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Approximation and relaxation of perimeter in the Wiener space

We characterize the relaxation of the perimeter in an infinite dimensional Wiener space, with respect to the weak L^2-topology. We also show that the rescaled Allen-Cahn functionals approximate this relaxed functional in the sense of Gamma-convergence.


Introduction
Extending the variational methods and the geometric measure theory from the Euclidean to the Wiener space has recently attracted a lot of attention. In particular, the theory of functions of bounded variation in infinite dimensional spaces started with the works by Fukushima and Hino [21,22]. Since then, the fine properties of BV functions and sets of finite perimeter have been investigated in [4,5,3,1]. We point out that this theory is closely related to older works by M. Ledoux and P. Malliavin [26,27]. In the Euclidean setting it is well-known that the perimeter can be approximated by means of more regular functionals of the form when ε tends to zero, in the sense of Γ-convergence with respect to the strong L 1 -topology [29,28]. An important ingradient in this proof is the compact embedding of BV in L 1 . A natural question is whether a similar approximation property holds in the infinite dimensional case. The main goal of this paper is answering to this question by computing the Γ-limit, as ε → 0, of the Allen-Cahn-type functionals (see Section 2 for precise definitions) In the Wiener space there are two possible definitions of gradient, and consequently two different notions of Sobolev spaces, functions of bounded variation and perimeters [4,1]. In one definition the compact embedding of BV γ (X) in L 1 γ (X) still holds [4,Th. 5.3] and the Γ-limit of F ε is, as expected, the perimeter up to a multiplicative constant. We do not reproduce here the proof of this fact, since it is very similar to the Euclidean one. A more interesting situation arises when we consider the other definition of gradient, which gives rise to a more invariant notion of perimeter and is therefore commonly used in the literature [21,22,4]. In this case, the compact embedding of BV γ (X) in L 1 γ (X) does not hold anymore. In particular sequences with uniformly bounded F ε -energy are not generally compact in the (strong) L 1 γ -topology, even though they are bounded in L 2 γ (X), and hence compact with respect to the weak L 2 γ (X)-topology. This suggests that the right topology for considering the Γ-convergence should rather be the weak L 2 γ (X)-topology. A major difference with the finite dimensional case is the fact that the perimeter function defined by is no longer lower semicontinuous in this topology, and therefore cannot be the Γ-limit of the functionals F ε . The problem is that the sets of finite perimeter are not closed under weak convergence of the characteristic functions. However, it is possible to compute the relaxation F of F (Theorem 4.6), which reads: Such functional is quite familiar to people studying log-Sobolev and isoperimetric inequalities in Wiener spaces [6,7,10]. Our main result is to show that the Γ-limit of F ε , with respect to the weak L 2 γ (X)topology, is a multiple of F (Theorem 5.3). The proof relies on the interplay between symmetrization, semicontinuity and isoperimetry. The plan of the paper is the following. In Section 2 we recall some basic facts about Wiener spaces and functions of bounded variation. In Section 3 we give the main properties of the Ehrhard symmetrizations. We also prove a Pólya-Szegö inequality and a Bernsteintype result in the Wiener space (Propositions 3.12 and 3.5), which we believe to be interesting in themselves. In Section 4, we use the Ehrhard symmetrization to compute the relaxation of the perimeter (Theorem 4.6). Finally, in Section 5 we compute the Γ-limit of the functionals F ε (Theorem 5.3) and discuss some consequences of this result.

