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Recent works on my thesis

The study of semilinear elliptic partial differential equations over the past 50 years, as well as the main focus of my thesis, can be broadly divided into two main areas: the static profile and the dynamical model. The former originated from the seminal works of Caffarelli, Gidas, and Spruck in the 1980s on the classification of semilinear equations, particularly Lane-Emden equations in the whole space $\mathbb{R}^N$ and the half-space $\mathbb{R}_+^N$. These studies have significantly influenced the development of semilinear equation theory and often reduce to the so-called Liouville-type theorems, which are typically concerned with unbounded domains1, representing a major area of research. Although there has been substantial progress in this extensive program, the question remains unresolved even for the "simplest" model, the Lane-Emden equation. The second direction focuses on dynamic models featuring double-well potentials, known as the Cahn-Hilliard model, including their vector-valued versions. These were studied by Rubinstein, Stenberg, and Keller in the 1990s. Their work suggested that the convergence properties of parabolic Allen-Cahn-type equations are connected to a geometric profiles known as mean curvature flow. Subsequently, this connection was established for the scalar case and some special vector-valued cases using various methodologies. However, the general vector-valued case remains unresolved.

1 This explains the emphasis on $\mathbb{R}^N$ and $\mathbb{R}_+^N$.

The following three major problems constitute the central focus of my work.

Liouville-type theorem and BCN conjecture

The Liouville theorem in elliptic PDEs is well-known and fundamental. It asserts that any bounded entire harmonic function—i.e., a solution of $\Delta u=0$—must be constant, a result that extends to general uniformly elliptic equations $a_{ij}D_{ij}u=0$; see [Qian25]. Indeed, various so-called Liouville-type theorems exist in elliptic PDEs and are instrumental in classifying solutions to certain types of elliptic equations; for details, see monographs [Dupaigne11], [Quittner19] and surveys [Chan18], [Wang21]. Our focus lies on Liouville-type theorems2 for the semilinear equations in unbounded domains:

$\Delta u+f(u)=0,\quad \text{in}\ \Omega.$
2 which can be viewed as Bernstein-type problems in geometry or De Giorgi-type problems.

We first recall a classic model established by Gidas and Spruck [Gidas81], [Gidas81b] and Caffarelli, Gidas, and Spruck [Caffarelli89], which states some results about the positive solutions to Lane-Emden equation in entire space and half-spaces.

Theorem (Gidas-Spruck, 1981; Caffarelli-Gidas-Spruck, 1989)
Let $\Omega=\mathbb{R}^N$ and $p>1$, the Lane-Emden equation
$\Delta u+u^p=0,\quad\text{in}\ \mathbb{R}^N,$
  1. (i) does not possess any positive classical solution if, $p
  2. (ii) any positive classical solution is radially symmetric with respect to some point. Precisely, it must be of the form of the Talenti solutions
    $u(x)=\left(\frac{\lambda}{\lambda^2+|x-x_0|^2}\right)^{\frac{N-2}{2}},\quad \lambda>0,\ x_0\in\mathbb{R}^N.$
Let $\Omega=\partial\mathbb{R}_+^N$ and $1 $\begin{cases} \Delta u+u^p=0,&\text{in}\ \mathbb{R}_+^N\\ u=0,&\text{on}\ \partial\mathbb{R}_+^N \end{cases}$
does not possess any positive classical solution.
Remark
$p_S$ is called critical index.
Remark
The study of problems in half-spaces is also significant, as half-spaces are not only the simplest unbounded domains (recall the concept of coercive epigraphs, which can be regarded as bounded domains to some extent; see [Berestycki97]), but they also serve as blow-up profiles for so-called singular perturbation models in smooth bounded domains:
$\epsilon^2\Delta u+f(u)=0,\quad \text{in}\ \Omega.$

Therefore, the classification (or Liouville-type theorems) of semilinear equations in unbounded domains can be roughly divided into two main categories: those in the entire space and those in half-spaces, namely

$\Delta u+f(u)=0,\quad \text{in}\ \mathbb{R}^N$

and

$\begin{cases} \Delta u+f(u)=0,&\text{in}\ \mathbb{R}_+^N\\ u=0,&\text{on}\ \partial\mathbb{R}_+^N, \end{cases}$

