On the Gamma Convergence of Functionals Defined Over Pairs of Measures and Energy-Measures

A novel general framework for the study of Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-convergence of functionals defined over pairs of measures and energy-measures is introduced. This theory allows us to identify the Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-limit of these kind of functionals by knowing the Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Gamma $$\end{document}-limit of the underlying energies. In particular, the interaction between the functionals and the underlying energies results, in the case these latter converge to a non-continuous energy, in an additional effect in the relaxation process. This study was motivated by a question in the context of epitaxial growth evolution with adatoms. Interesting cases of application of the general theory are also presented.

of the crystal) play an important role in the description of the evolution of the interface. For this reason, Fried and Gurtin (2004) introduced a model that includes the effect of adatoms, an additional variable whose evolution in time is coupled to the evolution in time of the interface of the crystal. In the simple case of a crystal growing on a general shape, i.e., not graph constrained, but without considering elastic effects or surface stress, the free energy of the system reads as: (1.1) Here E ⊂ R d is a set with finite perimeter representing the shape of the crystal, and u ∈ L 1 (∂ * E; [0, + ∞)) is the density of the adatoms. The Borel function ψ: [0, + ∞) → (0, + ∞) is assumed to be non-decreasing and satisfying inf ψ > 0. From the physical point of view, this latter hypothesis is motivated by the fact that, energetically, even an interface without adatoms cannot be created for free.
The interest in considering model (1.1) lies in the intriguing and challenging mathematical questions that are connected to the related evolution equations. In order to perform numerical simulations of the evolution equations obtained formally as the gradient flow of (1.1), Rätz and Voigt (2006) (see also Burger 2006) considered, for ε > 0, the following phase field model inspired by the Modica-Mortola functional (Modica 1987;Modica and Mortola 1977) (1.2) Here φ ∈ H 1 (R d ) is the phase variable, W : R → [0, + ∞) is a continuous double well potential vanishing at 0 and 1, and u ∈ C 0 (R d ; [0, + ∞)). The authors worked with the special case ψ(t) := 1 + t 2 2 . Since the Modica-Mortola functional approximates, in the sense of -convergence, the perimeter functional, it is expected that the phase field model (1.2) approximates the sharp interface energy (1.1). The identification of the correct -limit in the weakest topology ensuring compactness along sequences of uniformly bounded energy is currently missing in the literature. The main aim of this work consists in such an identification. The identification of the -limit is an important ingredient for the study of the convergence of the solutions of the gradient flow of the phase field energy (1.2) to the solution of the gradient flow of the sharp interface energy (1.1). In Rätz and Voigt (2006), this convergence was justified by using formal matched asymptotic expansions. We would like to draw the attention to the fact that energies of type (1.1) are also used in model the evolution and equilibria configuration of surfactants in which a chemical additive is considered to be present on the interface of a bubble, driving the evolution (Acerbi and Bouchitté 2008;Alicandro et al. 2012;Baía et al. 2017;Fonseca et al. 2007b).
As observed in Caroccia et al. (2018), energy (1.1) is not lower semi-continuous with respect to the L 1 × w * topology, where the density u is seen as the measure uH d−1 ¬ ∂ * E. (Note that this topology allows for very general cracks in the crystal.) Therefore, the functional G cannot be the -limit of the functionals G ε . Note that the L 1 convergence for sets only implies the weak* convergence of the distributional derivative of the characteristic functions of the sets. Therefore, the relaxation of the functional G does not follow from the results of Buttazzo and Freddi (1991), for which the weak* convergence of the total variation of the distributional derivative of the characteristic functions of the sets would be required. In Caroccia et al. (2018), the authors identified the relaxed functional G of G with respect to the L 1 × w * topology. The question is then the following: given that the Modica-Mortola functional is known to -converge to the perimeter functional F, is it true that the functionals G ε , which can be seen as adatom-density weighted versions of the functionals F ε , -converge to G, the adatom-density weighted versions of the functional F? The above problem was the motivation to undertake the study of the Gamma convergence of such kind of functionals in a more general framework. The advantage in doing so is in getting a better insight on the technical reasons leading to the answer of the question, other than developing a theory comprehending a variety of other interesting situations. We now introduce this general framework. Let ⊂ R d be a bounded open set with Lipschitz boundary and denote by A( ) the family of open subsets of . For ε > 0, consider the functionals where each F ε is lower semi-continuous in the first variable on each open set A ∈ A( ). For every φ ∈ L 1 , the maps over couples (φ, μ), where φ ∈ L 1 ( ) and μ is a finite nonnegative Radon measure on absolutely continuous with respect to the measure F φ ε . G F ε is set to be + ∞ otherwise. In the same spirit is defined The question we want to investigate is the following: is the -limit of the family {G F ε } ε>0 related to the relaxation of the functional G F in the L 1 × w * topology?
This question is reminiscent of a classical problem studied by Buttazzo and Freddi (1991) (see also Buttazzo 1989) on the -convergence of functionals defined over pairs of measures. Given a sequence of nonnegative Radon measures ν n on , they studied the -limit of functionals of the form where μ = dμ dν ν + μ ⊥ is the Radon-Nikodym decomposition of μ with respect to ν, and f ∞ is the recession function of f . Our framework includes their result for scalar-valued measures, and with f independent of x ∈ and sub-additive in the second variable. Indeed, it is possible to reduce the study of functionals (1.4) to our setting by taking F ε and F constants.
The novelty of this paper is in the treatment of the problem in this general setting, where the convergence ν n * ν is replaced by the -convergence of the underlying functionals F ε to F. Since for each φ ∈ L 1 ( ) we will ask that there exists {φ n } ⊂ L 1 ( ) with φ n → φ in L 1 ( ) such that F φ n n * F φ , in a sense the result in Buttazzo and Freddi (1991) can be seen as a pointwise convergence in our setting. When F is not continuous in L 1 , the interaction between the underlying functionals F ε and the function ψ gives rise, for a class of non-continuous functionals F, to an additional relaxation effect for G F . Because of the technical difficulties we have to deal with, the techniques we employ to prove our results, except for the liminf inequality, are independent from the ones present in Buttazzo and Freddi (1991).

