1.1 Main results

Hypothesis on $W$:

(H1) $W \in C^{1,\alpha }_{loc}({\mathbb {R}}^m ; [0,\,+ \infty )) \ ,\,\ \lbrace W=0 \rbrace = \lbrace a_1,\,a_2,\,...,\,a_N \rbrace \ ,\,\ N \in \mathbb {N} \ ,\, a_i$ are the global minima of $W$. Assume also that

\[ W_u(u) \cdot u > 0 \quad \textrm{and} \quad W(u)\geq c_1 |u|^2 \ ,\ \textrm{if}\ |u |>M . \]

Hypothesis on the Dirichlet data:

(H2)(i) $|g_{\varepsilon }| \leq M \ ,\, g_\varepsilon \stackrel {L^1(\Omega )}{\longrightarrow } g_0$ and $J_\varepsilon (g_\varepsilon,\, \Omega _{\rho _0} \setminus \Omega ) \leq C \ ,\,$ where $\partial \Omega$ is Lipschitz and $\Omega _{\rho _0}$ is a small dilation of $\Omega \ ,\,\ \rho _0 >1$, in which $g_\varepsilon$ is extended $(C ,\,\ M \ \textrm {indep. of}\ \varepsilon )$.

And either

(ii) $g_\varepsilon \in C^{1,\alpha }(\overline {\Omega }) \ ,\,\ | g_{\varepsilon } |_{1,\alpha } \leq \frac {M}{\varepsilon }$ and $\partial \Omega$ is $C^2$, where we denote with $| \cdot |_{1, \alpha }$ as the $C^{1, \alpha }$ norm.

Or (ii’) $g_\varepsilon \in H^1( \Omega )$ and $J_\varepsilon ( {u_\varepsilon },\, \Omega ) \leq C$.

For $i \neq j \ ,\,\ i,\,j \in \lbrace 1,\,2,\,...,\,N \rbrace$, let $U \in W^{1,2}( \mathbb {R};\mathbb {R}^m)$ be the 1D minimizer of the action

(1.4)\begin{align} & \sigma_{ij}:= \min \int_{-\infty}^{+\infty} \left( \frac{1}{2} |U'|^2 + W(U) \right) \,{\rm d}t <{+}\infty\ , \nonumber\\ & \lim_{t \rightarrow - \infty} U(t) = a_i \ ,\ \lim_{t \rightarrow + \infty} U(t) = a_j \ ,\ U( \mathbb{R}) \in \mathbb{R}^m \setminus \lbrace W=0 \rbrace \end{align}

where $U$ is a connection that connects $a_i$ to $a_j \ ,\,\ i,\,j \in \lbrace 1,\,2,\,...,\,N \rbrace$.

The existence of such geodesics has been proved under minimal assumptions on the potential $W$ in [Reference Zuniga and Sternberg38].

Let $J_\varepsilon$ defined in (1.1), we define

(1.5)\begin{align} \tilde{J}_{\varepsilon}(u, \Omega) := \begin{cases} J_{\varepsilon}(u,\Omega) \ ,\ \textrm{if} \ u = g_{\varepsilon} \ \textrm{on} \ \Omega_{\rho_0} \setminus \Omega\ ,\ u \in H^1_{loc}(\mathbb{R}^n;\mathbb{R}^m) \\ + \infty\ , \ \textrm{otherwise} \end{cases} \end{align}

where $\Omega \subset \Omega _{\rho _0}$ as in (H2)(i) and let

(1.6)\begin{align} J_0(u, \Omega) := \sum_{1 \leq i < j \leq N} \sigma_{ij} \mathcal{H}^{n-1} (\partial^* \Omega_i \cap \partial^* \Omega_j \cap \Omega) = \sum_{1 \leq i < j \leq N} \sigma_{ij} \mathcal{H}^{n-1} (S_{ij}(u) \cap \Omega), \end{align}

where $S_{ij}(u):= \partial ^* \lbrace u=a_i \rbrace \cap \partial ^* \lbrace u=a_j \rbrace \ ,\,\ u \in BV(\Omega ;\lbrace a_1,\,a_2,\,...,\,a_N \rbrace )$ and we denote as $\partial ^* \Omega _k$ the reduced boundary of $\Omega _k$.