The essential method used here is the moving plane method, which is based on the so-called maximum principle (also known as the comparison principle). The original application of the moving plane method can be traced back to Alexandrov [Alexandrov62] and Serrin [Serrin71]. This method was further developed and applied by Gidas, Ni, and Nirenberg [Gidas79], [Gidas81c], who used it to prove certain monotonicity and radial symmetry properties for semilinear equations. Subsequently, Berestycki and Nirenberg [Berestycki91] refined the method. Later, Dancer [Dancer92] utilized this enhanced approach to demonstrate the nonexistence of bounded positive solutions to the half-space Lane-Emden equation for $1 < p < p_D := \frac{N+1}{N-3}$. Moreover, following a series of studies by Berestycki, Caffarelli, and Nirenberg [Berestycki93], [Berestycki94], [Berestycki96], [Berestycki97a], [Berestycki97], the authors established both monotonicity and one-dimensional symmetry results of the half-space problem in certain unbounded domains as well as in bounded domains. Briefly, we present two well-known results from [Berestycki97], which inspired research on the monotonicity and one-dimensional symmetry of the half-space problem. These findings also motivated the study of rigidity properties of semilinear equations in half-spaces over the following thirty years and constitute one of the main subjects of the first part of my thesis.

Proposition (Berestycki-Caffarelli-Nirenberg, 1993)
Suppose that $u$ be a classical bounded positive solution to the half-space problem with $f\in C^{0,1}$ and
$M=\sup u<\infty.$
Then if $f(M)\le 0$, then $u$ is one-dimensional symmetric, i.e. $u=u(x_N)$ and it is also increasing. Furthermore, $f(M)=0$.
Remark
The condition boundedness is necessary. For example, if we take $f(s)=-s$, then $u=x_1 e^{x_N}$.
Proposition (Berestycki-Caffarelli-Nirenberg, 1996-97)
Suppose that $u$ be a classical positive solution to the half-space problem with $f\in C^{0,1}$ and
$f(0)\ge 0.$
Then $u$ is increasing with respect to $x_N$.
Remark
The condition positiveness and $f(0)\ge 0$ is necessary. For example, if we take $f(s)=s-1$, then $u=1+\cos x_N$.
Theorem (Berestycki-Caffarelli-Nirenberg, 1997)
Suppose that $u$ be a classical bounded positive solution to the half-space problem, then
  1. (i) if $f\in C^{0,1}$, $u$ is one-dimensional symmetric as $N=2$;
  2. (ii) if $f\in C^1$ and $f(0)\ge 0$ (the blue condition), the same conclusion holds as $N=3$.

This result is based on following contribution:

Theorem (Berestycki-Caffarelli-Nirenberg, 1997)
Suppose that $u$ be a classical positive solution to the half-space problem, with $f\in C^{0,1}$, then
  1. (i) $u$ is increasing with respect to $x_N$ as $N=2$;
  2. (ii) the same conclusion holds as $N\ge 3$ if, $f(0)\ge 0$ (the blue condition).
Remark
These results were improved by Farina [Farina20], who extended them to locally Lipschitz nonlinear terms, specifically $f\in C^{0,1}_{loc}$, with the additional assumption that $u$ is bounded on any finite strip. More recently, Sciunzi and Vuono [Sciunzi25] further relaxed the requirement on $u$, allowing it to be bounded only on two-codimensional finite strips. However, the condition imposed on the nature of $f$ at the origin, namely $f(0)\ge 0$ (the blue condition), still can NOT be dropped.

Both theorems lead to the first two conjectures in the case of half-spaces.

Conjecture (Berestycki-Caffarelli-Nirenberg, 1993-1997)
If a bounded positive solution $u$ of the half-space problem exists, then $u$ is one-dimensional symmetric and necessarily $f(M)=0$, where $M=\sup u$.
Conjecture (Berestycki-Caffarelli-Nirenberg, 1997)
If a positive solution $u$ of the half-space problem with
$f(0)<0,$
exists, then $u$ is increasing with respect to $x_N$ as $N\ge 3$.

However, the situation when $f(0)<0$ (the purple condition) is different and still OPEN in higher dimensions. Fortunately, the case $f(0)>0$ has been fully classified; see [Dupaigne22]. Conjecture 1 (BCN conjecture 1) was proved by the authors themselves for $N=2, 3$, and by Farina and Valdinoci [Farina10] for $N\leq5$ under certain conditions on $f$. For $N\geq6$, to the best of our knowledge, the problem remains OPEN. Conjecture 2 (BCN conjecture 2) was extended by Farina and Sciunzi [Farina16], [Farina17] to the case of a locally Lipschitz nonlinear term, and they also classified the nonnegative solutions. Notably, one should pay attention to the nonnegative periodic solution $u=1-\cos x_2$ for $\Delta u+u-1=0$ when $N=2$. Therefore, thanks to [Farina16], [Farina17], [Beuvin26], for $N=2$, there is no issue remaining.