Wiener space and functions of bounded variation
A clear and comprehensive reference on the Wiener space is the book by Bogachev [8] (see also [27]). We follow here closely the notation of [4]. Let X be a separable Banach space and let X * be its dual. We say that X is a Wiener space if it is endowed with a non-degenerate centered Gaussian probability measure γ. That amounts to say that γ is a probability measure for which x * ♯γ is a centered Gaussian measure on R for every x * ∈ X * . The non-degeneracy hypothesis means that γ is not concentrated on any proper subspace of X. As a consequence of Fernique's Theorem [8,Th. 2.8.5], for every x * ∈ X * , the function . Let H be the closure of R * X * in L 2 γ (X); the space H is usually called the reproducing kernel of γ. Let R, the operator from H to X, be the adjoint of R * that is, forĥ ∈ H, where the integral is to be intended in the Bochner sense. It can be seen that R is a compact and injective operator. We will let Q = RR * . We denote by H the space RH. This space is called the Cameron-Martin space. It is a separable Hilbert space with the scalar product given by if h i = Rĥ i . We will denote by | · | H the norm in H. The space H is a dense subspace of X, with compact embedding, and γ(H) = 0 if X is of infinite dimension. For x * 1 , .., x * m ∈ X * we denote by Π x * 1 ,..,x * m the projection from X to R m given by We will also denote it by Π m when specifying the points x * i is unnecessary. Two elements x * 1 and x * 2 of X * will be called orthonormal if the corresponding h i = Qx * i are orthonormal in H. We will fix in the following an orthonormal base of H given by h i = Qx * i . We also denote by H m = span(h 1 , .., h m ) ≃ R m and X ⊥ m = Ker(Π m ) = H ⊥ m X , so that Gaussian measures on R m , X ⊥ m respectively.
is a Gaussian in R m . If theĥ i are orthonormal, then such measure is the standard Gaussian measure on R m .
Given u ∈ L 2 γ (X), we will consider the canonical cylindrical approximation E m given by Notice that E m u is a cylindrical functions depending only on the first m variables, and E m u converges to u in L 2 γ (X). We will denote by FC 1 b (X) the space of cylindrical C 1 bounded functions that is the functions of the form v(Π m (x)) with v a C 1 bounded function from R m to R. We denote by FC 1 b (X, H) the space generated by all functions of the form Φh, with Φ ∈ FC 1 b (X) and h ∈ H. We now give the definitions of gradients, Sobolev spaces functions of bounded variation. Given u : X → R and h = Rĥ ∈ H, we define whenever the limit exists, and We define ∇ H u : X → H, the gradient of u by The operator div γ is the adjoint of the gradient so that for every u ∈ FC 1 b (X) and every Φ ∈ FC 1 b (X, H), the following integration by parts holds: The ∇ H operator is thus closable in L 2 γ (X) and we will denote by H 1 γ (X) its closure in L 2 γ (X). From this, formula (1) still holds for u ∈ H 1 γ (X) and Φ ∈ FC 1 b (X, H). Following [21,4], given u ∈ L 1 γ (X) we say that u ∈ BV γ (X) if We will also denote by |D γ u|(X) the total variation of u. If u = χ E is the characteristic function of a set E we will denote P γ (E) its total variation and say that E is of finite perimeter if P γ (E) is finite. As shown in [4] we have the following properties of BV γ (X) functions.
Theorem 2.2. Let u ∈ BV γ (X) then the following properties hold: • D γ u is a countably additive measure on X with finite total variation and values in H (we will note the space of these measures by M(X, H)), such that for every Φ ∈ FC 1 b (X) we have: Proposition 2.4 (Coarea formula [2]). If u ∈ BV γ (X) then for every borel set B ⊂ X, In Proposition 3.12, we will need the following extension of Proposition 2.4.
Lemma 2.5. For every function u ∈ BV γ (X) and every non-negative Borel function g, Proof.
where the A i ⊂ X are Borel sets. Using the coarea formula (2), we then get In [4] it is also shown that sets with finite Gaussian perimeter can be approximated by smooth cylindrical sets.
Proposition 2.6. Let E ⊂ X be a set of finite Gaussian perimeter then there exists Note that, for half-spaces, the perimeter can be exactly computed [4,Cor. 3.11].
Proposition 2.7. Let h = Rĥ ∈ H and c ∈ R then the half-space

The Ehrhard symmetrization
The Ehrhard symmetrization has been introduced by Ehrhard in [18] for studying the isoperimetric inequality in a Gaussian setting. We recall the definition and the main properties of such symmetrization.