Main Results and Idea of the Proofs
In the main result of this paper, Theorem 3.8 we are able to prove that the -limit of the functionals G F ε is G: the relaxation of the functional G F in the L 1 × w * topology. In particular, an application of Theorem 3.8 (Proposition 5.5) is used to prove the -convergence of G ε to G. We focus on a particular class of functionals for which continuity fails. Since we are assuming lower semi-continuity for the functional F, continuity at some φ ∈ L 1 ( ) fails when The class of functionals we consider are those for which the above loss of upper semi-continuity holds for all φ ∈ L 1 ( ) and locally in a quantitative way. Namely, we consider the family of functionals (Definition 3.1) for which for all φ ∈ L 1 ( ) and for all r ≥ 1 it is possible to find a sequence {φ n } n∈N ⊂ L 1 ( ) such that φ n → φ in L 1 ( ), and This class of functions contains some interesting cases, such as the perimeter functional and the total variation functional (Sects. 5.1, 5.2, respectively). In the former case, the validity of (1.5) was proved in Caroccia et al. (2018, Theorem 2) by using a wriggling construction: given a set of finite perimeter F ⊂ , local oscillations of the boundary of φ = 1 F , whose intensity is determined by the factor r , were used in order to get (1.5). For this class of functionals (Theorem 3.8), the -limit of the family Here ψ cs is the convex sub-additive envelope of ψ (Definition 2.1) and t .
Note that cs < + ∞, since ψ cs has at most linear growth at infinity (Lemma 2.5). Moreover, ψ cs ≤ ψ c . Therefore, the quantitative loss of upper semi-continuity of the functional F results in having a lower energy density for the limiting functional. We report here the ideas behind the proof of Theorem 3.8. The main technical difficulty of the paper is the fact that the underlying measures F φ we consider come from the energy F. The assumptions we require do not seem to be too restrictive.
The liminf inequality (Proposition 4.1) follows easily from classical results on lower semi-continuity of functionals defined over pairs of measures originally proved in Buttazzo and Freddi (1991).
The construction of the recovery sequence is done by using several approximations. In particular, we pass from ψ to ψ cs in two steps: first from ψ to ψ c and then from ψ c to ψ cs . This is possible because (ψ c ) cs = ψ cs : the convex sub-additive envelope of the convex envelope corresponds to the convex sub-additive envelope of the function itself (Lemma 2.5).
Given φ ∈ L 1 ( ) and a nonnegative Radon measure μ on , we consider its decomposition with respect to F φ . We first treat the singular part and we show that it can be energetically approximated by a finite sum of Dirac deltas whose, in turn, can be approximated by regular functions (Proposition 4.2). The main technical difficulty here is in having to deal with general Radon measures F φ .
We then turn to the absolutely continuous part of the measure μ. After having showed that it is possible to assume the density u to be a piecewise constant function (Proposition 4.4), we prove that it suffices to approximate the energy with density ψ c (Proposition 4.5).
Finally, in Proposition 4.6, given a couple (φ, u) we construct a sequence of pairs The technical construction is based on a measure-theoretical result, Lemma 6.1. This result allows to disintegrate in sub-domains containing, asymptotically, a certain percentage of F φ ( ), and such that F φ does not charge mass on their boundaries. We also present applications of our general results to some interesting cases: the perimeter functional, a weighted total variation functional and the classical Dirichlet energy (Sects. 5.1-5.3, respectively). In the former case, the lack of lower semicontinuity was already provided in Caroccia et al. (2018, Theorem 2). Therefore, using the general theory we developed, we can answer the question raised by the application in Continuum Mechanics (Proposition 5.5).
In the second case, the family of approximating functionals we consider is the one introduced by Slepčev and García-Trillos in the context of point clouds , and that are of wide interest for the community (Bresson et al. 2013;Calatroni et al. 2017;Chambolle et al. 2010;Cristoferi and Thorpe 2018;Slepčev 2015, 2016;Szlam and Bresson 2010;Thorpe and Theil 2017;van Gennip and Schönlieb 2017;Caroccia et al. 2020). The main technical result in studying this case is a wriggling result for the weighted total variation functional (Proposition 5.8), that allows us to use Theorem 3.8 to identify the -limit in Proposition 5.14.
This paper is organized as follows: In Sect. 3, we state the main hypotheses and results of the paper. After introducing the main notation in Sect. 2, we devote Sect. 4 to the proofs of Theorem 3.8. Finally, the above-mentioned applications are treated in Sect. 5. and f cs (t) := sup {ϕ(t) | ϕ ≤ f , ϕ convex and sub-additive} , respectively. Moreover, we set The first result is the key one that allows us to construct the recovery sequence in two steps. Proof It is immediate that, if g ≤ f is convex and sub-additive (since it is in particular convex), we have g ≤ f c . Henceforth f cs ≤ f c , yielding On the other hand, if g ≤ f c is convex and sub-additive function, it holds in particular g ≤ f . Hence, from g ≤ f cs we get ( f c ) cs ≤ f cs yielding the desired equality.