To simplify, we present a special case, i.e. Lane-Emden equation that has been mentioned above, which is linked to a Dancer's conjecture on half-spaces.

Conjecture (Dancer, 1992)
There is no nontrivial nonnegative solution to the half-space Lane-Emden equation.

It remains an OPEN problem; see [Sciunzi25], but several significant contributions have been made; see [Dancer92], [Chen14], [Farina20]. The weaker case for positive solutions was solved by Dupaigne, Sirakov, and Souplet [Dupaigne22], who proved the nonexistence of positive monotone solutions. Indeed, combined with Theorem 2 (BCN theorem 2) for $f(s)=s^p$, this result fully resolves the weaker version of the problem. More recently, there are several works; see [Dupaigne23], [Sciunzi25].

Regarding the case in the entire space, we focus on so-called stable solutions

$\int f'(s)\varphi^2\le \int|\nabla\varphi|^2,\quad \forall \varphi\in C_c^\infty$

and solutions with finite Morse index, seeing [Wang21].

For research in this area, we refer readers to Dancer's survey [Dancer10], the monographs [Dupaigne11], [Quittner19], and other surveys [Chan18], [Takahashi21], [Wang21]. Since this topic (the entire space case) is closely connected to the Bernstein problem and the De Giorgi conjecture discussed below, we primarily present results for general nonlinearities $f$, and reserve the discussion of special cases for subsequent sections.

Firstly, we present Dancer's conjecture about stable solutions in entire spaces.

Conjecture (Dancer, 2010)
Assume $N\le 7$ and $u$ is a bounded stable solutions of the entire space problem. Then either $u$ is constant on $\mathbb{R}^N$ or, after a rotation of axis, $u=u(x_N)$.

The investigation of this conjecture can be separated into two distinct cases: when the nonlinear term $f$ is nonnegative, and when the nonlinear term $f$ is sign-changing. While results concerning the latter case are scarce, we direct readers to the seminal work of Liu, Wang, Wei, and Wu [Liu24].

In the following discussions, we merely talk about the nonnegative nonlinear term, namely $f\ge0$ (noting that $f(0)\ge0$), noting that Cabré, Figalli, Ros-Oton, and Joaquim [Cabré20] and Cabré [Cabré22] obtained a prior estimate for stable solutions that was further used in further works. For example, Dupaigne and Farina [Dupaigne22b] used these estimates to establish the following Liouville theorem that partially answered Dancer's conjecture.

Theorem (Dupaigne-Farina, 2022)
Assume that $f$ is nonnegative and locally Lipschitz. Suppose $u$ is a stable solution of the entire space problem and it is bounded below. If $N\le 10$, then $u$ is constant.

We refer the readers to [Farina07], [Beuvin26] and reference therein for more recent works.

Bernstein problem and De Giorgi conjecture

We now turn to a specific model.

In 1978, De Giorgi [DeGiorgi79] conjectured that any bounded, monotone entire solution to the Allen–Cahn equation depends on only one variable for $n \le 8$. Since $u$ is monotone, all its level sets $\{u=s\}_{-1\le s\le1}$ are evidently graphs. We will later explain why De Giorgi believed these level sets are flat, which is equivalent to the solution depending on one variable for $n \le 8$. First, we state the conjecture formally.

Conjecture (De Giorgi, 1978)
Let $u$ be a bounded entire solution of the Allen–Cahn equation (ACE)
$\Delta u+u-u^3=0,\quad\text{in}\ \mathbb{R}^n,$
which is monotone in one direction, say $\frac{\partial u}{\partial x_n}>0$. Then $u$ depends on only one variable for $n\le 8$. Equivalently, all level sets of $u$ are hyperplanes.

To understand this conjecture, we introduce a more intuitive and geometric problem known as the Bernstein problem, which was fully resolved in the 20th century.

Conjecture (Bernstein, 1914)
Let $\Sigma$ be an entire minimal graph in $\mathbb{R}^N$. Then $\Sigma$ is a hyperplane.

