;n\ge1$. Here the $p$-sub-Laplace operator on the Heisenberg group is defined by
\begin{equation}\label{eq12}
 - {\Delta _{{\mathbb{H}},p}}u = di{v_H}\left( {{{\left| {{\nabla _H}u} \right|}^{p - 2}}{\nabla _H}u} \right),
\end{equation}
while the fractional $p$-sub-Laplace operator on the Heisenberg group is given by
\begin{equation}\label{eq13}
{\left( { - {\Delta _{{\mathbb{H}},p}}} \right)^s}u = {\rm{P}}{\rm{.V}}{\rm{.}}\int_{{{\mathbb{H}}^n}} {\frac{{{{\left| {u\left( \xi  \right) - u\left( \eta  \right)} \right|}^{p - 2}}\left( {u\left( \xi  \right) - u\left( \eta  \right)} \right)}}{{{{\left\| {{\eta ^{ - 1}} \circ \xi } \right\|}^{Q + sp}_{\mathbb{H}^n}}}}d\eta },
\end{equation}
where P.V. signifies the Cauchy principal value and $Q=2n+2$ is the homogeneous dimension of ${{\mathbb{H}}^n}$.

The main results are as follows:

The regularity theory for purely nonlocal equations involving fractional powers of the $p$-Laplacian on the Heisenberg group has witnessed remarkable progress in recent years, spurred by pivotal advancements such as the introduction of nonlocal tailing mechanisms and the establishment of nonlocal Harnack and H\"{o}lder regularity estimates. These contributions, as reported in Roncal and Thangavelu\cite{RT16}, Frank, Lieb, and Seiringer\cite{FLS08}, Ferrari and Franchi\cite{FF15}, Ferrari, Miranda, Pallara, Pinamonti and Sire\cite{FMPPS18}, Frank, Gonzalez, Monticelli and Tan\cite{FGMT15}, Manfredini, Palatucci, Piccinini and Polidoro\cite{MPPP23}, and Palatucci and Piccinini\cite{PP22}, collectively constitute a transformative body of work reshaping the analytical foundations of the field. Moreover, Palatucci and Piccinini in 2022 (see \cite{PP22}) and 2023 (see \cite{PP23}) proposed an open problem how to determine the regularity of nonlocal double-phase equations in the Heisenberg group. Fang, Zhang and Zhang provided a solution in \cite{FZZ24}.

Recently, Zhang, Niu and Wu in \cite{ZNW25} proved that both the derivative in the vertical direction and the derivative in the horizontal direction of weak solutions $u \in H{W^{1,p}}\left( \Omega \right)$ to mixed sub-Laplace and fractional sub-Laplace equations and $f \in {L^2}\left( \Omega \right)$ belonging to the first-order local Sobolev space $HW_{loc}^{1,p}\left( \Omega \right)$. Furthermore, under the assumption $f \in H{W^{m,p}}\left( \Omega \right)$, it follows that $u \in HW_{loc}^{m + 2,p}\left( \Omega \right)$. Subsequently, Zhang and Niu\cite{ZN26} achieved a significant breakthrough by establishing fundamental regularity properties for weak solutions to mixed local and nonlocal degenerate elliptic equations in $\mathbb{H}^n$. Specifically, they demonstrated the local boundedness of weak subsolutions, the H\"{o}lder continuity of weak solutions, and critically, both the Harnack inequality and the weak Harnack inequality. These results affirm that the classical difference and De Giorgi-Nash-Moser theory can be fruitfully extended to this hybrid setting, laying a robust foundation for the qualitative analysis of such equations. While H\"{o}lder continuity of the solution $u$ has been established, a central open question remains: does the horizontal gradient ${{\nabla _H}u}$ exhibit H\"{o}lder continuity, i.e., do solutions belong to the $C^{1,\alpha}$ class? This problem is exceptionally intricate in the context of degenerate ($p>2$) equations coupled with mixed operators, as standard linearization techniques prove inadequate, and the interplay between the local $p$-Laplacian structure and the nonlocal fractional term obfuscates the differentiation of the equation.

In this paper, we bridge this gap by investigating the $C^{1,\alpha}$-regularity of weak solutions to mixed local and nonlocal degenerate elliptic equations in the Heisenberg group. Building on the Harnack estimates and tail control developed in \cite{ZN26}, we aim to prove that under suitably structured assumptions, weak solutions not only exhibit H\"{o}lder continuity but also possess higher smoothness properties with their horizontal derivatives satisfying a H\"{o}lder condition. This result will extend the classical $C^{1,\alpha}$ estimates for $p$-sub-Laplace equations in $\mathbb{H}^n$ to the more intricate landscape of mixed local-nonlocal diffusion.

In this paper we focus on the analysis of the mixed local and nonlocal degenerate elliptic equations
\begin{equation}\label{eq0}
 - \Delta _{{{\mathbb{H}},p}}u + \Lambda{\left( { - \Delta _{{{\mathbb{H}},p}}} \right)^s}u = 0\;{\rm{in}}\;\Omega ,
\end{equation}
where $2\le p< \infty,\;0<s<1,\;0\le\Lambda \le 1 $, and $\Omega$ is a bounded open subset in the Heisenberg group ${{\mathbb{H}}^n},\;n\ge1$. Here the $p$-sub-Laplace operator on the Heisenberg group is defined by
\begin{equation}\label{eq12}
 - {\Delta _{{\mathbb{H}},p}}u = di{v_H}\left( {{{\left| {{\nabla _H}u} \right|}^{p - 2}}{\nabla _H}u} \right),
\end{equation}
while the fractional $p$-sub-Laplace operator on the Heisenberg group is given by
\begin{equation}\label{eq13}
{\left( { - {\Delta _{{\mathbb{H}},p}}} \right)^s}u = {\rm{P}}{\rm{.V}}{\rm{.}}\int_{{{\mathbb{H}}^n}} {\frac{{{{\left| {u\left( \xi \right) - u\left( \eta \right)} \right|}^{p - 2}}\left( {u\left( \xi \right) - u\left( \eta \right)} \right)}}{{{{\left\| {{\eta ^{ - 1}} \circ \xi } \right\|}^{Q + sp}_{\mathbb{H}^n}}}}d\eta },
\end{equation}
where P.V. signifies the Cauchy principal value and $Q=2n+2$ is the homogeneous dimension of ${{\mathbb{H}}^n}$.

The main results are as follows:

\begin{remark}
In contrast to \cite{ZN26}, where the H\"older index $\gamma \in (0,\frac{sp}{p-1})$ relies on the parameters $s$ and $p$, Theorem \ref{Th13} establishes a parameter-independent H\"older regularity with $\gamma \in (0,1)$. This holds under more general nonlinear frameworks and is particularly advantageous for subsequent regularity analysis.
\end{remark}

\begin{proposition}\label{Pro41}
Let $2\le p <\infty,\;0<s<1$ and $0\le \Lambda \le 1$. Assume $u \in HW_{{\rm{loc}}}^{1,p}\left( {{B_2}\left( {{\xi _0}} \right) \cap L_{sp}^{p - 1}\left( {{\mathbb{H}^n}} \right)} \right)$ is a weak solution of \eqref{eq0} in ${{B_2}\left( {{\xi _0}} \right)}$ satisfying
\begin{equation}\label{eq41}
{\left\| u \right\|_{{L^\infty }\left( {{B_1}\left( {{\xi _0}} \right)} \right)}} \le 1,\int_{{\mathbb{H}^n}\backslash {B_1}\left( {{\xi _0}} \right)} {\frac{{{{\left| {u\left( \eta \right)} \right|}^{p - 1}}}}{{\left\| \eta \right\|_{{\mathbb{H}^n}}^{Q + sp}}}d\eta } \le 1.
\end{equation}
If $0<h_0<\frac{1}{10}$ and $R$ satisfy $4{h_0}<R\le 1-5{h_0}$, and ${\nabla _H}u \in {L^q}\left( {{B_{R + 4{h_0}}}\left( {{\xi _0}} \right)} \right)$ for some $q \ge p$, then
\begin{equation}\label{eq42}
 \mathop {\sup }\limits_{0 < \left| h \right| < {h_0}} \int_{{B_{R - {4h_0}}}} {{{\left| {\frac{{\Delta _{Z,h}^2u}}{{{{\left| h \right|}^{\frac{1}{{q + 1}} + 1}}}}} \right|}^{q + 1}}d\xi } \le c(1+\Lambda) \left( {\int_{{B_{R + {4h_0}}}} {{{\left| {{\nabla _H}u} \right|}^q}d\xi } + 1} \right),
\end{equation}
where $c=c(Q,h_0,p,q,s)>0$.
\end{proposition}