Remark 2.3
Let us note that, in general, ( f c ) s = f cs . Here, with f s , we denote the sub-additive envelope of a function f . Indeed, in general ( f c ) s > f cs .
As an example, let us consider the function f (t) := max{2|t| − 1, 1}. Since f is convex, we have f c = f and thus ( f c ) s = f s . It is possible to check that f s is not convex. Therefore, it cannot coincide with the convex function f cs .
The following characterization of f c is well known [see, for instance, Braides for all t ∈ (0, + ∞).
A useful geometrical characterization of the convex sub-additive envelope of a convex function has been proved in Caroccia et al. (2018, Proposition A.9 and Lemma A.11).
Lemma 2.6 Let f : [0, + ∞) → (0, + ∞) be a Borel function. Then there exists t 0 ∈ (0, + ∞] such that -convergence We refer to Dal Maso (1993) for a comprehensive treatment ofconvergence. In this paper, we just need the sequential version of it for metric spaces.
In the proof of Theorem 3.8, we will make use of the following.
Remark 2.8 Let x ∈ Y . Assume that, for each δ > 0, there exists a sequence {x n } n∈N ⊂ Y such that x n → x and lim sup Then, by using a diagonal procedure, it is possible to find a sequence {y n } n∈N ⊂ Y with y n → x and lim sup Radon Measures We collect here the main properties of Radon measures we will need in the paper. For a reference, see Ambrosio et al. (2000, Section 1.4), and Maggi (2012, Section 2).

Definition 2.9
We denote by M + ( ) the space of finite nonnegative Radon measures on . We say that a sequence {μ n } n∈N ⊂ M + ( ) is weakly* converging to μ ∈ M + ( ), and we write μ n The following characterization of weak* convergence will be widely used in the paper.
In order to use the metric definition of -converge, we need a metric on the space M + ( ) that induces the weak* topology. This is possible because C 0 ( ) is separable. For a proof, see De Lellis (2008, Proposition 2.6).

Main Results
In this section, we state the main result of the paper, along with a corollary. In the following, ⊂ R d will always denote a bounded open set.
We say that F is an admissible energy if it satisfies the following conditions: We denote the class of admissible energies by Ad.
From now on, we will consider our functional F to be defined on X .

Remark 3.2 Note that if
Hypothesis (Ad3) is a non-degeneracy hypothesis needed to treat null sets for the measure F φ .

Remark 3.3
Note that if F satisfies Definition 3.1, it is indeed lower semi-continuous and it has the property that any element φ ∈ X can be approached in L 1 with a sequence {φ n } n∈N , which locally increases the energy of the prescribed amount f ≥ 1 (which acts as a Jacobian). Indeed, the convergence In particular, this also justifies in (Ad4) the name purely lower semi-continuous which encodes the fact that around any point φ ∈ X a liminf-type inequality for F is the sharpest bound that can be expected.
We now introduce the class of approximating energies.

Definition 3.4
We say that a sequence {F n } n∈N of functionals F n : such that F φ n n is non-atomic for all n ∈ N, and The class of good approximating sequences for F will be denoted by GA(F).

