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as the number n of atoms tends to infinity, which is crucial to overcome the lack of compactness outside the class of almost-connected sequences of energy-equibounded minimizers. In particular, it consists in proving that it is enough to select a connected component among those with largest cardinality for each solution of (4). This is achieved by proving compactness for almost-connected energy minimizers and then by defining a proper transformation \({\mathcal {T}}\) of configurations (based on iterated translations of connected components as detailed in Definition 2.1), which always allows to pass to an almost-connected sequence of minimizers.

In Sect. 2, we introduce the mathematical setting with the discrete models (expressed both with respect to lattice configurations and to Radon measures) and the continuum model, and the three main theorems of the paper. In Sect. 3, we treat the wetting regime and prove Theorem 2.2. In Sect. 4, we establish the compactness result for energy-\(E_n\)-equibounded almost-connected sequences. In Sect. 5 we prove the lower bound of the \(\Gamma \)-convergence result. In Sect. 6, we prove the upper bound of the \(\Gamma \)-convergence result. In Sect. 7, we study the convergence of almost-connected transformations of minimizers and present the proofs of both Theorems 2.3 and 2.4. In Sect. 8, we present some other positioning of \({\mathcal {L}}_F\) and \({\mathcal {L}}_S\) that can be reduced to the setting introduced in Sect. 2.

for some \(p,q\in \mathbb {N}\) without common factors, since the case of \(e_S=re_F\) for some \(r\in \mathbb {R}{\setminus }\mathbb {Q}\) is simpler, as the contribution of \(E_S\) is negligible (namely, in this case \(\# \partial {\mathcal {L}}_{FS}=1\)). More precisely, for \(e_S=re_F\) with \(r\in \mathbb {R}{\setminus }\mathbb {Q}\) the same analysis (or the one in Au Yeung et al. (2012) applies, and, up to rigid transformations, minimizers converge to a Wulff shape in \(\mathbb {R}^2{\setminus }S\) with the Wulff-shape boundary intersecting \(\partial S\) at least in a point.

In this section, the rigorous statements of the main theorems of the paper are presented. We begin with the following result that characterizes the wetting regime in terms of a condition only depending on \(v_{FF}\) and \(v_{FS}\), and the minimizers in such regime.

as the dewetting condition or the dewetting regime. The following result shows that connected components with the largest cardinality of minimizers incorporate the whole mass in the limit.

We rigorously prove by \(\Gamma \)-convergence that the discrete models converge to the continuum model, and in view of the previous result (even in the lack of a direct compactness result for general sequences of minimizers, possibly not almost connected), we prove convergence (up to passing to a subsequence and up to translations) of the minimizers of the discrete models to a bounded minimizer of the continuum model, which in turn it is also proven to exist. We do not discuss here further the minimality property of the Winterbottom shape for the energy \({\mathcal {E}}\) and the uniqueness of the minimizers of \(I_{\infty }\) in \({\mathcal {M}}_W\).

Every sequence \(\mu _n\in {\mathcal {M}}_n\) of minimizers of \(E_n\) admits, up to translation in the direction \({\varvec{t}}_1\) (i.e., up to replacing \(\mu _n\) with \(\mu _n(\cdot +c_n{\varvec{t}}_1)\) for chosen fixed integers \(c_n\in \mathbb {Z})\), a subsequence converging with respect to the weak* convergence of measures to a minimizer of \(I_\infty \) in \({\mathcal {M}}_W\).

for a constant \(C>0\). Then, there exist an increasing sequence \(n_r\), \(r\in \mathbb {N}\), and a measure \(\mu \in {\mathcal {M}}(\mathbb {R}^2)\) with \(\mu \ge 0\) and \(\mu (\mathbb {R}^2)=1\) such that \(\mu _{r}{\mathop {\rightharpoonup }\limits ^{*}} \mu \) in \({\mathcal {M}}(\mathbb {R}^2)\), where \(\mu _{r}:=\mu _{D_{n_r}(\,\cdot \,+a_{n_r})}\) for some translations \(a_n\in \mathbb {R}^2\) (see 14 for the definition of the empirical measures \(\mu _{D_{n_r}}\)). Moreover, if \(D_n\in {\mathcal {C}}_n\) are minimizers of \(V_n\) in \({\mathcal {C}}_n\), then we can choose \(a_n=t_n{\varvec{t}}_1\) for integers \(t_n\in \mathbb {Z}\).