For simplicity, we denote $u_R$ by $u$. Fix $0<\gamma<1$ and choose $i_\infty \in \mathbb{N}\setminus {0}$ such that
\begin{equation}\label{eq492}
1 - \gamma > \frac{Q}{{p + {i_\infty }}}.
\end{equation}
For $i = 0, \cdots ,{i_\infty }$, we define
\[{q_i} = p + i,\]
and
\[{h_0} = \frac{1}{{112{i_\infty }}},\;{R_i} = \frac{7}{8} - 4{h_0} - 14{h_0}i,\]
so
\[{R_0} + 4{h_0} = \frac{7}{8},\;{R_{{i_\infty }}} + 4{h_0} = \frac{3}{4},\;4{h_0} <R_i<1-5{h_0} .\]
By using Proposition \ref{Pro41} with
\[R = {R_i},q = {q_i},\;i = 0, \cdots ,{i_\infty },\]
and we have by \eqref{eq485}
\begin{equation}\label{eq486}
\mathop {\sup }\limits_{0 < \left| h \right| < {h_0}} \int_{{B_{R_0 - {4h_0}}}} {{{\left| {\frac{{\Delta _{Z,h}^2u}}{{{{\left| h \right|}^{\frac{1}{{{q_1}}} + 1}}}}} \right|}^{{q_1}}}d\xi } \le c(1 + \Lambda )\left( {\int_{{B_{\frac{7}{8}}}} {{{\left| {{\nabla _H}u} \right|}^p}d\xi } + 1} \right) \le C\left( {Q,p,s,\gamma} \right).
\end{equation}
Consider ${R_i} - 10{h_0} = {R_{i + 1}} + 4{h_0}$ for $i = 0, \cdots ,{i_\infty }$, we use Lemma \ref{Le213} in \eqref{eq486} to get
\begin{equation}\label{eq487}
\int_{{B_{{R_1} + 4{h_0}}}} {{{\left| {{\nabla _H}{u}} \right|}^{{q_1}}}d\xi } \le C\left( {Q,p,s,\gamma} \right).
\end{equation}
Applying Proposition \ref{Pro41} and \eqref{eq487} once again, we obtain
\begin{equation}\label{eq488}
\mathop {\sup }\limits_{0 < \left| h \right| < {h_0}} \int_{{B_{{R_1} - 4{h_0}}}} {{{\left| {\frac{{\Delta _{Z,h}^2u}}{{{{\left| h \right|}^{\frac{1}{{{q_2}}} + 1}}}}} \right|}^{{q_2}}}d\xi } \le c(1 + \Lambda )\left( {\int_{{B_{{{{R_1} + 4{h_0}}}}}} {{{\left| {{\nabla _H}u} \right|}^{{q_1}}}d\xi } + 1} \right) \le C\left( {Q,p,s,\gamma} \right).
\end{equation}
Moreover, by Lemma \ref{Le213} in \eqref{eq488}, we have
\begin{equation}\label{eq489}
\int_{{B_{{R_2} + 4{h_0}}}} {{{\left| {{\nabla _H}{u}} \right|}^{{q_2}}}d\xi } \le C\left( {Q,p,s,\gamma} \right).
\end{equation}
Repeating this procedure, it yields
\begin{equation}\label{eq490}
 \int_{{B_{{R_{i+1}} + 4{h_0}}}} {{{\left| {{\nabla _H}{u}} \right|}^{{q_{i+1}}}}d\xi } \le C\left( {Q,p,s,\gamma} \right)
\end{equation}
for all $i = 0, \cdots ,{i_\infty }-1.$ Taking $i={i_\infty }-1$ in \eqref{eq490} and using ${\left\| {{u}} \right\|_{{L^\infty }\left( {{B_1}} \right)}} \le 1$, we deduce
\begin{equation}\label{eq491}
{\left\| u \right\|_{H{W^{1,{q_{{i_\infty }}}}}\left( {{B_{{R_{{i_\infty }}} + 4{h_0}}}} \right)}} \le C\left( {Q,p,s,\gamma } \right).
\end{equation}
By means of \eqref{eq492}, we know ${q_{{i_\infty }}}>Q$, so we use ${R_{{i_\infty }}} + 4{h_0} = \frac{3}{4}$ and Lemma \ref{Le23} to get
$u \in C_{loc}^\gamma \left( {{B_{\frac{3}{4}}}} \right)$. In fact, we take a cut-off function $\psi$ between ${{B_{\frac{1}{2}}}}$ and ${{B_{\frac{3}{4}}}}$. Then
\begin{align}\label{eq493}
{\left\| u \right\|_{{C^\gamma }\left( {{B_{\frac{1}{2}}}} \right)}}& \le {\left\| {u\psi } \right\|_{{C^\gamma }\left( {{{\mathbb{H}}^n}} \right)}} \le c{\left\| {u\psi } \right\|_{H{W^{1,{q_{{i_\infty }}}}}\left( {{{\mathbb{H}}^n}} \right)}} = c{\left\| {u\psi } \right\|_{H{W^{1,{q_{{i_\infty }}}}}\left( {{B_{\frac{3}{4}}}} \right)}}\nonumber\\
& \le c{\left\| u \right\|_{H{W^{1,{q_{{i_\infty }}}}}\left( {{B_{\frac{3}{4}}}} \right)}} \le C\left( {Q,p,s,\gamma } \right).
\end{align}
Therefore, Theorem is proved.

\textbf{Proof of Theorem \ref{Th14}} After rescaling the variables as in the proof of Theorem \ref{Th13}, it suffices to show that
$${\left\| u_R \right\|_{{C^{1,\alpha }}\left( {{B_{\frac{1}{8}}}} \right)}} \le c\left( {Q,p,s,\gamma } \right),$$
where the function $u_R$ defined in \eqref{eq484} satisfies the following conditions:
\begin{equation}\label{eq52}
{\left\| {{u_R}} \right\|_{{L^\infty }\left( {{B_1}} \right)}} \le 1,\;{\int _{{\mathbb{H}^n}\backslash {B_1}}}\frac{{{{\left| {{u_R}\left( \eta \right)} \right|}^{p - 1}}}}{{\left\| \eta \right\|_{{\mathbb{H}^n}}^{Q + sp}}}d\eta \le 1.
\end{equation}
By Theorem \ref{Th13}, there exists $\gamma>\frac{sp}{p-1}$ such that
\[\mathop {\sup }\limits_{\xi \ne \eta \in {B_{\frac{1}{2}}}\left( {{\xi _0}} \right)} \frac{{\left| {{u_R}\left( \xi \right) - {u_R}\left( \eta \right)} \right|}}{{{{\left\| {{\eta ^{ - 1}} \circ \xi } \right\|}_{{\mathbb{H}^n}}^\gamma }}} \le c\left( {Q,p,s,\gamma } \right).\]
Now take any $\xi_0 \in {B_{\frac{1}{4}}}$, then we deduce from the above formula and the choice of $\gamma$ that
\[\int_{{B_{\frac{1}{4}}}\left( {{\xi _0}} \right)} {\frac{{{{\left| {{u_R}\left( {{\xi _0}} \right) - {u_R}\left( \eta \right)} \right|}^{p - 1}}}}{{{{\left\| {{\eta ^{ - 1}} \circ {\xi _0}} \right\|}_{{\mathbb{H}^n}}^{Q + sp}}}}d\eta } \le c\left( {Q,p,s,\gamma } \right)\int_{{B_{\frac{1}{4}}}\left( {{\xi _0}} \right)} {\frac{1}{{{{\left\| {{\eta ^{ - 1}} \circ {\xi _0}} \right\|}_{{\mathbb{H}^n}}^{Q + sp - \gamma \left( {p - 1} \right)}}}}d\eta } = c\left( {Q,p,s,\gamma } \right).\]
Moreover, we have by \eqref{eq52}
\[\int_{{\mathbb{H}^n}\backslash {B_{\frac{1}{4}}}\left( {{\xi _0}} \right)} {\frac{{{{\left| {{u_R}\left( {{\xi _0}} \right) - {u_R}\left( \eta \right)} \right|}^{p - 1}}}}{{{{\left\| {{\eta ^{ - 1}} \circ {\xi _0}} \right\|}_{{\mathbb{H}^n}}^{Q + sp}}}}d\eta } \le c\left( {Q,p,s} \right).\]
Hence
\[{\left\| {{{{{\left( { - \Delta _{{{\mathbb{H}},p}}} \right)}^s}}}{u_R}} \right\|_{{L^\infty }\left( {{B_{\frac{1}{4}}}} \right)}} \le c\left( {Q,p,s,\gamma } \right),\]
and also
\[{\left\| {{{{\left( { - \Delta _{{{\mathbb{H}},p}}} \right)}}}{u_R}} \right\|_{{L^\infty }\left( {{B_{\frac{1}{4}}}} \right)}} \le c\left( {Q,p,s,\gamma } \right),\]
which together with \eqref{eq52} and Theorem 1.2 in \cite{MZ21} imply
\[{\left\| {{u_R}} \right\|_{{C^{1,,\alpha }}\left( {{B_{\frac{1}{8}}}} \right)}} \le c\left( {Q,p,s,\gamma } \right).\]

\begin{remark}
In contrast to \cite{ZN26}, where the H\"older index $\gamma \in (0,\frac{sp}{p-1})$ relies on the parameters $s$ and $p$, Theorem \ref{Th13} establishes a parameter-independent H\"older regularity with $\gamma \in (0,1)$. This holds under more general nonlinear frameworks and is particularly advantageous for subsequent regularity analysis.
\end{remark}

\begin{theorem}\label{Th14}
Let $2\le p <\infty,\;0<s<1,\;sp<p-1$ and $0\le \Lambda \le 1$. Suppose $\Omega \in \mathbb{H}^n$ is a bounded open set and $u \in HW_{{\rm{loc}}}^{1,p}\left( \Omega \right) \cap{ L_{sp}^{p - 1}\left( {{\mathbb{H}^n}} \right)} $ is a weak solution of \eqref{eq0}. Then $u \in C_{loc}^{1,\alpha} \left( \Omega  \right)$ for some $0 < \alpha  < 1$. Moreover, for some $0 < \alpha  < 1$ and any Kor\'{a}nyi ball ${B_{2R}}\left( {{\xi _0}} \right) \subset  \subset \Omega $ with $0<R<1$, there exists a positive constant $c=c(Q,p,q,s,\gamma)$ such that
\begin{equation}\label{eq51}
 \mathop {\sup }\limits_{\xi  \ne \eta  \in {B_{\frac{R}{8}}}\left( {{\xi _0}} \right)} \frac{{\left| {{\nabla _H}u\left( \xi  \right) - {\nabla _H}u\left( \eta  \right)} \right|}}{{{{\left\| {{\eta ^{ - 1}} \circ \xi } \right\|}^\alpha _{\mathbb{H}^n}}}} \le \frac{c}{{{R^{1 + \alpha }}}}\left( {{{\left\| u \right\|}_{{L^\infty }\left( {{B_R}\left( {{\xi _0}} \right)} \right)}} + {\rm{Tai}}{{\rm{l}}_{p - 1,sp,p}}\left( {u;{\xi _0},R} \right)} \right),
\end{equation}
where the definition of ${\rm{Tai}}{{\rm{l}}_{p - 1,sp,p}}\left( {u;{\xi _0},R} \right)$ see \eqref{eq26} below.
\end{theorem}

The paper is organized as follows: In Section 2, we introduce relevant knowledge of the Heisenberg group, function spaces, and several necessary and new lemmas. In Section 3, we employ horizontal difference, along with certain difference estimates and the fractional Sobolev-type inequality on the Heisenberg group, to establish the crucial iteration scheme of Morrey-type for weak solutions to \eqref{eq0} required for the proof of the Theorem \ref{Th13}, i.e. Proposition \ref{Pro41}. In Section 4, we first establish a proposition regarding the gradient integrability of weak solutions by using Caccioppoli-type inequality, i.e. Proposition \ref{Pro42}. Then, by combining local boundedness, the Proposition \ref{Pro41}, the Proposition \ref{Pro42}, an iterative method, and the Morrey inequality, we prove Theorem \ref{Th13}. Finally, Theorem \ref{Th14} is derived by utilizing Theorem \ref{Th13} and Theorem 1.2 from \cite{MZ21}.