Remark 3.5
It is immediate from the definition that if {F n } n∈N is a good approximating sequence for F, then F n → F with respect to the L 1 topology. Hypothesis (GA3) is asking for the existence of a recovery sequence satisfying the additional requirement of recovering the energy also locally. From (GA1), we deduce the following closeness property: Remark 3.6 Notice that if F ∈ Ad is non-atomic, i.e., F φ is a non-atomic Radon measure for all φ ∈ X , then the constant sequence F n := F is a good approximating sequence for F.
We are now in the position to define the main objects of our study.
The main result of this paper concerns the behavior of sequences of F n -relative energies for a good approximating sequence {F n } n ∈ GA(F) of an admissible energy F. The interaction between the underlying functional F and the functional G F results in a lower limiting energy density, since ψ cs ≤ ψ c .
is defined as Remark 3.9 Note that cs < + ∞, since ψ cs is at most linear at infinity (Lemma 2.6).
In particular, combining the above theorem with Remark 3.6, allows to identify, for certain energies in Ad, the relaxation of the F-relative energy G F in the L 1 × w * topology.
Corollary 3.10 Let F ∈ Ad be non-atomic. Then, the relaxation of G F with respect to the L 1 × w * topology is G F lsc . Remark 3.11 From the properties of ψ c and ψ cs and using Remark 3.5, it is possible to deduce the following compactness property. Assume that a good approximating sequence {F n } n∈N enjoys this compactness property: it is possible to extract a subsequence (not relabeled) in such a way that φ n → φ in L 1 and μ n * μ, for some φ ∈ X and μ ∈ M + ( ).

Liminf Inequality
The proof of the liminf inequality for the -convergence result of Theorem 3.8 follows from the argument in Proposition (Ambrosio et al. 2000, Lemma 2.34). For the reader's convenience, we report it here.
Therefore, μ n = g n F φ n n for all n ∈ N. Since inf ψ > 0, using Remark 3.5 we have that φ ∈ X . Note that ψ ≥ ψ cs , and that, by Lemma 2.5 there exist two sequences By summing over j = 1, . . . , M, taking the limit as n → + ∞, exploiting the lower semi-continuity of F, and using the fact that b j > 0 together with μ n * μ, we get lim inf being N a set where μ ⊥ is concentrated and λ = F φ + μ ⊥ . Using Ambrosio et al. (2000, Lemma 2.35), we can infer Hence, using (4.2), we get This concludes the proof.

Limsup Inequality
The proof of the limsup inequality uses several approximations that we prove separately. We start by proving a density result that will allow us to construct the recovery sequence only for absolutely continuous couples, namely for pairs . 1).
Proof Let n ∈ N and consider a grid of open cubes {Q n j } j∈N of edge length 1/n. Without loss of generality, we can assume F φ (∂ Q n j ∩ ) = 0 for every j ∈ N. Write μ = gF φ + μ ⊥ , where g := dμ dF φ . We divide the proof in four steps.
Step 1 We claim that there exists a sequence {λ n } n∈N ⊂ M + ( ) of the form We identify the points of type "a" (in red) as those points x k j ∈ spt F φ and the points of type "b" (in green) as (c) We find suitably small balls j where we suitably modify the function φ.

(d) We accordingly define functions
)) for points of type "a" and as Note that Since η > 0 is arbitrary, from (4.7) and (4.8) we get This proves the claim.
Step 2 Let {λ n } n∈N ⊂ M + ( ) be the sequence provided by Step 1. Then . , x n M n ∈ spt μ ⊥ . Note that, for every n ∈ N, the cubes {Q n j } M n j=1 are pairwise disjoint. The idea is to locally deform the function φ around each point x n j and define a corresponding density h n j in a neighborhood of x n j . Let I n a := j = 1, . . . , M n | Q n j ∩ = ∅, x n j ∈ spt F φ , Fix n ∈ N. We show how to recursively define h n j and φ n j . Set φ n 0 := φ, and h n 0 := g. Assume φ n j−1 and h n j−1 are given and define φ n j , and h n j as follows. Case one j ∈ I n a . In this case, for any r > 0 we have F φ (B r (x n j )) > 0. Moreover using Ambrosio et al. (2000, Theorem 2.22 In particular, we can find r n 1 such that B r n (x n j ) ⊂ Q n j and for all j ∈ I n a . Then we define φ n j := φ n j−1 , and Case two x j ∈ I n b . Therefore, there exists r 0 > 0 such that for all r 0 > r > 0 we have Fix r n << 1 such that B 2r n (x n j ) ⊂ Q n j , and invoke property (Ad3) of Definition 3.1 with A = B 2r n (x n j ), U = B r n (x n j ), ε n := (M n n) −1 min j=1,...,M n {μ ⊥ (Q n j )} to find φ n j ∈ X such that φ n j = φ n j−1 on \B r n (x j ), F φ n j (∂ B r n (x j )) = 0 and Set ϕ n := φ n M n and h n := h n M n . Note that ϕ n = φ that h n = g outside M n j=1 B r n (x n j ), that h n ≥ nM n on M n j=1 B r n (x n ), (4.11) and that, by construction, ϕ n ∈ X . Moreover, F ϕ n (∂ B r n (x n j )) = 0 and Step 3 We claim that Indeed, recalling Remark 3.2, we get Fix δ 0 > 0. Using (4.11), it is possible to take n large enough so that The arbitrariness of δ 0 > 0, together with (4.6), yields Now, set and note that from (4.9) and (4.10) we get that (4.14) as n → + ∞. Since g ∈ L 1 ( ; F φ ), using (4.4), we get ζ(g) ∈ L 1 ( ; F φ ). Therefore, the Lebesgue-dominated convergence theorem yields (4.15) Using (4.12), (4.13) and (4.15), we get the claim.
Step 4 To complete the proof, it remains to show that h n F ϕ n * μ. Note that sup n∈N h n dF ϕ n < + ∞.
Let E ⊂⊂ be a bounded Borel set such that μ(∂ E) = 0. Fix η > 0 and take an and take n 0 ∈ N large enough so that j ∈ J n ∂ E implies Q n j ⊂⊂ U for all n > n 0 . Then Recalling (4.5), we get Therefore, using (4.14) and the arbitrariness of η > 0, we get The claim follows at once by (4.5), (4.16) and the triangle inequality.
The next result will allow us to consider only a special class of absolutely continuous couples. Let us introduce the following notation for partition of into cubes. Let Q := (− 1 2 , 1 2 ) d . For p ∈ R d and > 0, we define We denote by R( ) the class of all regular absolutely continuous couples.