This problem was initially solved by Bernstein for $N\le 3$ [Bernstein17], [Bernstein27]. Specifically, the surface $\Sigma$ is the graph of an entire solution $v$ to the minimal surface equation (MSE),

$\mathrm{div}\left(\frac{\nabla v}{\sqrt{1+|\nabla v|^2}}\right)=0,\quad\text{in}\ \mathbb{R}^N,$

which arises from the variation of the area functional,

$A(v,\mathbb{R}^N)=\int_{\mathbb{R}^N}\sqrt{1+|\nabla v|^2}\,\mathrm{d}x.$

However, applying Bernstein's cutoff method encounters difficulties in higher dimensions. After about five decades, Fleming [Fleming62] introduced a groundbreaking approach using blowing-down and minimal cones to address the problem in higher dimensions. This work not only motivated subsequent research in the 1960s to fully solve the problem but also gave rise to the seminal field of geometric measure theory. Subsequently, De Giorgi [DeGiorgi65] discovered a "one-dimensional gap" property for minimal cones3, which implied that the Bernstein problem holds in $\mathbb{R}^4$. Then, from 1966 to 1968, Almgren [Almgren66] and Simons [Simons68] proved it for $N=5$ and $N\le 7$, respectively. Finally, in 1969, Bombieri, De Giorgi, and Simons [Bombieri69] demonstrated the existence of non-flat minimal cones in $\mathbb{R}^N$ for $N\ge 9$, such as the Simons cone. Thus, the Bernstein problem was completely settled. We refer readers to [Giusti84] for more detailed historical perspectives and discussions.

3 If no non-flat minimal cone exists in $\mathbb{R}^{N-1}$, then the analogue of Bernstein's theorem holds for graphs in $\mathbb{R}^N$.

Returning to De Giorgi's conjecture, we recall the Ginzburg–Landau energy

$J(u,\Omega)=\int_\Omega \frac{1}{2}|\nabla u|^2+W(u)\,\mathrm{d}x,$

where $W$ is a double-well potential. We say $u$ is a local minimizer of $J(\cdot,\Omega)$ if

$J(u,\Omega)\le J(u+\varphi,\Omega),\quad \forall \varphi\in C^\infty_c(\Omega),$

and a global minimizer if $\Omega=\mathbb{R}^n$.

For simplicity, let $\Omega=B_1=\{x\in\mathbb{R}^n:|x|<1\}$. Introduce the rescaling $u_\epsilon(x):=u\left(x/\epsilon\right)$ for $x\in B_1$ and $\epsilon>0$. If $u$ minimizes $J(\cdot,B_{\epsilon^{-1}})$, then $u_\epsilon$ minimizes the rescaled functional $J_\epsilon$ in $B_1$, defined by

$J_\epsilon(v,B_1)=\int_{B_1}\frac{\epsilon}{2}|\nabla v|^2+\frac{1}{\epsilon}W(v)\,\mathrm{d}x.$

Intuitively, for fixed $v$, the main contribution to $J_\epsilon$ as $\epsilon\searrow 0^+$ comes from the potential $W(v)$, which is minimized when $v$ equals either $1$ or $-1$. Hence, $v$ may transition sharply from the region where $v=1$ to where $v=-1$ across an interface.

Let $\Sigma_s:=\{v=s\}$. By the Cauchy–Schwarz inequality and the co-area formula, we obtain

$J_\epsilon(v,B_1)\ge\int_{B_1}\sqrt{2W(v)}\cdot|\nabla v|\,\mathrm{d}x=\int_{-1}^1\sqrt{2W(v)}\cdot\mathcal{H}^{n-1}(\Sigma_s)\,\mathrm{d}s,$

where $\mathcal{H}^{n-1}(\Sigma_s)$ is the $(n-1)$-dimensional Hausdorff measure of the level set $\Sigma_s$. Minimizing $J_\epsilon$ requires each level set $\Sigma_s$ to be a minimal surface minimizing $\mathcal{H}^{n-1}(\Sigma_s)$. This provides a heuristic justification for conjecturing that all level sets of $u$ are flat for $n\le 8$.