Definition 3.1. We define the functions Φ and α by Notice that Φ(t) is the volume of the half-space {ĥ(x) < t} and that U (x) is the perimeter of a half-space of volume x.
Proof. Assume by contradictionĥ 1 =ĥ 2 , and let η > 0 be such that |ĥ 1 −ĥ 2 | L 2 γ (X) ≥ η. We shall bound from below by a positive constant the quantity thus contradicting the inclusion Letting h be a unitary vector in H orthogonal to h 1 , we can write Asĥ 1 andĥ are orthogonal we have Πĥ 1 ,ĥ ♯γ = γ 2 and thus Let us start by computing the Fourier transform of Πĥ 1 ,ĥ 2 ♯γ. Denoting byμ the Fourier transform of a measure µ (see [8,Sec. 1.2]) and letting Π := Πĥ 1 ,ĥ 2 , for every (z 1 , z 2 ) ∈ R 2 we have Thus, if we set K : It follows that Π♯γ is a centered Gaussian measure with density We now compute Finally, when λz 1 z 2 ≥ 0, we can bound e λz 1 z 2 from below by 1, and when λz 1 z 2 ≤ 0 we can bound it form below by e − 1 2 |z 1 z 2 | so that we can always bound from below by a positive constant.
We now define the Ehrhard symmetrization.
Definition 3.3. Let E ⊂ X and let m ∈ N. The Ehrhard symmetral of E along the first m variables is defined as (see Figure 1): x Figure 1: The Ehrhard symmetrization.
The interest of this symmetrization is that it decreases the Gaussian perimeter, while keeping the volume fixed.
Proposition 3.4. Let E be a set of finite perimeter and E * be an Ehrhard symmetral of In particular, we have the isoperimetric inequality with equality if and only if E is a half-space.
For the proof we refer to [7,10], and to [4] for the extension to infinite dimensions. We can also prove a stronger result which is a kind of Bernstein Theorem in this setting.
Proposition 3.5. The half-spaces are the only local minimizers of the Gaussian perimeter with volume constraint.
Proof. Let E ⊂ X be a local minimizer of the (Gaussian) perimeter and let v = γ (E). This means that, for every R > 0 and every set F of finite perimeter, with γ (F ) = v and E∆F ⊂ B R (where B R denotes the ball of radius R centered at 0), we have If E is not an half space then, by Proposition 3.4, there exists η > 0 such that Let α R be such that We have that α R tends to α(v) when R goes to infinity and we get where we used various time the inequality (see [23]) and where ε(R) is a function which goes to zero when R goes to infinity. We thus found a contradiction.
Remark 3.6. In the Euclidean setting, half-spaces are the only local minimizers of the perimeter only in dimension lower than 8 (see [23]). Notice also that if we drop the volume constraint, half spaces are no longer local minimizers for the Gaussian perimeter, since there are no nonempty local minimizers.
In the sequel we will also need another transformation which from a finite dimensional function gives an Ehrhard symmetric set whose sections have volume prescribed by the original function. More precisely: Given a measurable cylindrical function u : The link between Ehrhard sets and Ehrhard symmetrization is the following: Proposition 3.8. Let E be a set of finite perimeter and E * be its Ehrhard symmetrization with respect to the first (m + 1) variables, then In the next proposition we compute the perimeter of Ehrhard sets. It slightly extends a result in [15].