Proposition 4.4 Let
Proof For n ∈ N let p n ∈ R d be such that be the cubes such that G p n ,1/n = {Q n j ∩ } M n j=1 and set and for j / ∈ J n F φ set α n j := 1 nM n .
Define g n := M n j=1 α n j 1 Q n j .
We claim that g n F φ * gF φ . Note that sup n∈N g n dF φ < + ∞.

Let E ⊂⊂ be a bounded Borel set with gF
and take n ∈ N large enough so that j ∈ J n ∂ E implies Q n j ⊂ U for all n ≥ n 0 . Then Since η > 0 is arbitrary, we get the claim. To conclude the proof, we have to show that Using the convexity of f , we have On the other hand, using g n F φ * gF φ , the convexity of f , and Ambrosio et al. (2000, Theorem 2.34), we get Using (4.18) and (4.19) we conclude.

Proposition 4.5 Let F ∈ Ad, and (φ, gF φ ) ∈ R( ). Then there exists
Moreover, if we write h n = M i=1 κ n i 1 n i , we can ensure that κ n i ≤ t 0 for all i = 1, . . . , M n , and n ∈ N, where t 0 > 0 is given by Lemma 2.6.
Proof If t 0 = + ∞, then there is nothing to prove, since this would mean that ψ cs = ψ c . Thus, assume t 0 ∈ (0, + ∞). Write g = M i=1 α i 1 i , where α i > 0, and let Define the function Let {φ n } n∈N ⊂ X be the sequence given by Definition 3.1 such that φ n → φ in X , and (4.21) In particular, for all i = 1, . . . , M, it holds F φ n (∂ i ) → 0. It is then possible, for all n sufficiently large, to choose δ n > 0 such that By definition of regular absolutely continuous couple, we have that { i } M i=1 = G p, for some p ∈ R d , > 0. Since F φ n is a Radon measure, it is possible to slightly translate the underline grid of cubes of a small vector so that G p+v, do not charge energy F φ n on ∂G p+v, ( ). More precisely, we can find a sequence {v n } n∈N ⊂ R d with |v n | < δ n , such that for all i = 1, . . . , M and n ∈ N. Define, for n ∈ N, We claim that h n F φ n * gF φ . Indeed, sup n∈N h n dF φ n < + ∞
Proposition 4.6 Let F ∈ Ad, and {F n } n∈N ∈ GA(F). Then, for every (φ, such that and where δ n → 0 as n → + ∞. Up to a subsequence, not relabeled, we can assume λ i n → λ i ∈ [0, 1], as n → + ∞. By invoking Lemma 6.1, for all i = 1, . . . , M, there exists a sequence of Borel sets {R i m } m∈N with R i m ⊂ i having the following properties: Using the fact that F φ n n * F φ and that F φ (∂ i ) = 0 for all i = 1, . . . , M, it is possible to select a subsequence {m n } n∈N with m n → + ∞ as n → + ∞ such that as n → + ∞ for all i = 1, . . . , M. Define We claim that Let E ⊂⊂ be a bounded Borel set with gF φ (∂ E) = 0. Then, using (4.26) and property c, we get Thanks to (4.28), (4.29) and the fact that λ i n → λ i , we conclude that Since sup n∈N h n dF φ n < + ∞, we obtain (4.31). Finally, we note that Thus, taking the limit as n → + ∞ and using (4.28), (4.29) together with the fact that λ i n → λ i , we get This concludes the proof.

Selected Applications
In this section, we assume the open set ⊂ R d to have Lipschitz boundary.