Furthermore, equality in the above inequality holds if $\epsilon|\nabla v|=\sqrt{2W(v)}$. Let

$g_0\left(\frac{d_{\Sigma_0}(x)}{\sqrt{2}\epsilon}\right):=v(x),$

where $d_{\Sigma_0}:B_1\to \mathbb{R}_{\ge0}$ is the signed distance function from $x$ to the level set $\Sigma_0$. Then clearly

$g_0'=\sqrt{2W(g_0)},\quad g_0(0)=0,$

where $g_0$ is the unique one-dimensional minimizer of $J_\epsilon$ and is increasing. Note also that $g_0$ is bounded since $W$ is a double-well potential. In particular, if $W(s)=\frac{1}{4}(1-s^2)^2$ and $\epsilon=1$, then $g_0(x)=\tanh{(x/\sqrt{2})}$, which is precisely the unique (up to translation) one-dimensional solution to the Allen–Cahn equation,

$g_0''+g_0-g_0^3=0.$

Thus, the conjecture can be restated more precisely as follows.

Conjecture (De Giorgi, 1978)
Let $u$ be a bounded entire solution of the Allen–Cahn equation
$\Delta u+u-u^3=0,\quad\text{in}\ \mathbb{R}^n,$
which is monotone in one direction, say $\frac{\partial u}{\partial x_n}>0$. Then $u$ depends on only one variable for $n\le 8$. More precisely, $u$ must have the form
$u(x)=\tanh\left(\frac{a\cdot x-b}{\sqrt{2}}\right),$
where $a\in \mathbb{R}^n$ is a unit vector and $b\in\mathbb{R}$.

Motivated by the preceding discussion, one can pose an analogous but more general question for an arbitrary double-well potential $W$.

Conjecture (General case)
Let $u$ be a bounded entire solution of the semilinear elliptic equation
$\Delta u-W'(u)=0,\quad\text{in}\ \mathbb{R}^n,\quad \text{where $W$ is a double-well potential},$
which is monotone in one direction, say $\frac{\partial u}{\partial x_n}>0$. Then $u$ depends on only one variable for $n\le 8$. More precisely, $u$ must have the form
$v(x)=g_0\left(\frac{d_{\Sigma_0}(x)}{\sqrt{2}}\right),$
where $g_0'=\sqrt{2W(g_0)}$ and $g_0(0)=0$.

We now present several milestone results on De Giorgi's conjecture (Conjecture 1). First, inspired by the methods of Berestycki, Caffarelli, and Nirenberg [Berestycki97], Ghoussoub and Gui [Ghoussoub98] proved the conjecture for general $W$ in dimension $n=2$. Subsequently, by exploiting an "energy gap" estimate, Ambrosio and Cabré [Ambrosio00] proved it for $n=3$; later, Giovanni Alberti, Ambrosio, and Cabré [Alberti01] extended the proof to general $W$ in the same dimension using a calibration argument. Building on the idea of asymptotic flatness from [Alberti01], Savin [Savin09] resolved the conjecture for $4\le n\le 8$ under the technical assumption

$\lim_{x_n\to\pm\infty}u(x)=\pm1,\quad \text{pointwise}.$

by refining a flatness convergence theorem of De Giorgi (see [Giusti84]), originally derived from the work of Modica [Modica78]. Finally, del Pino, Kowalczyk, and Wei [Kowalczyk11] constructed a counterexample for $n\ge 9$, which had long been anticipated based on the work of Bombieri, De Giorgi, and Simons [Bombieri69]. Subsequently, Wang [Wang17] provided a new proof for Savin's flatness theorem by improved Allard regularity and Savin [Savin17] simplified the original proof. More recently, Savin and Zhang [Savin25], [Savin25b] extended the results to nonlocal models and the rigidity results with sub-quadratic growth assumption (recall the corresponding condition in Bernstein problem). To this day, the conjecture remains OPEN for $4\le n\le 8$ without the assumption of pointwise convergence at infinity. We refer the readers to Wei's survey [Wei12] for more discussions about the connection between Bernstein problem and De Giorgi conjecture.

A weaker version of the problem, obtained by strengthening the pointwise convergence to uniform convergence, is known as the Gibbons conjecture. This was completely solved independently by Farina [Farina99], Barlow, Bass and Gui [Barlow00], and by Berestycki, Hamel and Monneau [Berestycki00].

Finally, there are plenty of extensions on Allen-Cahn equation and De Giorgi conjecture, such as stable De Giorgi conjecture, finite Morse index, fractional Allen-Cahn equation, and nonlocal equations, etc. We refer the readers to Wang's survey [Wang21].