In the sequel, we denote by $c$ a generic positive constant which may vary from line to line. Relevant dependencies on parameters shall be emphasised utilizing parentheses, i.e., $c= c(n, p, q)$ means that $c$ depends on $n, p, q$.

To prove Theorem~\ref{Th13}, the paper proceeds as follows. In Section~3, it uses “horizontal difference, along with certain difference estimates and the fractional Sobolev-type inequality on the Heisenberg group” to “establish the crucial iteration scheme of Morrey-type for weak solutions to \eqref{eq0} required for the proof of the Theorem~\ref{Th13}, i.e. Proposition~\ref{Pro41}.” In Section~4, it “first establish[es] a proposition regarding the gradient integrability of weak solutions by using Caccioppoli-type inequality, i.e. Proposition~\ref{Pro42}.” Then, “by combining local boundedness, the Proposition~\ref{Pro41}, the Proposition~\ref{Pro42}, an iterative method, and the Morrey inequality, we prove Theorem~\ref{Th13}.”

To prove the main theorem, the paper proceeds as follows. Next, it uses “horizontal difference, along with certain difference estimates and the fractional Sobolev-type inequality on the Heisenberg group” to “establish the crucial iteration scheme of Morrey-type for weak solutions to
\begin{equation}\label{eq0}
 - \Delta _{{{\mathbb{H}},p}}u + \Lambda{\left( { - \Delta _{{{\mathbb{H}},p}}} \right)^s}u = 0\;{\rm{in}}\;\Omega ,
\end{equation}
required for the proof of the main theorem, i.e. the following proposition.

\begin{proposition}\label{Pro41}
Let $2\le p <\infty,\;0<s<1$ and $0\le \Lambda \le 1$. Assume $u \in HW_{{\rm{loc}}}^{1,p}\left( {{B_2}\left( {{\xi _0}} \right) \cap L_{sp}^{p - 1}\left( {{\mathbb{H}^n}} \right)} \right)$ is a weak solution of \eqref{eq0} in ${{B_2}\left( {{\xi _0}} \right)}$ satisfying
\begin{equation}\label{eq41}
{\left\| u \right\|_{{L^\infty }\left( {{B_1}\left( {{\xi _0}} \right)} \right)}} \le 1,\int_{{\mathbb{H}^n}\backslash {B_1}\left( {{\xi _0}} \right)} {\frac{{{{\left| {u\left( \eta  \right)} \right|}^{p - 1}}}}{{\left\| \eta  \right\|_{{\mathbb{H}^n}}^{Q + sp}}}d\eta }  \le 1.
\end{equation}
If $0<h_0<\frac{1}{10}$ and $R$ satisfy $4{h_0}<R\le 1-5{h_0}$, and ${\nabla _H}u \in {L^q}\left( {{B_{R + 4{h_0}}}\left( {{\xi _0}} \right)} \right)$ for some $q \ge p$, then
\begin{equation}\label{eq42}
 \mathop {\sup }\limits_{0 < \left| h \right| < {h_0}} \int_{{B_{R - {4h_0}}}} {{{\left| {\frac{{\Delta _{Z,h}^2u}}{{{{\left| h \right|}^{\frac{1}{{q + 1}} + 1}}}}} \right|}^{q + 1}}d\xi } \le c(1+\Lambda) \left(  {\int_{{B_{R + {4h_0}}}} {{{\left| {{\nabla _H}u} \right|}^q}d\xi } + 1} \right),
\end{equation}
where $c=c(Q,h_0,p,q,s)>0$.
\end{proposition}

Next, it “first establish[es] a proposition regarding the gradient integrability of weak solutions by using Caccioppoli-type inequality, i.e.” the following.

\begin{proposition}\label{Pro42}
Let $2\le p <\infty,\;0<s<1$ and $0\le \Lambda \le 1$. If $u \in HW_{{\rm{loc}}}^{1,p}\left( {{B_2}\left( {{\xi _0}} \right) \cap L_{sp}^{p - 1}\left( {{\mathbb{H}^n}} \right)} \right)$ is a weak solution of \eqref{eq0} in ${{B_{2}}\left( {{\xi _0}} \right)}$ satisfying \eqref{eq41}, then
\begin{equation*}
\int_{{B_{\frac{7}{8}}}\left( {{\xi _0}} \right)} {{{\left| {{\nabla _H}u} \right|}^p}d\xi }  \le C\left( {Q,p,s} \right).
\end{equation*}
\end{proposition}

Then, “by combining local boundedness, the proposition above, the preceding proposition, an iterative method, and the Morrey inequality, we prove the main theorem.”

\label{Th13}
Let $2\le p <\infty,\;0<s<1$ and $0\le \Lambda \le 1$. Suppose $\Omega \in \mathbb{H}^n$ is a bounded open set and $u \in HW_{{\rm{loc}}}^{1,p}\left( \Omega \right) \cap{ L_{sp}^{p - 1}\left( {{\mathbb{H}^n}} \right)} $ is a weak solution of
\begin{equation}\label{eq0}
 - \Delta _{{{\mathbb{H}},p}}u + \Lambda{\left( { - \Delta _{{{\mathbb{H}},p}}} \right)^s}u = 0\;{\rm{in}}\;\Omega ,
\end{equation}
Then $u \in C_{loc}^\gamma \left( \Omega  \right)$ for any $0 < \gamma  < 1$. Moreover, for any $0 < \gamma  < 1$ and any Kor\'{a}nyi ball ${B_{2R}}\left( {{\xi _0}} \right) \subset  \subset \Omega $ with $0<R<1$, there exists a positive constant $c=c(Q,p,s,\gamma)$ such that
\begin{equation}\label{eq482}
  \mathop {\sup }\limits_{\xi  \ne \eta  \in {B_{\frac{R}{2}}}\left( {{\xi _0}} \right)} \frac{{\left| {u\left( \xi  \right) - u\left( \eta  \right)} \right|}}{{{{\left\| {{\eta ^{ - 1}} \circ \xi } \right\|}^\gamma _{\mathbb{H}^n}}}} \le \frac{c}{{{R^\gamma }}}\left( {{{{\left\| u \right\|}_{{L^\infty }\left( {{B_R}\left( {{\xi _0}} \right)} \right)}}} + {\rm{Tai}}{{\rm{l}}_{p - 1,sp,p}}\left( {u;{\xi _0},R} \right)} \right),
\end{equation}
where ${\rm{Tai}}{{\rm{l}}_{p - 1,sp,p}}\left( {u;{\xi _0},R} \right)$ is given by
\begin{equation}\label{eq26}
 {\rm{Tai}}{{\rm{l}}_{q,\alpha ,\beta }}\left( {u;{\xi _0},R} \right) = {\left[ {{R^\beta }\int_{{{\mathbb{H}}^n}\backslash {B_R}\left( {{\xi _0}} \right)} {\frac{{{{\left| u \right|}^q}}}{{{{\left\| {\xi _0^{ - 1} \circ \xi } \right\|}^{Q + \alpha }_{\mathbb{H}^n}}}}d\xi } } \right]^{\frac{1}{q}}}.
\end{equation}

Question: Let \(\mathbb{H}^n\) be the Heisenberg group with homogeneous dimension \(Q=2n+2\), Kor\u00e1nyi gauge \(\|\cdot\|_{\mathbb{H}^n}\), and Kor\u00e1nyi balls \(B_r(\xi_0)=\{\xi\in\mathbb{H}^n:\|\xi_0^{-1}\circ\xi\|_{\mathbb{H}^n}<r\}\). Define the \(p\)-sub-Laplacian by \(-\Delta_{\mathbb{H},p}u=\operatorname{div}_H(|\nabla_Hu|^{p-2}\nabla_Hu)\), and the fractional \(p\)-sub-Laplacian by \(( -\Delta_{\mathbb{H},p})^s u(\xi)=\mathrm{P.V.}\int_{\mathbb{H}^n}\frac{|u(\xi)-u(\eta)|^{p-2}(u(\xi)-u(\eta))}{\|\eta^{-1}\circ\xi\|_{\mathbb{H}^n}^{Q+sp}}\,d\eta\). Assume \(2\le p<\infty\), \(0<s<1\), \(0\le \Lambda\le1\), \(\Omega\subset\mathbb{H}^n\) is bounded and open, and \(u\in HW^{1,p}_{\mathrm{loc}}(\Omega)\cap L_{sp}^{p-1}(\mathbb{H}^n)\) is a weak solution of \(-\Delta_{\mathbb{H},p}u+\Lambda(-\Delta_{\mathbb{H},p})^s u=0\) in \(\Omega\). For \(q,\alpha,\beta>0\), define \(\operatorname{Tail}_{q,\alpha,\beta}(u;\xi_0,R)=\left[R^{\beta}\int_{\mathbb{H}^n\setminus B_R(\xi_0)}\frac{|u(\xi)|^q}{\|\xi_0^{-1}\circ\xi\|_{\mathbb{H}^n}^{Q+\alpha}}\,d\xi\right]^{1/q}\). Under these assumptions, which quantitative estimate holds?