Perimeter Functional
As a first application of the general theory developed in the previous sections, we consider the perimeter functional. In the following, we will identify the space X := BV ( ; {0, 1}) with the space of sets with finite perimeter in . We define the functional F: Using the above result, it is possible to obtain the following.

Proposition 5.2 The functional F is a purely lower semi-continuous admissible energy.
Proof In order to show that the functional F is an admissible energy, we just need to prove property (Ad3), since the others are trivially satisfied. Let E ⊂ be a set of finite perimeter such that |D1 E |(A) = 0 for some open set A ⊂ . Then, 1 E is constant on A. Assume φ := 1 E = 0 on A. The other case can be treated similarly. Let U ⊂⊂ A be an open subset, and ε > 0. Pick B R (x) ⊂⊂ U and for r ∈ (0, R) set we get the desired result.
Finally, the fact that the energy F is purely lower semi-continuous follows by Theorem 5.1 localized in . Indeed, the wriggling procedure used in the proof of Theorem 5.1 is a local construction.

The Modica-Mortola Approximation of the Perimeter
We now consider, for ε > 0, the Modica-Mortola functional F ε : where W ∈ C 0 (R) is nonnegative potential with at least linear growth at infinity, and such that {W = 0} = {0, 1}. We report here the classical result by Modica (Modica 1987;Modica and Mortola 1977).
A careful analysis of the proof of the above results yields the following.
Proposition 5.4 Let {ε n } n∈N be such that ε n → 0, and set F n := F ε n for n ∈ N. Then, the sequence {F n } n∈N is a good approximating sequence for F.

Proof
We just have to prove property (GA3), being the others trivially satisfied. The statement of Theorem 5.3 holds for every open set U ⊂ with Lipschitz boundary, and such that |Dφ|(∂U ) = 0. Therefore, by using Lemma 6.2, we get (GA3).
Proposition 5.4 allows us to use the abstract results proved in the previous section. In particular, we obtain the following.

Remark 5.6
The functional G ε has been used in Rätz and Voigt (2006) as a phase field diffuse approximation for a model describing the evolution of interfaces in the epitaxial growth with adatoms. The authors worked with the special case ψ(t) := 1 + t 2 2 . In the same paper, it has been claimed that the solutions of the gradient flow of the phase field model converge to the solution of the sharp interface one. This claim was supported by formal matching asymptotics. It is worth noticing that the evolution equations for the sharp model do not account for the recession part and neither for the convex sub-additive envelope of ψ.
Proposition 5.5 answers the question posed in the introduction.