and D γ u = ∇u γ + D s γ u is the Radon-Nikodym decomposition of D γ u. Proof. By [15,Th. 4.3] the result holds for u ∈ H 1 γm (R m ). We will show by approximation that the same holds for u ∈ BV γm (R m ). Let E = ES m (u), then we can find sets E n such that γ(E n ∆E) → 0 and P γ (E n ) → P γ (E) as n → +∞, and all the E n have smooth boundary and are Ehrhard symmetric. Thus, for every n ∈ N, there exists a smooth function u n such that 0 ≤ u n ≤ 1, E n = ES m (u n ), u n → u in L 1 γm (R m ), and Since, by Proposition 4.4, the functional R m U (u) 2 + |D γm u| 2 dγ m is lower semicontinuous in L 1 γm (R m ), we get The other inequality follows as in [15]. Let E = Π m+1 (E) ⊂ R m+1 and observe that γ m+1 ( E) = γ(E) and P γ m+1 ( E) = P γ (E). By Vol'pert Theorem [2, Th. 3.108] there exists a set B ⊂ R m such that for every x ∈ B, ν E m+1 (x, α(u E (x))) exists and is not equal to zero, where ν E m+1 denotes the last coordinate of the unit external normal to ∂ * E. By [15,Lemma 4.4], γ m -almost every x ∈ B is a point of approximate differentiability for u. By Lemma 4.5 and 4.6 of [15] we then have As γ m (B c ) = 0, we find that The last transformation that we consider is the analog of the Schwarz symmetrization in the Gaussian setting, and was first introduced by Ehrhard in [19].
Definition 3.10. Let u ∈ X → R be a measurable function and let m ∈ N be fixed. We define the m-dimensional Ehrhard symmetrization u * of u as follows: • for all t ∈ R we let E * t be the Ehrhard symmetrization of E t := {u > t} with respect to the first m variables; • we let u * (x) := inf{t : x ∈ E * t }. As (4) implies γ({u * > t}) = γ({u > t}) for all t ∈ R, from the Layer Cake formula it follows that, if u ∈ L 2 γ (X), then u * ∈ L 2 γ (X) and Indeed, we have Proof. The proof is a straightforward adaptation of the analogous proof for the Schwarz symmetrization [25,Th. 3.4].
Recalling (6) with p = 2, we have only to show that Again by the Layer Cake formula we have Thus (8) would follow from the same inequality for sets, that is, Let x m ∈ R m and assume that then by definition of the Ehrhard symmetrization we have and therefore which gives (9).
As for the Schwarz symmetrization, a Pólya-Szegö principle holds for the Ehrhard symmetrization.
Proposition 3.12. Let u ∈ H 1 γ (X), let m ∈ N and let u * be the m-dimensional Ehrhard symmetrization of u. Then u * ∈ H 1 γ and Moreover, if m = 1 and equality holds in (10) (10) is proven for Lipschitz functions, in finite dimensions. We extend it by approximation to Sobolev functions. We can assume u ≥ 0, since we have (u ± ) * = (u * ) ± , where u ± , (u * ) ± denote the positive and negative part of u and u * , respectively. Let u n ∈ FC 1 c (X) be positive functions converging to u in H 1 γ (X), then by (7), u * n converges to u * in L 2 γ (X) and thus by the lower semicontinuity of the H 1 γ (X) norm we have We now turn to the equality case for one-dimensional symmetrizations. For this we closely follow [10] and give an alternative proof of (10), based on ideas of Brothers and Ziemer [9] for the Schwarz symmetrization. Let u ∈ H 1 γ (X) and µ(t) = γ({u > t}) = γ({u * > t}). By the coarea formula (3), for all t ∈ R we have Since u * is a function depending only on one variable, arguing as in [14] we get As u * is monotone we have that |∇ H u * | H is constant on {u * = t} ∩ {∇ H u * = 0}. Observe also that, being u * one-dimensional, {u * = t} has a well defined meaning. We thus find: which implies, recalling (11), Let us note that as in [10, Lem. 4.2], using (3) with g = χ {∇ H u=0} we find and thus for almost every t ∈ R, This shows that for almost every t ∈ R, ∇ H u(x) = 0 for |D γ χ Et |-almost every x ∈ X and thus By (3), (5), (12) and (13), we eventually get As a consequence, if equality holds in (10), then equality holds in the Gaussian isoperimetric inequality, that is, This implies that almost every level-set of u is a half-space, i.e. for almost every t ∈ R there existsĥ t ∈ H such that {u > t} = {ĥ t < α(µ(t))}, and without loss of generality we can assume that |h t | H = 1. Such half-spaces being nested, by Lemma 3.2 we have that h t does not depend on t and thus u(x) = v(ĥ(x)).