For every \(0<\gamma<1\), \(u\in C^\gamma_{\mathrm{loc}}(\Omega)\). More precisely, for every \(0<\gamma<1\) and every Kor\u00e1nyi ball \(B_{2R}(\xi_0)\Subset\Omega\) with \(0<R<1\), there exists a constant \(c=c(Q,p,s,\gamma)>0\) such that \[\sup_{\xi\ne\eta\in B_{R/2}(\xi_0)}\frac{|u(\xi)-u(\eta)|}{\|\eta^{-1}\circ\xi\|_{\mathbb{H}^n}^{\gamma}}\le \frac{c}{R^{\gamma}}\left(\|u\|_{L^\infty(B_R(\xi_0))}+\operatorname{Tail}_{p-1,sp,p}(u;\xi_0,R)\right).\]

Choices:
B. For every \(0<\gamma<1\), \(u\in C^\gamma_{\mathrm{loc}}(\Omega)\). More precisely, for every \(0<\gamma<1\) and every Kor\u00e1nyi ball \(B_{2R}(\xi_0)\Subset\Omega\) with \(0<R<1\), there exists a constant \(c=c(Q,p,s,\gamma)>0\) such that \[\sup_{\xi\ne\eta\in B_R(\xi_0)}\frac{|u(\xi)-u(\eta)|}{\|\eta^{-1}\circ\xi\|_{\mathbb{H}^n}^{\gamma}}\le \frac{c}{R^{\gamma}}\left(\|u\|_{L^\infty(B_R(\xi_0))}+\operatorname{Tail}_{p-1,sp,p}(u;\xi_0,R)\right).\]
C. There exists some \(\gamma\in(0,1)\) such that \(u\in C^\gamma_{\mathrm{loc}}(\Omega)\). More precisely, for every Kor\u00e1nyi ball \(B_{2R}(\xi_0)\Subset\Omega\) with \(0<R<1\), there exists a constant \(c=c(Q,p,s,\gamma)>0\) such that \[\sup_{\xi\ne\eta\in B_{R/2}(\xi_0)}\frac{|u(\xi)-u(\eta)|}{\|\eta^{-1}\circ\xi\|_{\mathbb{H}^n}^{\gamma}}\le \frac{c}{R^{\gamma}}\left(\|u\|_{L^\infty(B_R(\xi_0))}+\operatorname{Tail}_{p-1,sp,p}(u;\xi_0,R)\right).\]
D. For every \(0<\gamma<1\), \(u\in C^\gamma_{\mathrm{loc}}(\Omega)\). More precisely, for every \(0<\gamma<1\) and every Kor\u00e1nyi ball \(B_{2R}(\xi_0)\Subset\Omega\) with \(0<R<1\), there exists a constant \(c=c(Q,p,s)>0\), independent of \(\gamma\), such that \[\sup_{\xi\ne\eta\in B_{R/2}(\xi_0)}\frac{|u(\xi)-u(\eta)|}{\|\eta^{-1}\circ\xi\|_{\mathbb{H}^n}^{\gamma}}\le \frac{c}{R^{\gamma}}\left(\|u\|_{L^\infty(B_R(\xi_0))}+\operatorname{Tail}_{p-1,sp,p}(u;\xi_0,R)\right).\]
E. For every \(0<\gamma<1\), \(u\in C^\gamma_{\mathrm{loc}}(\Omega)\). More precisely, for every \(0<\gamma<1\) and every Kor\u00e1nyi ball \(B_{2R}(\xi_0)\Subset\Omega\) with \(0<R<1\), there exists a constant \(c=c(Q,p,s,\gamma)>0\) such that \[\sup_{\xi\ne\eta\in B_{R/2}(\xi_0)}\frac{|u(\xi)-u(\eta)|}{\|\eta^{-1}\circ\xi\|_{\mathbb{H}^n}^{\gamma}}\le \frac{c}{R^{\gamma}}\left(\|u\|_{L^\infty(B_R(\xi_0))}+\operatorname{Tail}_{p,sp,p}(u;\xi_0,R)\right).\]

The stem gives assumptions and definitions but does not state or strongly hint at the exact conclusion. The correct estimate is not leaked explicitly.

The item is essentially a theorem-recall question: it asks directly which quantitative estimate holds under the stated assumptions, with choices that are near-variants of the theorem statement.

Some reasoning is needed to distinguish subtle variants involving interior shrinkage, quantification over b3, dependence of constants, and the tail exponent. However, the task is closer to recognizing the exact theorem statement than generating a conclusion from first principles.

The distractors are mathematically plausible and target realistic failure modes: overstrong domain of the seminorm, weakened existential quantifier, incorrect uniformity in b3, and a wrong tail exponent. They are distinct and well-aligned with common confusions.

A technically well-crafted but theorem-recall-heavy MCQ. It avoids answer leakage and has strong distractors, but it is largely a restatement/recognition task rather than a genuinely generative reasoning question.


\label{main th}

There exists a minimizing couple $(u,\ell)$ of the following problem in both variables
\begin{equation}
    \label{pb min}
    \min\left \{E_\varepsilon(u,\ell), \, u \in  H^{1}({\mathcal C}) \text{ such that } 0\leqslant u \leqslant 1 , \, u = 1 \text{ on } \partial{{\mathcal C}} \text{ and } \ell \in \mathrm{Hom}^\Lambda(\gamma_0,\gamma_1) \right\}.
\end{equation}

Moreover, the minimizer $u$ is $\alpha$-Hölder continuous for all $0<\alpha<1$ with the estimate:
\[\|u\|_{C^{0,\alpha}(\mathcal{C})} \leqslant C_\alpha \frac{1+\Lambda c_\varepsilon ^{-1}}{\varepsilon^{\alpha}}.\]

\begin{proposition}
\label{prop 1.2}
For all $u \in H^1(\C)\cap C(\overline{\Ccal})$ there exists a minimizer for the problem \eqref{pb min u fixe}.
\end{proposition}

\begin{theorem}
\label{main th}

There exists a minimizing couple $(u,\ell)$ of the following problem in both variables
\begin{equation}
    \label{pb min}
    \min\left \{E_\eps(u,\ell), \, u \in  H^{1}(\C) \text{ such that } 0\leqslant u \leqslant 1 , \, u = 1 \text{ on } \partial{\C} \text{ and } \ell \in \curve^\Lambda(\gamma_0,\gamma_1) \right\}.
\end{equation}

Moreover, the minimizer $u$ is $\alpha$-Hölder continuous for all $0<\alpha<1$ with the estimate:
\[\|u\|_{C^{0,\alpha}(\mathcal{C})} \leqslant C_\alpha \frac{1+\Lambda c_\varepsilon ^{-1}}{\varepsilon^{\alpha}}.\]

\end{theorem}

This paper is devoted to the study of the functional introduced in \cite{bonnivard2025phasefieldapproximationplateaus} to approximate a Plateau problem, which consists in finding a surface of minimal area spanning a collection of closed curves $\gamma_0,..., \gamma_n$ contained in the boundary of an open, bounded, convex set $\Ccal_0 \subset \mathbb{R}^3$.

More precisely, the competitor surfaces are defined as the images of homotopies connecting the given curves. We define below the set of admissible homotopies connecting a curve $\gamma_i$ to $\gamma_j$:
\[
\mathrm{Hom}(\gamma_i,\gamma_j) : = \{ \ell \in \mathrm{Lip}([0,1] \times \mathbb{S}^{1}, \overline{\Ccal_0}) \text{ such that } \ell(0) = \gamma_i \text{ and } \ell(1) = \gamma_{j} \}.
\]
Note that we require more than mere homotopies: we specifically consider Lipschitz homotopies. For any homotopy $\ell \in \mathrm{Hom}(\gamma_i,\gamma_j)$, we define the associated surface $\surf_\ell$ as its image:
\begin{equation}\label{Def:surface_image}
	\surf_{\ell} := \ell([0,1] \times \mathbb{S}^{1}) \subset \overline{\Ccal_0}.
\end{equation}
We emphasize that the surface $\surf_{\ell}$ is $\mathcal{H}^{2}$-rectifiable and has finite $\mathcal{H}^{2}$-measure. Moreover, for any $u\in H^1(\mathbb{R}^3)$, the trace of $u$ on $\surf_{\ell}$ is well defined, since the set of points that are not Lebesgue points of $u$ is $\mathcal{H}^2$-negligible.
However, to obtain the existence result, we need to impose additional properties on the competitors.
For $\Lambda>0$, we consider
\[
\mathrm{Hom}^\Lambda(\gamma_i,\gamma_j) : = \{ \ell \in \mathrm{Hom}(\gamma_i,\gamma_j) \text{ such that }  \;\mathrm{Lip}(\ell)\leq \Lambda \text{ and } \surf_\ell \text{ is } \Lambda-\text{upper Ahlfors regular} \ \}.\]

Let us recall the definition of upper Ahlfors regularity.
\begin{definition}
Let $E\subset \mathbb{R}^3$ and $\Lambda>0$. The set $E$ is said to be $\Lambda$-upper Ahlfors regular if for all $x\in E$ and all $r>0$ we have
\[\mathcal{H}^{2}(E\cap B(x,r)) \leqslant \Lambda \pi r^2.\]
\end{definition}

The definition of our functional is based on the notion of a generalized geodesic distance between curves, associated with a given weight function $u$, which is defined as follows.
\begin{definition}\label{Def:geoDistance}
	Let $(\delta_\varepsilon)_{\varepsilon>0}$ be a sequence of positive numbers and $u \in C(\overline{\Ccal_0})$. We define the geodesic distance between $\gamma_i$ and $\gamma_j$:
	\begin{align*}
			&d_{u}(\gamma_i,\gamma_j) := \inf\left \{  \int_{\surf_{\ell}}( u^2 + \delta_\varepsilon) d\mathcal{H}^{2} \ |\ \ell \in \mathrm{Hom}^\Lambda(\gamma_i,\gamma_j)\right \}.
	\end{align*}
\end{definition}
Note that, in \cite{bonnivard2025phasefieldapproximationplateaus}, the geodesic distance is defined for Lipschitz homotopies $\ell$. The assumptions of uniform Lipschitz regularity and uniform upper Ahlfors regularity of the images are not required there. However, in the present work, we need to impose these assumptions in order to obtain the compactness result necessary for the existence theorem.