Total Variation Functional
In this section, we generalize the result of Sect. 5.1 by considering the total variation functional defined over the whole class of functions of bounded variation. Let ρ ∈ and consider the energy F: Proposition 5.7 F ∈ Ad.
We start by proving that the total variation is a purely lower semi-continuous functional.
Proposition 5.8 (A wriggling result for total variation) Let φ ∈ BV ( ) and f ∈ L 1 ( , |Dφ|) with f ≥ 1. Then there exists a sequence {φ k } k∈N ⊂ BV ( ) such that The main technical step needed to get the above proposition is given by the following result.
Lemma 5.9 Fix p ≥ 1 and let Q L ⊂ R d be a cube centered at the origin of edge length L > 0. Let φ ∈ C ∞ (Q L ) ∩ C 0 (Q L ) and let r > 1. Then, there exists a sequence of piecewise C 1 maps S n : Q L → Q L such that S n = Id on ∂ Q L , S n → Id uniformly on Q L ,
Proof We divide the proof in several steps.
Step one φ affine Write φ(y) = y · ν + φ 0 . First of all we note that it suffices to show that, given β > 1 and a cube Q ⊂ Q L with two of its faces orthogonal to ν, there exists a sequence of maps T n : Q → Q such that T n = Id on ∂ Q , T n → Id uniformly on Q and Indeed, by simply extending the map T n to the whole cube as the identity outside Q we get is surjective on [1, + ∞), given r > 1 it is possible to find β > 1 such that Thus, using (5.1) we conclude. Therefore, we can assume without loss of generality that ν = ρe d , L = 1, Q L = Q. For b > 0, let s b : R → R be the periodic extension of the function for t ∈ [− 1, 1]. For n ∈ N, define the function g n : [− 1, 1] → [0, + ∞) as g n (t) := 1 2n + 1 s b ((2n + 1)t).
(5.2) Moreover, let Q n ⊂ Q be a cube of side length 1 − 1 n , and fix a smooth cutoff function ϕ n : R → [0, 1] with ϕ n (t) = 1 for |t| ≤ 1 − 1 n and ϕ n (t) = 0 for |t| ≥ 1, and such that |∇ϕ n | ≤ Cn, (5.3) for some constant C > 0 independent of n. Define the function T n : Q → Q as Note that T n (Q) = Q, T n = Id on ∂ Q and that Hence, (5.5) On the other hand, using the definition of g n , of Q n , and (5.3), we get is surjective on [1, + ∞), given r > 1, it is possible to find b > 1 such that, using (5.5) and (5.6), we get The required convergence on T n → Id follows at once.
Step two φ ∈ C ∞ (Q L ). For n ∈ N consider the grid {Q n i } n d i=1 of cubes of size 1 n , with centers x n i partition Q. Fix δ > 0 and observe that there exists n 0 ∈ N such that for all n ≥ n 0 it holds Set ν n i := ∇φ(x n i ) and define ψ n i (y) := y · ν n i + φ(x n i ).
For every i = 1, . . . n d , thanks to the previous step, it is possible to find a map T n i : Q n i → Q n i with T n i = Id on ∂ Q n i such that Using (5.4) and (5.8), we get max y∈Q L ∇T n (y)∇φ(T n (y)) − ∇T n (y)ν n i ≤ where the constant C > 0 depends only on L and r . Therefore, from (5.8), (5.9), and (5.10), we get where in the last step we used the inequality |a p − b p | ≤ |a − b| (|a| p−1 + |b| p−1 ).
It is now useful to introduce the following notation. For a given Radon measure μ, we define the class of regular multipliers m( , μ) as the family of f ∈ L 1 ( , μ) of the form By standard arguments of measure theory, it is possible to prove the following density result.
Lemma 5.10 Let μ ∈ M + ( ) be a Radon measure and pick f ∈ L 1 ( , μ) with f ≥ 1 μ-a.e. on . Then there exists a sequence Therefore, we just need to provide the wriggling construction for functions f ∈ m( , |Dφ|). This will be done in the next result.
Proof We divide the proof in three steps.
Step one Fix an open set A ⊂ , ϕ ∈ C ∞ ( ) ∩ C 0 ( ) and r ≥ 1. The goal of this step is to construct a sequence {ϕ k } k∈N ⊂ W 1,1 ( ) such that ϕ k = ϕ on \A and For each k ∈ N, consider a grid of cubes {Q k j } j∈N with side length 1/k and define the finite set of indexes Using Lemma 5.9, for each Q k j there exists a smooth map S k j : → such that S k j = Id on \Q k j and Note that ϕ k ∈ W 1,1 ( ), ϕ k = ϕ on \A and by construction ϕ k → ϕ in L 1 . We now need to show that |Dϕ k | ¬ Let E ⊂⊂ be a bounded Borel set such that |Dϕ|(∂ E) = 0. Take an open set U ⊂ A with U ⊃ ∂ E such that |Dϕ|(U ) ≤ η and define Then, Take k large enough so that j ∈ J k ∂ E implies Q k j ⊂ U , and Then, from the definition of ϕ k and (5.11) we get (5.12) by taking the limit as k → + ∞ in (5.12) we get Since η > 0 is arbitrary, we conclude that Step two Let φ ∈ BV ( ), and f ∈ L 1 ( , |Dφ|) of the form |Dφ n | = |∇φ n |L n * |Dφ|.
We are now in position to prove that the total variation is purely lower semicontinuous.
We can now conclude the proof of Proposition 5.7 Proof of Proposition 5.7 Properties (Ad1) and (Ad2) follow easily by the definition of F.
In order to prove property (Ad3), we argue as follows: Let φ ∈ BV ( ), and A ⊂ an open set such that F(φ; A) = 0. Fix ε > 0 and an open set U ⊂⊂ A. Let x ∈ U and R > 0 such that B R (x) ⊂ U . Then φ ≡ c on B R (x). Let r ∈ (0, R) that will be chosen later and set φ := φ + 1 B r (x) . Then Taking r < ε max ρ 2 dω d Finally, we show that F is purely lower semi-continuous. Fix φ ∈ BV ( ), f ≥ 1, f ∈ L 1 ( , F φ ). Let {φ n } n∈N be the sequence given by Proposition 5.8 such that φ n → φ in L 1 ( ) and |Dφ n | * f |Dφ|. (5.13) We claim that for all j = 1, . . . , M k . Note that, for any open set A ⊂⊂ , the following estimates hold Take a bounded Borel set E ⊂⊂ such that F φ (∂ E) = 0. Consider bounded open sets E 1 ⊂⊂ E ⊂⊂ E 2 . Then, using (5.14), we get By taking the limit as n → + ∞ on both sides, and using (5.13) and (5.14), we have Similar computation shows also Being the above valid for all k ∈ N yields Moreover, since E 1 and E 2 are arbitrary and F φ (∂ A) = 0, we conclude that we conclude that F φ n * f F φ .

A Non-Local Approximation
The non-local approximation of the weighted total variation we consider in this section is the one used by Trillos and Slepčev (2016) in the context of total variation on graphs. Let η: [0, + ∞) → [0, + ∞) be a compactly supported smooth function with where η ε (t) := η(tε −1 )ε −d . Fix a sequence {ε n } n∈N such that ε n → 0 as n → + ∞, and set F n := F ε n . In Trillos and Slepčev (2016), the authors proved the following -convergence result.
It is possible to improve the above -convergence result and obtain the following.