Remark 3.13. We notice that the fact that equality in (10) implies that u is onedimensional is a specific feature of the Gaussian setting, and the analogous statement does not hold for the Schwarz symmetrization in the Euclidean case [9]. Indeed, this property is a consequence of the fact that Gaussian measures, differently from the Lebesgue measure, are not invariant under translations.

Relaxation of perimeter
In this section we compute the relaxation of the perimeter functional with respect to the weak L 2 γ (X)-topology. The fact that F is not lower semicontinuous can be easily checked by taking the sequence E n = { x * n , x < 0}. Indeed, the characteristic functions of these sets weakly converge to the constant function 1/2, which is not a characteristic function, while the perimeter of E n is constantly equal to 1/ √ 2π. We will show that the relaxation of F is equal to Observe that the functional F already appears in the seminal work of Bakry and Ledoux [6] and in the earlier work of Bobkov [7] in the context of log-Sobolev inequalities. This functional has been also studied in [10]. See also [4,Remark 4.3] where it appears in a setting closer to ours.
Let us first recall the definition of the lower semicontinous envelope of a function (see [16] for more details).
Definition 4.1. Let X be a topological vector space. For every function F : X → R, its lower semicontinuous envelope (or relaxed function) is the greatest lower semicontinuous function that lies below F . When X is a metric space, the following caracterization holds.
Proposition 4.2. Let X be a metric space. For every function F : X → R, and every x ∈ X, the relaxed function F is given by We now show a representation formula for F which is reminiscent of the definition of the total variation and of the nonparametric area functional (see [23]). We start with a preliminary result.
Proof. The proof is adapted from [17].
We will show that By the observation at the beginning of the proof and the triangle inequality we get

From this, setting Φ = m
Taking Φ equal to Φ 2 on a sufficiently small neighborhood of the support of µ s and equal to Φ 1 outside this neighborhood, we get which gives the opposite inequality.
Step 3. In order to conclude the proof, it is enough to notice that for every Φ ∈ L 1 µ (X, H), with |Φ| H ≤ 1, we have Proof. We apply Lemma 4.3 with µ = Du and g = U (u). Since µ is tight [4], the space FC 1 b (X, H) is dense in L 1 µ (X, H) so that we can restrict the supremum in (16) to smooth cylindrical functions Φ, ξ.
Remark 4.5. Since U is concave, the duality formula (16) is not sufficient to prove that F is lower semicontinuous for the weak L 2 γ (X)-topology. It shows however the lowersemicontinuity of F in the strong L 2 γ (X)-topology.
We now prove that F is the lower semicontinuous envelope of F . Theorem 4.6. F is the relaxation of F in the weak L 2 γ (X)-topology. Proof. Let us first notice that F takes finite values only on functions of the closed unit ball of L 2 γ (X) which is metrizable for the weak convergence. Therefore the relaxation and the sequential relaxation in the weak topology of L 2 γ (X) coincide. Let χ En be a sequence of sets weakly converging in L 2 γ (X) to u ∈ BV γ (X), with uniformly bounded perimeter. We shall show that Notice that, by weak convergence, we necessarily have 0 ≤ u ≤ 1 a.e. on X. For all n ≥ 1 and k ≥ 2, we let E k n be the Ehrhard symmetral of E n with respect to the first k variables. Recalling the notation of Section 3, we have As X |D γ E k (χ En )| H ≤ P γ (E n ) and E k (χ En ) depends only on the first k variables, by the compact embedding of BV γ k (R k ) into L 1 γ k (R k ) we can extract a subsequence from E k (χ En ) which converges strongly to u k := E k (u). From this we get that E k+1 n = ES k (E k χ En ) tends strongly to E k+1 := ES k (u k ). By the lower semicontinuity of the perimeter we then have lim For every ϕ ∈ FC 1 b (X), with ϕ depending only of the j ≤ k first variables, there holds which implies that the sequence χ E k+1 tends weakly to u. In order to conclude the proof it remains to show that lim Notice that, by Proposition 3.9, there holds For every Φ ∈ FC 1 b (X, H) and ξ ∈ FC 1 b (X), depending on the first k variables and such that the range of Φ is included in H k , by Proposition 4.4, we have Taking the supremum in Φ, ξ and recalling (16), we then get for all k.