Let $\mathcal{C}$ an smooth open bounded convex set containing the closure $\overline{\Ccal_0}$. This set $\mathcal{C}$ will serve as our domain of study. Indeed, since we rely on PDE techniques, it is convenient to work within a smooth bounded domain. We now recall the definition of the functional introduced in \cite{bonnivard2025phasefieldapproximationplateaus}:
	\begin{equation}
		\label{functionalGeneral}
		F_{\varepsilon}(u) := \varepsilon \int_{\mathcal{C}}|\nabla u|^{2} dx + \frac{1}{4\varepsilon}\int_{\mathcal{C}}(1-u)^{2}dx + \frac{1}{c_{\varepsilon}}\sum_{i=1}^{n}d_{u}(\gamma_0, \gamma_{i}).
	\end{equation}
When, the sequences of positive numbers $(\delta_\varepsilon)$ and $(c_\varepsilon)$ are assumed to converge to zero, and to satisfy that $\delta_\varepsilon/c_\varepsilon$ converges to zero as $\varepsilon \to 0$, \cite{bonnivard2025phasefieldapproximationplateaus} establishes that this functional approximate some Plateau's problem through a $\Gamma$-convergence result.

For simplicity, we will assume in the following that the prescribed boundary consists of only two curves, $\gamma_0$ and $\gamma_1$. However, the results established in the present article remain valid when the boundary contains more than two curves.

In this paper, we are specifically interested in the decoupled functional defined below.
\begin{definition}
	For $u \in H^{1}({\mathcal C})$ such that $0\leqslant u \leqslant 1$, with $u = 1$ on $\partial{{\mathcal C}}$, and $\ell \in \mathrm{Hom}^\Lambda(\gamma_0,\gamma_1)$,
	\begin{align*}
		&E_{\varepsilon}(u,\ell) := \varepsilon\int_{\mathcal{C}} |\nabla u|^{2} dx + \frac{1}{4\varepsilon}\int_{\mathcal{C}}(1-u)^{2} dx + \frac{1}{c_{\varepsilon}}\int_{\surf_{\ell}} (u^2 + \delta_\varepsilon) d\mathcal{H}^{2}.
	\end{align*}
\end{definition}

\begin{remark}
	From Definition~\ref{Def:geoDistance} of the geodesic distance between closed curves, we know that for all $u\in H^{1}(\mathcal{C})$ such that $0\leqslant u \leqslant 1$ and $u = 1$ on $\partial{\mathcal{C}}$,
		\[F_{\varepsilon}(u) = \inf\left \{ E_{\varepsilon}(u,\ell)\ ,\ell \in \mathrm{Hom}^\Lambda(\gamma_0,\gamma_1)  \right \}.\]
\end{remark}

The main result of this paper is the existence of a pair that minimizes this decoupled energy $E_\varepsilon$
\begin{equation*}
    \inf_{(u,\ell)} E_\varepsilon(u,\ell).
\end{equation*}

More precisely, the competitor surfaces are defined as the images of homotopies connecting the given curves. We define below the set of admissible homotopies connecting a curve $\gamma_i$ to $\gamma_j$:
\[
\mathrm{Hom}(\gamma_i,\gamma_j) : = \{ \ell \in \mathrm{Lip}([0,1] \times \mathbb{S}^{1}, \overline{\Ccal_0}) \text{ such that } \ell(0) = \gamma_i \text{ and } \ell(1) = \gamma_{j} \}.
\]
Note that we require more than mere homotopies: we specifically consider Lipschitz homotopies. For any homotopy $\ell \in \mathrm{Hom}(\gamma_i,\gamma_j)$, we define the associated surface $\surf_\ell$ as its image:
\begin{equation}\label{Def:surface_image}
 \surf_{\ell} := \ell([0,1] \times \mathbb{S}^{1}) \subset \overline{\Ccal_0}.
\end{equation}
We emphasize that the surface $\surf_{\ell}$ is $\mathcal{H}^{2}$-rectifiable and has finite $\mathcal{H}^{2}$-measure. Moreover, for any $u\in H^1(\mathbb{R}^3)$, the trace of $u$ on $\surf_{\ell}$ is well defined, since the set of points that are not Lebesgue points of $u$ is $\mathcal{H}^2$-negligible.
However, to obtain the existence result, we need to impose additional properties on the competitors.
For $\Lambda>0$, we consider
\[
\mathrm{Hom}^\Lambda(\gamma_i,\gamma_j) : = \{ \ell \in \mathrm{Hom}(\gamma_i,\gamma_j) \text{ such that } \;\mathrm{Lip}(\ell)\leq \Lambda \text{ and } \surf_\ell \text{ is } \Lambda-\text{upper Ahlfors regular} \ \}.\]

The definition of our functional is based on the notion of a generalized geodesic distance between curves, associated with a given weight function $u$, which is defined as follows.
\begin{definition}\label{Def:geoDistance}
 Let $(\delta_\varepsilon)_{\varepsilon>0}$ be a sequence of positive numbers and $u \in C(\overline{\Ccal_0})$. We define the geodesic distance between $\gamma_i$ and $\gamma_j$:
 \begin{align*}
 &d_{u}(\gamma_i,\gamma_j) := \inf\left \{ \int_{\surf_{\ell}}( u^2 + \delta_\varepsilon) d\mathcal{H}^{2} \ |\ \ell \in \mathrm{Hom}^\Lambda(\gamma_i,\gamma_j)\right \}.
 \end{align*}
\end{definition}
Note that, in \cite{bonnivard2025phasefieldapproximationplateaus}, the geodesic distance is defined for Lipschitz homotopies $\ell$. The assumptions of uniform Lipschitz regularity and uniform upper Ahlfors regularity of the images are not required there. However, in the present work, we need to impose these assumptions in order to obtain the compactness result necessary for the existence theorem.

Let $\mathcal{C}$ an smooth open bounded convex set containing the closure $\overline{\Ccal_0}$. This set $\mathcal{C}$ will serve as our domain of study. Indeed, since we rely on PDE techniques, it is convenient to work within a smooth bounded domain. We now recall the definition of the functional introduced in \cite{bonnivard2025phasefieldapproximationplateaus}:
 \begin{equation}
 \label{functionalGeneral}
 F_{\varepsilon}(u) := \varepsilon \int_{\mathcal{C}}|\nabla u|^{2} dx + \frac{1}{4\varepsilon}\int_{\mathcal{C}}(1-u)^{2}dx + \frac{1}{c_{\varepsilon}}\sum_{i=1}^{n}d_{u}(\gamma_0, \gamma_{i}).
 \end{equation}
When, the sequences of positive numbers $(\delta_\varepsilon)$ and $(c_\varepsilon)$ are assumed to converge to zero, and to satisfy that $\delta_\varepsilon/c_\varepsilon$ converges to zero as $\varepsilon \to 0$, \cite{bonnivard2025phasefieldapproximationplateaus} establishes that this functional approximate some Plateau's problem through a $\Gamma$-convergence result.

In this paper, we are specifically interested in the decoupled functional defined below.
\begin{definition}
 For $u \in H^{1}({\mathcal C})$ such that $0\leqslant u \leqslant 1$, with $u = 1$ on $\partial{{\mathcal C}}$, and $\ell \in \mathrm{Hom}^\Lambda(\gamma_0,\gamma_1)$,
 \begin{align*}
 &E_{\varepsilon}(u,\ell) := \varepsilon\int_{\mathcal{C}} |\nabla u|^{2} dx + \frac{1}{4\varepsilon}\int_{\mathcal{C}}(1-u)^{2} dx + \frac{1}{c_{\varepsilon}}\int_{\surf_{\ell}} (u^2 + \delta_\varepsilon) d\mathcal{H}^{2}.
 \end{align*}
\end{definition}

\begin{remark}
 From Definition~\ref{Def:geoDistance} of the geodesic distance between closed curves, we know that for all $u\in H^{1}(\mathcal{C})$ such that $0\leqslant u \leqslant 1$ and $u = 1$ on $\partial{\mathcal{C}}$,
 \[F_{\varepsilon}(u) = \inf\left \{ E_{\varepsilon}(u,\ell)\ ,\ell \in \mathrm{Hom}^\Lambda(\gamma_0,\gamma_1) \right \}.\]
\end{remark}

The main result of this paper is the existence of a pair that minimizes this decoupled energy $E_\varepsilon$
\begin{equation*}
 \inf_{(u,\ell)} E_\varepsilon(u,\ell).
\end{equation*}

\begin{definition}\label{Def:geoDistance}
	Let $(\delta_\eps)_{\eps>0}$ be a sequence of positive numbers and $u \in C(\overline{\Ccal_0})$. We define the geodesic distance between $\gamma_i$ and $\gamma_j$:
	\begin{align*}
			&d_{u}(\gamma_i,\gamma_j) := \inf\left \{  \int_{\surf_{\ell}}( u^2 + \delta_\eps) d\Haus \ |\ \ell \in \curve^\Lambda(\gamma_i,\gamma_j)\right \}.
	\end{align*}
\end{definition}

The main result of this paper is the existence of a pair that minimizes this decoupled energy $E_\varepsilon$
\begin{equation*}
 \inf_{(u,\ell)} E_\varepsilon(u,\ell).
\end{equation*}

There exists a minimizing couple $(u,\ell)$ of the following problem in both variables
\begin{equation}
 \label{pb min}
 \min\left \{E_\eps(u,\ell), \, u \in H^{1}(\C) \text{ such that } 0\leqslant u \leqslant 1 , \, u = 1 \text{ on } \partial{\C} \text{ and } \ell \in \curve^\Lambda(\gamma_0,\gamma_1) \right\}.
\end{equation}

The study of this energy is motivated by the $\Gamma$-convergence result established in \cite{bonnivard2025phasefieldapproximationplateaus}, where this decoupled functional was introduced both in the proof of the $\Gamma$-convergence and for numerical applications. Recall that the phase-field approach they proposed is a generalization to Plateau’s problem of the method introduced in \cite{bonnivard2015approximation} for Steiner’s problem. A similar study of the existence of solutions for the decoupled functional in the Steiner case was carried out in \cite{bonnivard2018phase}. The present work is inspired by this paper, but there are some differences that we explain below.