Finally, we define the limiting functional subject to the limiting boundary conditions

(1.7)\begin{align} \tilde{J}_0 (u, \Omega) := \begin{cases} J_0(u,\Omega) \ ,\ \textrm{if} \ u \in BV(\Omega;\lbrace a_1,a_2,...,a_N \rbrace) \quad \textrm{and}\ u= g_0 \quad \textrm{on} \quad \Omega_{\rho_0} \setminus \Omega\ \\ + \infty\ ,\ \textrm{otherwise} \end{cases} \end{align}

We can write $J_\varepsilon,\,J_0,\, \tilde {J}_\varepsilon,\, \tilde {J}_0 : L^1(\Omega ; \mathbb {R}^n) \rightarrow \overline {\mathbb {R}}$, where $\overline {\mathbb {R}} = \mathbb {R} \cup \lbrace \infty \rbrace$ and the $\Gamma$-convergence will be with respect to the $L^1$ topology.

Our first main result is the following

Theorem 1.1 Let $J_{\varepsilon }$ be defined by (1.1) and $\tilde {J}_{\varepsilon } \ ,\,\ \tilde {J}_0$ defined in (1.5) and (1.7) respectively.

Then

(1.8)\begin{equation} \Gamma- \lim_{\varepsilon \rightarrow 0} \tilde{J}_{\varepsilon}(u,\Omega) = \tilde{J}_0(u,\overline{\Omega}) . \end{equation}

Remark 1.2 Note that the domain of $\tilde {J}_0$ is the closure of $\Omega$, which means that there is a boundary term (see also (2.9) in [Reference Owen, Rubinstein and Sternberg32] for the analogue in the scalar case). More precisely, by proposition 3.5 and theorem 5.8 in [Reference Evans and Gariepy16] we can write

(1.9)\begin{align} & \tilde{J}_0(u, \overline{\Omega}) = \frac{1}{2} \sum_{i=1}^N \int_{\overline{\Omega}} | D ( \phi_i \circ u) | \ = \frac{1}{2} \sum_{i=1}^N \int_{\Omega} | D ( \phi_i \circ u) |\nonumber\\ & \qquad + \frac{1}{2} \sum_{i=1}^N \int_{\partial \Omega} | T(\phi_i \circ u) - T(\phi_i \circ g_0) | \ d \mathcal{H}^{n-1} \nonumber\\ & \quad \textrm{where} \ {\phi_i \ \textrm{defined in} \ (3.2) \ } \textrm{and} \ T \ \textrm{is the trace operator for} \ BV \ \textrm{functions}. \end{align}

The overview of the strategy of the proof of theorem 1.1 is as follows. First we observe that the $\Gamma$-limit established in [Reference Baldo7], in particular theorem 2.5, holds also without the mass constraint (see theorem 2.2 in Preliminaries section). Next, we apply a similar strategy to that of [Reference Ansini, Braides and Chiadò Piat6, Theorem 3.7] in which there is a $\Gamma$-convergence result with boundary conditions in the scalar case which states that we can incorporate the constraint of Dirichlet values in the $\Gamma$-limit, provided that this $\Gamma$-limit is determined. Since by theorem 2.2 we have that $J_\varepsilon \Gamma$-converges to $J_0$, we establish the $\Gamma$-limit of $\tilde {J}_\varepsilon$, that is, the $\Gamma$-limit of the functional $J_\varepsilon$ with the constraint of Dirichlet values. For the proof of the $\Gamma$-limit we can assume either (H2)(ii) or (H2)(ii’).

Next, we study the solution of the geometric minimization problem that arises from the limiting functional.