Repeating the same argument with u k+1 instead of u, we obtain that F (u k ) is nondecreasing in k. Therefore there exists ℓ ≥ 0 such that Assume by contradiction that ℓ < F (u). Then there exists δ > 0 such that F (u k ) ≤ F (u) − δ for all k, hence there exist N ∈ N, Φ ∈ FC 1 b (X, H) and ξ ∈ FC 1 b (X), depending only on the first N variables, such that which leads to a contradiction.
Remark 4.7. Theorem 4.6 provides an example of a nonconvex functional, namely F , which is lower semicontinuous for the weak L 2 γ (X)-topology. We also know that semicontinuity does not holds for general functional of the form since if we take for instance f (u, p) := g 2 (u) + |p| 2 with g such that g(1/2) > U (1/2) and g(0) = g(1) = 0, then, letting u n := { x * n , x < 0}, we have u n ⇀ u = 1/2 weakly in L 2 γ (X), so that One could wonder what are the right hypotheses for a functional of this form to be lower semicontinuous with respect to the weak topology.

Γ-limit for the Modica-Mortola functional
Let us briefly recall the definition of Γ-convergence. We refer to [16] for a comprehensive treatment of the subject.
Definition 5.1. Let X be a topological space, and let F n : X → R be a sequence of functions. The Γ-lower limit and the Γ-upper limit of the sequence F n is defined as where N (x) denotes the set of all open neighbourhoods of x in X. When the Γ-lower limit and the Γ-upper limit coincide, we say that the sequence F n Γ-converges.
As for the relaxation, if X is a metric space we have a sequential caracterization of the Γ-convergence.
Theorem 5.2. Let X be a metric space. A sequence of functions F n Γ-converges to F : X → R if and only if the following two conditions hold: • for every sequence x n converging to x, it holds lim • for every x ∈ X there exists a sequence x n converging to x with lim Let now W ∈ C 1 (R) be a double-well potential with minima in {0, 1}, that is, W (t) ≥ 0 for all t ∈ R, and W (t) = 0 iff t ∈ {0, 1}. We also assume W (t) ≥ C(t 2 − 1) for some C > 0 and t ∈ R. A typical example of such potential is W (t) = t 2 (t − 1) 2 .
For any ε > 0 we define the functionals F ε : We are ready to prove our main Γ-convergence result. Proof. Notice first that the Γ-limit does not change if we restrict the domain of F ε to the functions u ∈ H 1 γ (X) such that 0 ≤ u ≤ 1. This follows from the following two facts: -for all u ∈ H 1 γ (X), lettingũ = min(max(u, 0), 1), we have F ε (ũ) ≤ F ε (u); ε dγ for all u ∈ H 1 γ (X), which implies that the Γ-limit is concentrated on the functions u ∈ L 2 γ (X) such that u(x) ∈ {0, 1} for a.e. x ∈ X.
Since the restricted domain is contained in the unit ball of L 2 γ (X), which is metrizable for the weak L 2 γ (X)-topology, by Theorem 5.2 the Γ-limit and the sequential Γ-limit of F ε coincide.
The Γ-limsup is done similarly to the (Euclidean) finite dimensional case [29,28]. Since F is the relaxation of F in the weak L 2 γ (X)-topology and since we can approximate sets of finite perimeter by smooth cylindrical sets by Proposition 2.6, for every u ∈ BV γ (X) with 0 ≤ u ≤ 1 there exists a sequence E n of smooth cylindrical sets with χ En converging weakly to u and such that P γ (E n ) tends to F (u). This shows that we can restrict ourselves to smooth cylindrical sets for computing the Γ-limsup of F ε . Let m ∈ N and E = Π −1 m (E m ), where E m ⊂ R m is a smooth set with finite Gaussian perimeter, and let which gives the desired inequality letting δ → 0 and m → +∞.