We begin by studying the following minimization problem
\begin{equation}
\label{pb min l fixe}
 \inf_{u} E_\varepsilon(u,\ell), \text{ where } \ell \in \curve^\Lambda(\gamma_0,\gamma_1).
\end{equation}
The following Proposition will be proven in Section~\ref{section l fixe}.
\begin{proposition}
For any given $\ell \in \curve^\Lambda(\gamma_0, \gamma_1)$, there exists a unique minimizer $u \in H^1(\Ccal)$ of $ E_\varepsilon(\cdot,l)$ which is globally $C^{0,\alpha}$, for all $0< \alpha <1$: there exists a constant $C_\alpha>0$ such that
\[\|u \|_{C^{0,\alpha}(\mathcal{C})} \leqslant C_\alpha \frac{1+\Lambda c_\varepsilon ^{-1}}{\varepsilon^{\alpha}}.\]
\end{proposition}
The assumption that the surface $S_\ell$ is $\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity, and hence the existence of minimizers in this setting. We then apply classical results from elliptic regularity to derive Hölder estimates for the minimizer. Note that these estimates depend explicitly on $\Lambda$ and $\varepsilon$, but are independent of the specific surface $S_\ell$. This part is strongly inspired by \cite{bonnivard2018phase}.

\begin{proposition}
\label{prop holder}
Let $\ell \in~\curve^\Lambda(\gamma_0,\gamma_1)$ and let $u$ be the minimizer of $E_\varepsilon^\ell$. Then, $u$ is $\alpha$-Hölder continuous for all $0<\alpha<1$ with the estimate:
\[\|u\|_{C^{0,\alpha}(\mathcal{C})} \leqslant C_\alpha \frac{1+\Lambda c_\varepsilon ^{-1}}{\varepsilon^{\alpha}}.\]
\end{proposition}

\begin{lemma}
\label{lem 2.10}
Recall the definition of the parameter $\eta_0 = \dist(\partial \Ccal_0,\partial \Ccal)$ introduced in the previous Lemma.
Let $\ell \in \curve^\Lambda(\gamma_0, \gamma_1)$, $0< \rho < \min(1,\frac{\eta_0}{4})$ and $x_0 \in \mathcal{C}$ such that $\dist(x_0,\partial \mathcal{C}) \geqslant \frac{\eta_0}{2} $. Then, $u \in W^{1,p}(B(x_0,\rho))$ for all $3\leqslant p < \infty$ and we have the following estimate on the rescaled function $u_\rho (x) := u(x_0+\rho x)$
\begin{equation}
 \|\nabla u_\rho\|_{L^p(B_1)} \leqslant C_p\left( \frac{\rho^2}{\varepsilon^2}+1+\frac{\Lambda \rho}{c_\varepsilon \varepsilon}\right).
\end{equation}
\end{lemma}

\medskip
\emph{Step 3.} We define the rescaled function $u_\rho$ on $B_2$ by $u_\rho(x) := u(x_0+\rho x)$. Then, for all $\varphi \in C^\infty_c(B_2)$
\begin{align*}
 \int_{B_2}\nabla u_\rho \cdot \nabla\varphi \dm x &= \frac{1}{\rho}\int_{B(x_0,2\rho)}\nabla u \cdot \nabla\varphi_\rho \dm x,\\
 &= \frac{1}{\rho4\varepsilon^2}\int_{B(x_0,2\rho)}(1- u)\varphi_\rho \dm x - \frac{1}{\rho c_\varepsilon \varepsilon}B[\ell](u,\varphi_\rho),\\
 &= \frac{\rho^2}{4\varepsilon^2}\int_{B_2}(1- u_\rho)\varphi \dm x - \frac{1}{\rho c_\varepsilon \varepsilon}\langle T_\rho,\varphi \rangle.
\end{align*}
Hence, $u_\rho$ satisfies the following PDE in $\mathcal{D}'(B_2)$
\[-\Delta u_\rho = \frac{\rho^2}{4\varepsilon^2}(1- u_\rho) - \frac{1}{\rho c_\varepsilon \varepsilon} T_\rho.\]
We denote $w_\rho := u_\rho + \frac{1}{\rho c_\varepsilon \varepsilon}v_\rho \in H^1(B_2)\cap L^\infty(B_2)$. And we deduce that
\[-\Delta w_\rho = \frac{\rho^2}{4\varepsilon^2}(1- u_\rho) \text{ in } \mathcal{D}'(B_2). \]
Thus, \cite[Corollary 8.36]{gilbargelliptic} yields that $w_\rho \in C_{\mathrm{loc}}^{1,\alpha}(B_2)$ for all $\alpha >0$ and
\begin{align*}
 \|\nabla w_\rho\|_{L^\infty(B_1)} \leqslant \|w_\rho\|_{C^{1,\alpha}(B_1)} &\leqslant C\left(\| w_\rho\|_{L^\infty(B_1)} +\frac{\rho^2}{4\varepsilon^2}\|1- u_\rho\|_{L^\infty(B_1)}\right),\\
 &= C\left(\| u_\rho\|_{L^\infty(B_1)}+ \frac{1}{\rho c_\varepsilon \varepsilon}\| v_\rho\|_{L^\infty(B_1)} +\frac{\rho^2}{4\varepsilon^2}\right),\\
 &\leqslant C\left(1 + \frac{\Lambda \rho}{c_\varepsilon \varepsilon} + \frac{\rho^2}{4\varepsilon^2}\right).
\end{align*}
Finally, we can conclude that $u_\rho = w_\rho - \frac{1}{\rho c_\varepsilon \varepsilon}v_\rho \in W^{1,p}(B_1)$ with the desired estimate
\[ \|\nabla u_\rho\|_{L^p(B_1)} \leqslant C_p\left( \frac{\rho^2}{\varepsilon^2}+1+\frac{\Lambda \rho}{c_\varepsilon \varepsilon}\right).\]
This concludes the proof of this Lemma.
\end{proof}

Now let's consider the case where $\dist(x_0, \partial \mathcal{C}) > \frac{\eta_0}{2}> \varepsilon$, in particular this implies that $B(x_0,\varepsilon) \subset \C$. In this case, Lemma~\ref{lem 2.10}, applied with $\rho = \varepsilon$ and $p = \frac{3}{1-\alpha}$ yields the estimate on the rescaled function $u_\varepsilon$
 \[\|\nabla u_\varepsilon\|_{L^p(B_1)} \leqslant C_p\left(1+ \frac{\Lambda}{c_\varepsilon}\right).\]
 Since we have shown that the $L^\infty$ norm of $u$ is smaller than $1$, we deduce that the $L^p$ norm of the scaled function $u_\varepsilon$ is bounded by the volume of the domain. And we conclude by applying the Sobolev embedding Theorem (see for instance \cite[Theorem 4.12]{adams2003sobolev}),
 \begin{equation}
 \|u_\varepsilon\|_{C^{0,\alpha}(B_1)} \leqslant C_\alpha\left(1+ \frac{\Lambda}{c_\varepsilon}\right).
 \end{equation}
 Hence, scaling back leads to
 \begin{equation}
 \frac{|u(x)-u(y)|}{|x-y|^\alpha} \leqslant \frac{1}{\varepsilon^\alpha} \|u_\varepsilon\|_{C^{0,\alpha}(B_1)} \leqslant \frac{C_\alpha}{\varepsilon^\alpha}\left(1+ \frac{\Lambda}{c_\varepsilon}\right).
 \end{equation}
\end{itemize}
Finally, since $\|u\|_{L^\infty(\Ccal)} \leqslant 1$, we get the desired estimate on the Hölder norm of $u$
\[ \|u\|_{C^{0,\alpha}(\mathcal{C})} \leqslant C_\alpha \frac{1+\Lambda c_\varepsilon ^{-1}}{\epsilon^{\alpha}}.\]
This achieves the proof of the Hölder regularity of the solution $u$ of $\min E_\eps(u,\ell)$, when $\ell$ is fixed in $\curve^\Lambda$.
\end{proof}

The study of this energy is motivated by the $\Gamma$-convergence result established in \cite{bonnivard2025phasefieldapproximationplateaus}, where this decoupled functional was introduced both in the proof of the $\Gamma$-convergence and for numerical applications. Recall that the phase-field approach they proposed is a generalization to Plateau’s problem of the method introduced in \cite{bonnivard2015approximation} for Steiner’s problem. A similar study of the existence of solutions for the decoupled functional in the Steiner case was carried out in \cite{bonnivard2018phase}. The present work is inspired by this paper, but there are some differences that we explain below.

To prove Theorem~\ref{main th}, we first consider the optimization problems with either $u$ or $\ell$ fixed. As usual, establishing the existence of minimizers requires both a compactness result and the lower semi-continuity of the functional. The compactness follows from the choice of the class of competitors, so the main issue is to establish lower semi-continuity.

We begin by studying the following minimization problem
\begin{equation}
\label{pb min l fixe}
    \inf_{u} E_\varepsilon(u,\ell), \text{ where } \ell \in \mathrm{Hom}^\Lambda(\gamma_0,\gamma_1).
\end{equation}
The following Proposition will be proven in Section~\ref{section l fixe}.
\begin{proposition}
For any given $\ell \in \mathrm{Hom}^\Lambda(\gamma_0, \gamma_1)$, there exists a unique minimizer $u \in H^1(\mathcal{C})$ of $ E_\varepsilon(\cdot,l)$ which is globally $C^{0,\alpha}$, for all $0< \alpha <1$: there exists a constant $C_\alpha>0$ such that
\[\|u \|_{C^{0,\alpha}(\mathcal{C})} \leqslant C_\alpha \frac{1+\Lambda c_\varepsilon ^{-1}}{\varepsilon^{\alpha}}.\]
\end{proposition}
The assumption that the surface $S_\ell$ is $\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity, and hence the existence of minimizers in this setting. We then apply classical results from elliptic regularity to derive Hölder estimates for the minimizer. Note that these estimates depend explicitly on $\Lambda$ and $\varepsilon$, but are independent of the specific surface $S_\ell$. This part is strongly inspired by \cite{bonnivard2018phase}.