In order to conclude the proof it suffices to prove that \(\mu (\mathbb {R}^2)=1\), and this directly follows from the fact that the support of \(\mu _r\) are contained in a compact set of \(\mathbb {R}^2\). The last claim follows from the fact that if \(D_n \in {\mathcal {C}}_n\) is a minimizer of \(V_n\) then we have that \({\mathcal {T}}_1 (D_n)=D_n\), i.e., all connected components of \(D_n\) are connected with the substrate and \(D_n\) is almost-connected. \(\square \)

(Compactness) Assume (32). Let \(D_n\in {\mathcal {C}}_n\) be configurations satisfying (48) and let \(\mu _{n}:=\mu _{{\mathcal {T}}(D_n)}\) be the empirical measures associated with the transformed configurations \({\mathcal {T}}(D_n)\in {\mathcal {C}}_n\) associated with \(D_n\) by Definition 2.1. Then, up to translations (i.e., up to replacing \(\mu _{n}\) by \(\mu _{n}(\cdot +a_n)\) for some \(a_n\in \mathbb {R}^2 )\) and a passage to a non-relabeled subsequence, \(\mu _{n}\) converges weakly * in \({\mathcal {M}}({\mathbb {R}}^2)\) to a measure \(\mu \in {\mathcal {M}}_W\), where \({\mathcal {M}}_W\) is defined in (35). Furthermore, if \(D_n\in {\mathcal {C}}_n\) are minimizers of \(V_n\) in \({\mathcal {C}}_n\), then we can choose \(a_n=t_n{\varvec{t}}_1\) for integers \(t_n\in \mathbb {Z}\).

We notice that Theorem 7.1 is not enough to conclude Assertion 3. of Theorem 2.4. In fact, the compactness provided for energy equi-bounded sequences \(D_n\in {\mathcal {C}}_n\) by Theorem 4.4 of Sect. 4 holds only for almost-connected configurations \(D_n\). Therefore, as detailed in the following result, we can deduce the convergence of a subsequence of minimizers only after performing (for example) the transformation \({\mathcal {T}}\) given by Definition 2.1, which does not change the property of being a minimizer.

Assume (32). For every sequence of minimizers \(\mu _n\in {\mathcal {M}}_n\) of \(E_n\), there exists a ( possibly different) sequence of minimizers \(\widetilde{\mu }_n\in {\mathcal {M}}_n\) of \(E_n\) that admits a subsequence converging with respect to the weakly *convergence of measures to a minimizer of \(I_\infty \) in

Let \(\mu _n\in {\mathcal {M}}_n\) be minimizers of \(E_n\). By (15), (16), and (105) there exist configurations \(D_n\in {\mathcal {C}}_n\) such that \(\mu _n:=\mu _{D_n}\). Let \({\mathcal {T}}(D_n)\in {\mathcal {C}}_n\) be the transformed configurations associated with \(D_n\) by Definition 2.1. We notice that the sequence of measures

Therefore, by Theorems 7.1 and 4.4 we obtain that there exist a sequence of vectors \(a_n:=t_n{\varvec{t}}_1\) for \(t_n\in \mathbb {Z}\), an increasing sequence \(n_k\), \(k\in \mathbb {N}\), and a measure \(\mu \in {\mathcal {M}}_W\) (being a minimizer of \(I_{\infty }\)) such that \(\widetilde{\mu }_{n_k}\rightharpoonup ^*\mu \) in \({\mathcal {M}}(\mathbb {R}^2)\), where

In view of Theorem 2.3, we can improve the previous result and in turns, prove the convergence of minimizers (up to a subsequence) directly without passing to an auxiliary sequence of minimizers obtained by performing the transformation \({\mathcal {T}}\) given by Definition 2.1. In fact, Theorem 2.3 allows to exclude the possibility that a sequence of (not almost-connected) minimizers \(\mu _n\in {\mathcal {M}}_n\) loses mass in the limit.

By Corollary 7.2, there exists a (possibly different) sequence of minimizers \(\widetilde{\mu }_{n_k}\in {\mathcal {M}}_{n_k}\) of \(E_{n_k}\) that (up to passing to a non-relabeled subsequence) converge with respect to the weak* convergence of measures to a minimizer \(\mu \in {\mathcal {M}}_W\) of \(I_\infty \). Therefore, there exists a bounded set \(D\subset \mathbb {R}^2{\setminus }S\) of finite perimeter with \(|D|=1/\rho \) such that \(\mu = \rho \chi _D\) and \({\widetilde{\mu }}_{n_k}\) converge with respect to the weak* convergence to \( \rho \chi _D\).

Let \(\mu _n\in {\mathcal {M}}_n\) be minimizers of \(E_n\). By Corollary 7.2, there exist another sequence of minimizers \({\widetilde{\mu }}_n\in {\mathcal {M}}_n\) of \(E_n\), an increasing sequence \(n_k\) for \(k\in \mathbb {N}\), and a measure \(\mu \in {\mathcal {M}}_W\) minimizing \(I_\infty \) such that

for some integers \(t_n\in \mathbb {Z}\), and for configurations \(D_n\in {\mathcal {C}}_n\) such that \(\mu _n:=\mu _{D_n}\), where \({\mathcal {T}}(D_n):={\mathcal {T}}_2({\mathcal {T}}_1(D_n))\) (see Definition 2.1). Furthermore, by (16) and (105) the configurations \(D_n\) are minimizers of \(V_n\) in \({\mathcal {C}}_n\) and hence, \({\mathcal {T}}_1(D_n)=D_n\) and by Theorem 2.3 we have that, up to a non-relabeled subsequence,

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