Remark 5.4. As in the Euclidean case, a similar result can be proven for the volume constrained problems. In this case, the proof of the Γ-liminf is exactly the same as in Theorem 5.3, and the Γ-limsup is also very similar. The only difference comes from the fact that we have to adapt the recovery sequence to have the right volume, and this can be done as in [28] by slightly translating η δ .
We now state some simple implications of the Γ-convergence result.
Proposition 5.5. Let m ∈ [0, 1] and u ε be a minimizer of then u ε = v ε (ĥ ε (x)) for someĥ ε ∈ H with |h ε | H = 1 and some v ε minimizer of the one-dimensional problem in particular, v ε (strongly) converges to the characteristic function of a half-line.
Proof. For every u ∈ H 1 γ (X), by Proposition 3.12, we have X u * dγ = X udγ and F ε (u * ) ≤ F ε (u), with equality only if u is of the form u(x) = v(ĥ(x)) for someĥ ∈ H with |h| H = 1. Using thatĥ is the limit in L 2 γ (X) of linear functions of the form R * x * i , it is readily seen that ∇ Hĥ = h, and thus we get Therefore problem (17) reduces to the one-dimensional problem (18). Using the compact embedding of H 1 γ 1 (R) in L 2 γ 1 (R) (see [4,Th. 4.10]) and the direct method of the calculus of variations, we get that (18) has a minimizer. Moreover, by the Γ-convergence of the one-dimensional functionals in the strong L 2 γ 1 (R)-topology towards the a multiple of the perimeter (which can be obtained exactly as in the classical Modica-Mortola Theorem since compact embedding of BV γ 1 (R) in L 1 γ 1 (R) holds), we find that every sequence of minimizers v ε of (18) has a subsequence strongly converging towards the characteristic of the half-line of measure m.
We finally give another convergence result for the prescribed curvature problem in case of uniqueness of minimizers.
Proposition 5.6. Let g ∈ L 2 γ (X), then the following assertions are equivalent: • the functional has a unique minimizer in the class of sets of finite perimeter; • the functional has a unique minimizer in BV γ (X).
Moreover, when this holds the two minimizers coincides. Finally, if u ε is a sequence in H 1 γ (X) satisfying sup ε F ε (u ε ) + X u ε gdγ ≤ C for some C > 0, then u ε has a subsequence strongly converging to χ E in L 2 γ (X), where E is the common minimizer of (19) and (20).
Proof. We first notice that the problem (19) always has a solution. Indeed, arguing as in [12], if E n is a minimizing sequence for (19), it has a subsequence weakly converging to some u ∈ BV γ (X). By the lower semicontinuity of the total variation and the coarea formula we then have and thus the sets {u > t} minimize F g for almost every t. As F is the relaxation of the perimeter we have that the minimum values in (19) and (20) are the same and thus any minimizer of F g is also a minimizer of F g . This shows that if uniqueness does not hold in (19) then it does not hold in (20), too. Now, if u is a minimizer of F g , applying the coarea formula once again we get As above, this implies that {u > t} solves (19) for almost every t. Therefore, if the minimizer of F g is not a characteristic function, then uniqueness does not hold neither in (19) nor in (20). This proves the first part of the Proposition. The second statement easily follows from Theorem 5.3. Indeed, as the functionals F ε (u)+ X ugdγ Γ-converge to F g in the weak L 2 γ (X)-topology, for every sequence u ε bounded in energy, there exists a subsequence weakly converging to χ E (where E is the unique minimizer of (19) and (20)). However, by the lower semicontinuity of the norm, Thus u ε L 2 γ (X) converges to χ E L 2 γ (X) , which implies the strong convergence of u ε . Remark 5.7. In [24], we provide an example of functionals for which uniqueness of minimizers holds, namely where g : X → R is convex and λ ∈ (0, +∞) is large enough.