Rubinstein-Sternberg-Keller conjecture and mean curvature flow

The rescaling $u_\epsilon$ and the blow-down limit as $\epsilon\searrow 0^+$ lead to the scaled version of equation $\partial_tu=\epsilon \Delta u-\frac{1}{\epsilon}W'(u)=\epsilon\Delta u+\frac{1}{\epsilon}(u-u^3)$. This is a typical reaction–diffusion equation [Rubinstein89], [Rubinstein89b]. Under a simple scaling, we obtain

$\partial_tu=\Delta u-\frac{1}{\epsilon^2}W'(u)=\Delta u+\frac{1}{\epsilon^2}(u-u^3).$

We are interested in the asymptotic behavior of $u$ under this flow as $\epsilon\searrow 0^+$, which was studied for the steady case $\Delta u-1/\epsilon^2W'(u)=0$ by Modica [Modica77], [Modica78], [Modica87]. In light of De Giorgi's conjecture, we expect the asymptotic profile of $u$ under the flow to match the static case, namely

$u(x)\sim g_0\left(\frac{d_{\Sigma^t}(x)}{\sqrt{2}\epsilon}\right),\quad \epsilon\searrow0^+,$

where $\Sigma^t$ denotes the interface $\Sigma_0$ at time $t$ and $d_{\Sigma^t}(x)$ is the signed distance from $x$ to $\Sigma^t$. Formal calculation yields the following mean curvature flow for $d_{\Sigma^t}(x)$:

$\partial_td_{\Sigma^t}(x)=\Delta d_{\Sigma^t}(x),\quad \text{on the interface $\Sigma^t$}.$

Geometrically, the velocity of the surface $\Sigma^t$ equals its mean curvature vector. The rigorous proof was given by de Mottoni and Schatzman [DeMottoni95] (by extension method), Bronsard and Kohn [Bronsard91] (in radial version), Evans, Soner and Souganidis [Evans92] (by viscosity solution) and Ilmanen [Ilmanen93] (by geometric measure theory), etc.

Finally, we extend the Allen–Cahn equation to the vector-valued case. Let $u:\Omega\subset \mathbb{R}^n\to \mathbb{R}^m$ be a vector-valued function and $W:\mathbb{R}^m\to \mathbb{R}_{\ge0}$ be a double-well potential that vanishes on two disjoint connected submanifolds $M_1$ and $M_2$, i.e.,

$W(M_1)=W(M_2)=0.$

This gives the vector-valued phase transition model:

$\partial_tu=\Delta u-\frac{1}{\epsilon^2}\partial_uW(u).$

Formal calculations [Rubinstein89], [Rubinstein89b] suggest that the interface $\Sigma^t$ evolves by mean curvature flow, while away from the interface, $u$ behaves as a harmonic map heat flow into either $M_1$ or $M_2$. However, a rigorous proof of this result is lacking, and the problem remains open, known as the Rubinstein-Sternberg-Keller problem; see [Fei23].

Conjecture (Rubinstein–Sternberg-Keller, 1989)
As $\epsilon\to 0$ in the vector-valued phase transition model
$\partial_tu=\Delta u-\frac{1}{\epsilon^2}\partial_uW(u),$
the interface $\Sigma^t$ moves by mean curvature flow, while $u$ behaves as a harmonic map heat flow away from the interface into either $M_1$ or $M_2$.

Several milestone results exist for this problem. For example, Bronsard and Stoth [Bronsard98] considered

$W(u):=(|u|-a)^2(b-|u|)^2,\quad u\in\mathbb{R}^2,$

and rigorously proved the conjecture in this special case. Subsequently, Lin, Pan, and Wang [Lin12] studied $W$ attaining its global minimum at two disjoint compact connected submanifolds $N^{\pm}$ in $\mathbb{R}^m$. They showed that the energy functional

$E_\epsilon(u)=\int_\Omega|\nabla u|^2+\frac{1}{\epsilon^2}W(u)\,\mathrm{d}x$

has the leading-order asymptotic expansion as $\epsilon\searrow0^+$:

$E_\epsilon\sim\frac{c}{\epsilon}\mathcal{H}^{n-1}(\Sigma^t)+D+O(1),$

where $c$ represents the energy of the minimal connecting orbit between $N^+$ and $N^-$, and $D$ is the energy of the minimal harmonic map into $N^{\pm}$. Many subsequent works have addressed special forms of $W$, such as [Fei18], [Lin19], [Laux21], [Lin23], [Fei23]. For the latest results, we refer readers to the work of Dong and Wang [Dong25].

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