Then, in Section~\ref{section u fixe} we consider the minimization problem
\begin{equation}
\label{pb min u fixe}
    \inf_{\ell \in \mathrm{Hom}^\Lambda} E_\varepsilon(u,\ell), \text{ where } u \in H^1(\mathcal{C})\cap C(\overline{\mathcal{C}}),
\end{equation}
and prove the following Proposition.
\begin{proposition}
\label{prop 1.2}
For all $u \in H^1({\mathcal C})\cap C(\overline{\mathcal{C}})$ there exists a minimizer for the problem \eqref{pb min u fixe}.
\end{proposition}

As previously mentionned, the main difficulty in the proof of Proposition~\ref{prop 1.2} lies in establishing the lower semicontinuity of the functional, which relies on the following Lemma.

 \begin{lemma}
 \label{golab}
Let $\ell_n : [0,1] \times  \mathbb{S}^1 \to \mathbb{R}^3$ be a sequence of uniformly Lipschitz functions, which converges uniformly to a Lipschitz function $\ell  : [0,1] \times  \mathbb{S}^1 \to \mathbb{R}^3$.
Then, for all open set $A$

$$\liminf \mathcal{H}^{2} \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} S_{\ell_n}(A) \geqslant \mathcal{H}^{2} \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} S_\ell(A),$$
where $S_\ell$ (respectively $S_{\ell_n}$) denotes the image of the Lipschitz function $\ell$ (respectively $\ell_n)$.
\end{lemma}

This Lemma, proven in Section~\ref{section u fixe}, constitutes a central and original contribution of the present article and relies crucially on the specific definition of the class of surfaces under consideration. It can be viewed as a generalization of Go\l{}\k{a}b’s theorem to the two-dimensional Hausdorff measure, and its proof follows a similar strategy. In Go\l{}\k{a}b’s theorem, the first step consists in establishing the rectifiability of the limit set; in our setting, this property is immediate, since the surfaces are assumed to be images of Lipschitz functions. We then study the weak limit of the measures $(\mathcal{H}^{2} \mathbin{\vrule height 1.6ex depth 0pt width 0.13ex\vrule height 0.13ex depth 0pt width 1.3ex} S_{\ell_n})$ and, using the tangent plane of the limit surface, derive a lower bound on the upper density of the limit measure to conclude. The key point of our proof relies on a topological argument, stating that a Lipschitz image, which is uniformly close to a disk, must have at least almost the density of a disk (see the proof of Lemma~\ref{golab}).

We emphasize that the proof of this Lemma does not rely on the specific definition set $\mathbb{S}^1 \times [0,1]$; it only assumes it to be a two-dimensional subset of $\mathbb{R}^3$. Moreover, the result is also expected to remain valid in dimensions higher than two, although the proof would then require more delicate topological arguments.

Finally, in Section~\ref{section rien fixe}, Theorem~\ref{main th} is proved by combining these results, in particular the Hölder regularity and the preceding Lemma.

\textbf{Acknowledgments.}
 I would like to thank Antoine Lemenant for suggesting this problem to me as well as Matthieu Bonnivard for their help in this project.
This work was partially supported by the IUF grant of Antoine Lemenant and by the ANR project STOIQUES.

To prove Theorem~\ref{main th}, the authors “first consider the optimization problems with either $u$ or $\ell$ fixed.” Existence requires “both a compactness result and the lower semi-continuity of the functional”; “the compactness follows from the choice of the class of competitors,” so “the main issue is to establish lower semi-continuity.”

1) **Fix $\ell$**: study \eqref{pb min l fixe}. For any $\ell\in\mathrm{Hom}^\Lambda(\gamma_0,\gamma_1)$ there is a “unique minimizer $u\in H^1(\mathcal C)$” which is “globally $C^{0,\alpha}$,” with the stated estimate. Here, “the assumption that the surface $S_\ell$ is $\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity,” and then “classical results from elliptic regularity” give Hölder estimates (depending on $\Lambda$ and $\varepsilon$ but “independent of the specific surface $S_\ell$”).

2) **Fix $u$**: study \eqref{pb min u fixe}. For $u\in H^1(\mathcal C)\cap C(\overline{\mathcal C})$ there exists a minimizer in $\ell$. The “main difficulty” is again lower semicontinuity, which “relies on” Lemma~\ref{golab}, a Go\l{}\k{a}b-type lower semicontinuity statement for $\mathcal H^2\llcorner S_{\ell_n}$ under uniform convergence of uniformly Lipschitz parametrizations. Its proof “follows a similar strategy” to Go\l{}\k{a}b: rectifiability is “immediate,” then one studies “the weak limit of the measures,” and “using the tangent plane of the limit surface, derive[s] a lower bound on the upper density of the limit measure to conclude”; the “key point” is “a topological argument” that a Lipschitz image uniformly close to a disk has “at least almost the density of a disk.”

3) **Conclusion**: “Finally, in Section~\ref{section rien fixe}, Theorem~\ref{main th} is proved by combining these results, in particular the Hölder regularity and the preceding Lemma.”

To prove the main theorem, the authors “first consider the optimization problems with either $u$ or $\ell$ fixed.” Existence requires “both a compactness result and the lower semi-continuity of the functional”; “the compactness follows from the choice of the class of competitors,” so “the main issue is to establish lower semi-continuity.”

1) **Fix $\ell$**: study \eqref{pb min l fixe}. For any $\ell\in\mathrm{Hom}^\Lambda(\gamma_0,\gamma_1)$ there is a “unique minimizer $u\in H^1(\mathcal C)$” which is “globally $C^{0,\alpha}$,” with the stated estimate. Here, “the assumption that the surface $S_\ell$ is $\Lambda$-upper Ahlfors regular is the key ingredient to establish lower semi-continuity,” and then “classical results from elliptic regularity” give Hölder estimates (depending on $\Lambda$ and $\varepsilon$ but “independent of the specific surface $S_\ell$”).

2) **Fix $u$**: study
\begin{equation}
\label{pb min u fixe}
    \inf_{\ell \in \curve^\Lambda} E_\varepsilon(u,\ell), \text{ where } u \in H^1(\Ccal)\cap C(\overline{\Ccal}),
\end{equation}
For $u\in H^1(\mathcal C)\cap C(\overline{\mathcal C})$ there exists a minimizer in $\ell$. The “main difficulty” is again lower semicontinuity, which “relies on” the following lemma.
\begin{lemma}
 \label{golab}
Let $\ell_n : [0,1] \times  \mathbb{S}^1 \to \R^3$ be a sequence of uniformly Lipschitz functions, which converges uniformly to a Lipschitz function $\ell  : [0,1] \times  \mathbb{S}^1 \to \R^3$.
Then, for all open set $A$

$$\liminf \Haus \mres S_{\ell_n}(A) \geqslant \Haus \mres S_\ell(A),$$
where $S_\ell$ (respectively $S_{\ell_n}$) denotes the image of the Lipschitz function $\ell$ (respectively $\ell_n)$.
\end{lemma}
Its proof “follows a similar strategy” to Go\l{}\k{a}b: rectifiability is “immediate,” then one studies “the weak limit of the measures,” and “using the tangent plane of the limit surface, derive[s] a lower bound on the upper density of the limit measure to conclude”; the “key point” is “a topological argument” that a Lipschitz image uniformly close to a disk has “at least almost the density of a disk.”

3) **Conclusion**: “Finally, next we conclude the proof of the main theorem by combining these results, in particular the Hölder regularity and the preceding lemma.”

Question:
Fix \(\varepsilon>0\), \(\Lambda>0\), positive numbers \(c_{\varepsilon}\) and \(\delta_{\varepsilon}\), a smooth bounded convex domain \(\mathcal C\), a set \(\mathcal C_0\) with \(\overline{\mathcal C_0}\subset \mathcal C\), and closed curves \(\gamma_0,\gamma_1\). For a Lipschitz homotopy \(\ell:[0,1]\times \mathbb S^1\to \overline{\mathcal C_0}\) with \(\ell(0)=\gamma_0\) and \(\ell(1)=\gamma_1\), let \(\Sigma_{\ell}=\ell([0,1]\times \mathbb S^1)\). Define \(\mathrm{Hom}^{\Lambda}(\gamma_0,\gamma_1)\) as the set of such \(\ell\) satisfying \(\mathrm{Lip}(\ell)\le \Lambda\) and such that \(\Sigma_{\ell}\) is \(\Lambda\)-upper Ahlfors regular (i.e. \(\mathcal H^2(\Sigma_{\ell}\cap B(x,r))\le \Lambda r^2\) for all balls). For admissible \(u\in H^1(\mathcal C)\) with \(0\le u\le 1\) and \(u=1\) on \(\partial\mathcal C\), define
\[
E_{\varepsilon}(u,\ell)=\varepsilon\int_{\mathcal C}|\nabla u|^2\,dx+\frac{1}{4\varepsilon}\int_{\mathcal C}(1-u)^2\,dx+\frac{1}{c_{\varepsilon}}\int_{\Sigma_{\ell}}(u^2+\delta_{\varepsilon})\,d\mathcal H^2.
\]
Which statement holds for the minimization problem over all admissible pairs \((u,\ell)\)?