In order to obtain precise information about the minimizer of the limiting functional $\tilde {J}_0 (u,\, \overline {B}_1) \ ,\, \ B_1 \subset \mathbb {R}^2$, we impose that the limiting boundary conditions $g_0$ have connected phases. So we assume,

(H2) (iii) Let $g_0 = \sum _{i=1}^3 a_i \ \chi _{I_i}(\theta ) \ ,\,\ \theta \in [0,\, 2 \pi ) \ ,\,\ I_i \subset [0,\, 2 \pi ) \ ,\,\ \cup _{i=1}^3 I_i = [0,\, 2 \pi )$ be the limit of $g_\varepsilon$. Assume that $I_i$ are connected and that

\begin{align*} & \theta_0 < \frac{2 \pi}{3} \ ,\ \textrm{where} \ \theta_0 \ \textrm{is the largest angle of the points} \ p_i = \partial I_k \cap \partial I_l \\ & k \neq l \ ,\ i \in \lbrace 1,2,3 \rbrace \setminus \lbrace k,l \rbrace . \end{align*}

The assumption $\theta _0 < \frac {2 \pi }{3}$ arises from Proposition 3.2 in [Reference Morgan30] that we utilize for the proof (see proposition 2.5 in Preliminaries section) and guarantees that the boundary of the partition defined by the minimizer will be line segments meeting at a point inside $B_1$.

Our second main result is the following

Theorem 1.3 Let $u_0 = a_1 \chi _{\Omega _1} +a_2 \chi _{\Omega _2} + a_3 \chi _{\Omega _3}$ be a minimizer of $\tilde {J}_0(u,\,\overline {B}_1)$ subject to the limiting Dirichlet values (H2)(iii).

Then the minimizer is unique and in addition,

(1.10)\begin{equation} \partial \Omega_i \cap \partial \Omega_j \ \textrm{are line segments meeting at} \ 120^o \ \textrm{in a point in} \ B_1 \ (i \neq j). \end{equation}

For proving theorem 1.3, we first prove that the partition defined by $u_0$ is $(M,\,0,\, \delta )$-minimal as in Definition 2.1 in [Reference Morgan30] (see definition 2.4). This is proved by a comparison argument by defining a Lipschitz perturbation of the partition of the minimizer with strictly less energy. Then, by utilizing a uniqueness result for $(M,\, 0 ,\,\delta )$-minimal sets in [Reference Morgan30] (see proposition 2.5), we can conclude that the minimizer of the limiting energy is unique and the boundaries of the partition that the minimizer defines are line segments meeting at $120^o$ degrees in an interior point of the unit disc.

In the last subsection, we note that the result in theorem 1.3 can be extended also to the mass constraint case (see [Reference Baldo7]). However, in this case the uniqueness will be up to rigid motions of the disc (see Theorems 3.6 and 4.1 in [Reference Canete and Ritore10]).

1.2 Previous fundamental contributions

We will now briefly introduce some of the well-known results in the scalar case. The notion of $\Gamma$-convergence was introduced by De Giorgi and Franzoni in [Reference De Giorgi and Franzoni14] and in particular relates phase transition-type problems with the theory of minimal surfaces. One additional application of $\Gamma$-convergence is the proof of existence of minimizers of a limiting functional, say $F_0$, by utilizing an appropriate sequence of functionals $F_\varepsilon$ that we know they admit a minimizer and the $\Gamma$-limit of $F_\varepsilon$ is $F_0$. And also vice versa ([Reference Kohn and Sternberg25]), we can obtain information for the $F_\varepsilon$ energy functional from the properties of minimizers of the limiting functional $F_0$. We can think of this notion as a generalization of the Direct Method in the Calculus of Variations i.e. if $F_0$ is lower semicontinuous and coercive we can take $F_\varepsilon = F_0$ and then $\Gamma$-lim $F_\varepsilon =F_0$.

There are many other ways of thinking of this notion, such as a proper tool in finding the limiting functional among a sequence of functionals.