Proposition 5.13
The sequence {F n } n∈N is a good approximating sequence for F.
Proof Property (GA1) follows from the liminf inequality of Theorem 5.12, while property (GA2) is immediate from the definition of F n . In order to prove property (GA3), we follow the same steps used in the proof of Theorem 5.12 (Trillos and Slepčev 2016, Section 4.2), that we briefly report here for the reader's convenience. Given φ ∈ BV ( ), we extended it to a BV function defined in the whole R d in such a way that the extension, still denoted by φ, is such that |Dφ|(∂ ) = 0. For any δ > 0, having set be a sequence such that φ n → φ in L 1 ( δ ), and ρ 2 |Dφ n | * ρ 2 |Dφ| on M + ( δ ).
Let E ⊂⊂ be a Borel set with |Dφ|(∂ E) = 0. Assume [0, M] is the support of η. Then it holds that where we used the change of variable h := x−y ε n and z = x + t(y − x). In the case ρ is Lipschitz, it is possible to prove that R n,k ≤ Cε n E εn |∇φ n |ρ 2 dz, for some constant C > 0 independent of k. Therefore, lim sup The result of Proposition 5.13 allows us to apply Theorem 5.7 to get the following.

Relaxation of the p-Dirichlet Energy
In this last section, we would like to note that the choice of the L 1 convergence is not fundamental for the validity of the main results of this paper. Indeed, Theorem 3.8 holds also if the L 1 converge is replaced by the L p convergence. Of course, in Definitions 3.1 and 3.4 one also has to replace the L 1 convergence with the L p one.
As an example of application of our results in the L p case, we consider, for p > 1, the energy F: L p ( ) × A( ) → [0, + ∞) given by F(φ; A) := A |∇φ| p dx.
Proof It is easy to see that assumptions (Ad1),(Ad2) and (Ad3) are satisfied. Thus, F ∈ Ad. The fact that F is purely lower semi-continuous is obtained by using a slight variation of the proof of Proposition 5.8 by applying Lemma 5.9 for p > 1, Lemma 5.10 on μ = |∇φ| p L n and Lemma 5.11 suitably adapted.
Noting that the constant sequence F n := F is a good approximating sequence for F and using Corollary 3.10 we obtain the following Proposition.
Proof Fix j ∈ N. It is possible to find x j ∈ R d such that G x j ,1/ j is such that μ(∂G x j ,1/ j ) = 0. Let {Q j i } M j j=1 denote the elements of the grid. For every i = 1, . . . , M j , since μ is not atomic, it is possible to use (Fonseca and Leoni 2007, Proposition 1.20) in order to find a μ-measurable set S For t > 0, let S j i := { f n ≥ t}. Using the fact that each μ n is a Radon measure, and thus for all but countably many t > 0 it holds μ n (∂ S j i ) = 0, it is possible to find t > 0 such that μ n (∂ S j i ) = 0 for all n ∈ N, and where we used (6.1). Therefore, we conclude by sending j → + ∞, and using the arbitrariness of η.
The second result is a test for the weak* convergence of measures. Proof Let E ⊂ be a Borel set with μ(∂ E) = 0. We want to show that μ n (E) → μ(E). Fix ε > 0. Using the outer regularity of μ, it is possible to find an open set V ⊃ E such that μ(V ) < μ(E) + ε 2 , and an open set W ⊃ ∂ E with μ(W ) < ε 2 . Since ∂ E is compact, there exists r > 0 such that B r (x) ⊂ W for all x ∈ ∂ E. Let U := V ∪ W . Using mollification, it is possible to find an open set U 2 with Lipschitz boundary, such that E ⊂ U 2 ⊂ U . Then μ(U 2 ) < μ(E) + ε. (6.2) Using the inner regularity of μ, it is possible to find a compact set K ⊂ E\∂ E such that μ(K ) > μ(E) − ε. Note that here we used the fact that μ(∂ E) = 0. In the case E\∂ E = ∅, we can just take K = ∅. Since K ⊂ E\∂ E, and is bounded, we have that dist(K , ∂ E) > 0. Therefore, using mollifications, it is then possible to find an open set U 1 with Lipschitz boundary, such that K ⊂ U 1 ⊂ E\∂ E. Then μ(E) − ε < μ(U 1 ). (6.3) Note that μ n (U 1 ) ≤ μ n (E) ≤ μ n (U 2 ). (6.4) Since by assumption we have that μ n (U 1 ) → μ(U 1 ), μ n (U 2 ) → μ(U 2 ), from (6.2) to (6.4), we get Using the arbitrariness of ε > 0, we conclude that μ n (E) → μ(E), getting the desired result.