There exists at least one admissible pair \((u,\ell)\) that minimizes \(E_{\varepsilon}(u,\ell)\) jointly among all pairs with \(u\in H^1(\mathcal C)\), \(0\le u\le 1\), \(u=1\) on \(\partial\mathcal C\), and \(\ell\in \mathrm{Hom}^{\Lambda}(\gamma_0,\gamma_1)\). Moreover, for such a minimizing pair, the function \(u\) belongs to \(C^{0,\alpha}(\mathcal C)\) for every \(0<\alpha<1\), and it satisfies
\[
\|u\|_{C^{0,\alpha}(\mathcal C)}\le C_{\alpha}\,\frac{1+\Lambda c_{\varepsilon}^{-1}}{\varepsilon^{\alpha}}.
\]

Choices:
B.
There exists at least one admissible pair \((u,\ell)\) that minimizes \(E_{\varepsilon}(u,\ell)\) jointly among all pairs with \(u\in H^1(\mathcal C)\), \(0\le u\le 1\), \(u=1\) on \(\partial\mathcal C\), and \(\ell\in \mathrm{Hom}(\gamma_0,\gamma_1)\) (that is, without imposing the uniform Lipschitz bound or the \(\Lambda\)-upper Ahlfors regularity on \(\Sigma_\ell\)). Moreover, for such a minimizing pair, the function \(u\) belongs to \(C^{0,\alpha}(\mathcal C)\) for every \(0<\alpha<1\), and it satisfies
\[
\|u\|_{C^{0,\alpha}(\mathcal C)}\le C_{\alpha}\,\frac{1+\Lambda c_{\varepsilon}^{-1}}{\varepsilon^{\alpha}}.
\]
C. There exists at least one admissible pair \((u,\ell)\) that minimizes \(E_{\varepsilon}(u,\ell)\) jointly among all pairs with \(u\in H^1(\mathcal C)\), \(0\le u\le 1\), \(u=1\) on \(\partial\mathcal C\), and \(\ell\in \mathrm{Hom}^{\Lambda}(\gamma_0,\gamma_1)\).
D.
For every admissible \(u\in H^1(\mathcal C)\) with \(0\le u\le 1\) and \(u=1\) on \(\partial\mathcal C\), there exists \(\ell\in \mathrm{Hom}^{\Lambda}(\gamma_0,\gamma_1)\) minimizing \(E_{\varepsilon}(u,\ell)\) over \(\ell\); and for every admissible \(\ell\in \mathrm{Hom}^{\Lambda}(\gamma_0,\gamma_1)\), there exists a minimizer \(u\) of \(E_{\varepsilon}(u,\ell)\) over \(u\). Consequently, there exists a pair \((u,\ell)\) such that \(u\) minimizes \(E_{\varepsilon}(\,b7\,,\ell)\) and \(\ell\) minimizes \(E_{\varepsilon}(u,\,\u00b7\,)\), and this pair minimizes \(E_{\varepsilon}\) jointly over all admissible pairs. Moreover, for such a pair,
\[
\|u\|_{C^{0,\alpha}(\mathcal C)}\le C_{\alpha}\,\frac{1+\Lambda c_{\varepsilon}^{-1}}{\varepsilon^{\alpha}}
\]
for every \(0<\alpha<1\).
E.
There exists at least one admissible pair \((u,\ell)\) that minimizes \(E_{\varepsilon}(u,\ell)\) jointly among all pairs with \(u\in H^1(\mathcal C)\), \(0\le u\le 1\), \(u=1\) on \(\partial\mathcal C\), and \(\ell\in \mathrm{Hom}^{\Lambda}(\gamma_0,\gamma_1)\). Moreover, for such a minimizing pair, the function \(u\) belongs to \(C^{0,\alpha}(\mathcal C)\) for every \(0<\alpha<1\), and it satisfies
\[
\|u\|_{C^{0,\alpha}(\mathcal C)}\le C_{\alpha}\,\frac{1+\Lambda c_{\varepsilon}^{-1}}{\varepsilon}.
\]

The stem does not state or strongly hint at the exact correct conclusion; it only sets up the functional and asks which existence result is valid.

The item is close to a theorem-recall question: choice A essentially states the full theorem, while the others are perturbations of it. Still, the options do force comparison of different strengths of conclusion.

Some reasoning is needed to reject the unrestricted class, unjustified uniqueness, and overstrong Lipschitz regularity, and to distinguish the strongest valid statement from a weaker true one. But the task remains largely recognition of the theorem statement.

The distractors are plausible and mathematically meaningful: one weakens the admissible class, one gives a weaker true statement, one overclaims uniqueness, and one overstates regularity. These reflect realistic failure modes.

A reasonably strong MCQ with good distractors and little answer leakage, but it leans toward theorem recognition rather than deep generative reasoning.


Question:
Fix \(\varepsilon>0\), \(\Lambda>0\), positive numbers \(c_{\varepsilon}\) and \(\delta_{\varepsilon}\), a smooth bounded convex domain \(\mathcal C\), a set \(\mathcal C_0\) with \(\overline{\mathcal C_0}\subset \mathcal C\), and closed curves \(\gamma_0,\gamma_1\). For a Lipschitz homotopy \(\ell:[0,1]\times \mathbb S^1\to \overline{\mathcal C_0}\) with \(\ell(0)=\gamma_0\) and \(\ell(1)=\gamma_1\), let \(\Sigma_{\ell}=\ell([0,1]\times \mathbb S^1)\). Define \(\mathrm{Hom}^{\Lambda}(\gamma_0,\gamma_1)\) as the set of such \(\ell\) satisfying \(\mathrm{Lip}(\ell)\le \Lambda\) and such that \(\Sigma_{\ell}\) is \(\Lambda\)-upper Ahlfors regular (i.e. \(\mathcal H^2(\Sigma_{\ell}\cap B(x,r))\le \Lambda r^2\) for all balls). For admissible \(u\in H^1(\mathcal C)\) with \(0\le u\le 1\) and \(u=1\) on \(\partial\mathcal C\), define
\[
E_{\varepsilon}(u,\ell)=\varepsilon\int_{\mathcal C}|\nabla u|^2\,dx+\frac{1}{4\varepsilon}\int_{\mathcal C}(1-u)^2\,dx+\frac{1}{c_{\varepsilon}}\int_{\Sigma_{\ell}}(u^2+\delta_{\varepsilon})\,d\mathcal H^2.
\]
Which statement holds for the minimization problem over all admissible pairs \((u,\ell)\)?

Choices:
B.
There exists at least one admissible pair \((u,\ell)\) that minimizes \(E_{\varepsilon}(u,\ell)\) jointly among all pairs with \(u\in H^1(\mathcal C)\), \(0\le u\le 1\), \(u=1\) on \(\partial\mathcal C\), and \(\ell\in \mathrm{Hom}(\gamma_0,\gamma_1)\) (that is, without imposing the uniform Lipschitz bound or the \(\Lambda\)-upper Ahlfors regularity on \(\Sigma_\ell\)). Moreover, for such a minimizing pair, the function \(u\) belongs to \(C^{0,\alpha}(\mathcal C)\) for every \(0<\alpha<1\), and it satisfies
\[
\|u\|_{C^{0,\alpha}(\mathcal C)}\le C_{\alpha}\,\frac{1+\Lambda c_{\varepsilon}^{-1}}{\varepsilon^{\alpha}}.
\]
C. There exists at least one admissible pair \((u,\ell)\) that minimizes \(E_{\varepsilon}(u,\ell)\) jointly among all pairs with \(u\in H^1(\mathcal C)\), \(0\le u\le 1\), \(u=1\) on \(\partial\mathcal C\), and \(\ell\in \mathrm{Hom}^{\Lambda}(\gamma_0,\gamma_1)\).
D.
For every admissible \(u\in H^1(\mathcal C)\) with \(0\le u\le 1\) and \(u=1\) on \(\partial\mathcal C\), there exists \(\ell\in \mathrm{Hom}^{\Lambda}(\gamma_0,\gamma_1)\) minimizing \(E_{\varepsilon}(u,\ell)\) over \(\ell\); and for every admissible \(\ell\in \mathrm{Hom}^{\Lambda}(\gamma_0,\gamma_1)\), there exists a minimizer \(u\) of \(E_{\varepsilon}(u,\ell)\) over \(u\). Consequently, there exists a pair \((u,\ell)\) such that \(u\) minimizes \(E_{\varepsilon}(\,b7\,,\ell)\) and \(\ell\) minimizes \(E_{\varepsilon}(u,\,\u00b7\,)\), and this pair minimizes \(E_{\varepsilon}\) jointly over all admissible pairs. Moreover, for such a pair,
\[
\|u\|_{C^{0,\alpha}(\mathcal C)}\le C_{\alpha}\,\frac{1+\Lambda c_{\varepsilon}^{-1}}{\varepsilon^{\alpha}}
\]
for every \(0<\alpha<1\).
E.
There exists at least one admissible pair \((u,\ell)\) that minimizes \(E_{\varepsilon}(u,\ell)\) jointly among all pairs with \(u\in H^1(\mathcal C)\), \(0\le u\le 1\), \(u=1\) on \(\partial\mathcal C\), and \(\ell\in \mathrm{Hom}^{\Lambda}(\gamma_0,\gamma_1)\). Moreover, for such a minimizing pair, the function \(u\) belongs to \(C^{0,\alpha}(\mathcal C)\) for every \(0<\alpha<1\), and it satisfies
\[
\|u\|_{C^{0,\alpha}(\mathcal C)}\le C_{\alpha}\,\frac{1+\Lambda c_{\varepsilon}^{-1}}{\varepsilon}.
\]

The stem gives only the setup and definitions of the admissible class and energy. It does not explicitly state the existence, Hölder regularity, or the precise estimate that appears in the correct choice.

The item is close to theorem recognition: the correct option essentially reproduces the theorem statement for the given setup. However, it is not a pure tautology because the alternatives modify key hypotheses and conclusions in nontrivial ways.

Selecting the correct answer requires some reasoning about compactness assumptions, strength of conclusions, and the scaling in the Hölder estimate. Still, the task is mainly identifying the exact theorem statement rather than deriving a conclusion from first principles.

The distractors are strong and mathematically meaningful: one drops needed compactness/regularity assumptions, one gives a weaker true statement, one confuses separate minimization with joint minimization, and one alters the epsilon scaling. These are plausible and distinct failure modes.

A solid MCQ with no answer leakage and high-quality distractors, but it leans toward theorem-statement recognition rather than deep generative reasoning.