Let $X$ be the space of the measurable functions $u : \Omega \subset \mathbb {R}^n \rightarrow \mathbb {R}$ endowed with the $L^1$ norm and

\begin{align*} & F_\varepsilon(u,\Omega) := \begin{cases} \int_\Omega \frac{\varepsilon}{2} | \nabla u|^2 + \frac{1}{\varepsilon}W(u) \,{\rm d}x \ ,\ u \in W^{1,2} (\Omega ; \mathbb{R}) \cap X \ \\ + \infty\ , \ \textrm{elsewhere in} \ X \end{cases} \\ & F_0(u,\Omega) := \begin{cases} \sigma \mathcal{H}^{n-1} (Su) \ ,\ u \in SBV(\Omega ; \lbrace -1,1 \rbrace) \cap X \ \\ + \infty\ , \ \textrm{elsewhere in} \ X \end{cases} \ \\ & \quad \textrm{where} \ W : \mathbb{R} \rightarrow [0, + \infty ) \ ,\ \lbrace W =0 \rbrace = \lbrace -1 , 1 \rbrace\ ,\ \sigma = \int_{{-}1}^1 \sqrt{2 W(u)}\,{\rm d}u \\ & \qquad {and} \ Su \ \textrm{is the singular set of the SBV function} \ u. \ \end{align*}

Let now $u_\varepsilon$ be a minimizer of $F_\varepsilon$ subject to a mass constraint, that is, $\int _{\Omega }u = V \in (0,\, |\Omega |)$. The asymptotic behaviour of $u_\varepsilon$ was first studied by Modica and Mortola in [Reference Modica and Mortola28] and by Modica in [Reference Modica27, Reference Modica29]. Also, later Sternberg [Reference Sternberg34] generalized these results for minimizers with volume constraint. Furthermore, Owen et al. in [Reference Owen, Rubinstein and Sternberg32] and Ansini et al. in [Reference Ansini, Braides and Chiadò Piat6], among others, studied the asymptotic behaviour of the minimizers subject to Dirichlet values for the scalar case.

As mentioned previously, one of the most important outcomes of $\Gamma$-convergence in the scalar phase transition-type problems is the relationship with minimal surfaces. More precisely, the well-known theorem of Modica and Mortola states that the $\varepsilon$-energy functional of the Allen–Cahn equation $\Gamma$-converges to the perimeter functional that measures the perimeter of the interface between the phases (i.e. $\Gamma$-$\textrm {lim} \ F_\varepsilon = F_0$). So the interfaces of the limiting problem will be minimal surfaces.

This relationship is deeper as indicated in the De Giorgi conjecture (see [Reference De Giorgi15]) which states that the level sets of global entire solutions of the scalar Allen–Cahn equation that are bounded and strictly monotone with respect to $x_n$ are hyperplanes if $n \leq 8$. The relationship with the Bernstein problem for minimal graphs is the reason why $n \leq 8$ appears in the conjecture. The $\Gamma$-limit of the $\varepsilon$-energy functional of the Allen–Cahn equation is a possible motivation behind the conjecture.

In addition, Baldo in [Reference Baldo7] and Fonseca and Tartar in [Reference Fonseca and Tartar18] extended the $\Gamma$-convergence analysis for the phase transition-type problems to the vector case subject to a mass constraint and the limiting functional measures the perimeter of the interfaces separating the phases, and thus there is a relationship with the problem of minimizing partitions. In § 5 we analyse this in the set up of Dirichlet boundary conditions. Furthermore, the general vector-valued coupled case has been thoroughly studied in the works of Borroso–Fonseca and Fonseca–Popovici in [Reference Cristina Barroso and Fonseca12] and [Reference Fonseca and Popovici17] respectively.

There are many other fundamental contributions on the subject, such as the works of Gurtin [Reference Gurtin21, Reference Gurtin22], Gurtin and Matano [Reference Gurtin and Matano23] on the Modica–Mortola functional and its connection with materials science, the work of Hutchingson and Tonegawa on the convergence of critical points in [Reference Hutchinson and Tonegawa24], the work of Bouchitté [Reference Bouchitté8] and of Cristoferi and Gravina [Reference Cristoferi and Gravina13] on space-dependent wells and extensions on general metric spaces in the work of Ambrosio in [Reference Ambrosio5]. Several extensions to the non-local case and fractional setting have also been studied by Alberti-Bellettini in [Reference Alberti and Bellettini1], by Alberti-Bouchitté-Seppecher in [Reference Alberti, Bouchitté and Seppecher2] and by Savin-Valdinoci in [Reference Savin and Valdinoci33] among others.