Journal of Convex Analysis Volume 4 (1997), No. 1, 91–108

Variational Approximation of Functionals with Curvatures and Related Properties

G. Bellettini

Dipartimento di Matematica Applicata “U. Dini”, Università di Pisa, via Bonanno 25 bis, 56100 Pisa, Italy. e-mail: bellettini@sns.it

Received September 8, 1995

We consider the problem of approximating via Γ-convergence a class of functionals depending on curvatures of smooth compact boundaries. We investigate the connections between the approximation problem and the lower semicontinuous envelope of the original functional. We provide some examples of lower semicontinuous functionals and their variational approximation.

1991 Mathematics Subject Classification: 49J45.

1. Introduction

Functionals depending on curvatures arise in different contexts, such as differential geometry [30], integral geometry [28], and singular perturbations in the theory of phase transitions [12]. The study of minimum problems related to functionals of this type is meaningful in some applications, for instance in the segmentation of images in computer vision [24,25]. In this article we study the L^{1}(IR^{N})-lower semicontinuity and the approximation, via Γ-convergence, of a functional F defined on the class C_{b}^{∞}(IR^{N}) of all bounded open sets E ∈ C^{∞}(IR^{N}), and consisting of two parts: the perimeter of E and an integral over ∂E of a suitable function f of the curvatures of ∂E. If F reduces to the perimeter functional then it is lower semicontinuous and it turns out [13,22,23] that the measures defined by

[h^{−1}|∇v|^{2} + hW (v)] dx,  (1.1)

where v is a smooth function, W (t) = t^{2}(1 − t)^{2} and h ∈ IN, provide an approximation of the perimeter as h → +∞. This result has been widely generalized [1,6,26]. The presence here of the curvatures in the energy F is clearly source of new difficulties. We start by expressing the curvature term through the signed distance function d_{E} from ∂E, and this reflects on the choice of the approximating functionals F_{h}. To be more precise the following result is a consequence of Theorems 4.2 and 4.3. Let F, F_{h} : L^{1}(IR^{N}) → [0, +∞] be the maps defined by

F (u) = ∫_{∂E}[1 + f(x, ∇d_{E}, ∇^{2}d_{E})] dH^{N−1} if u = 1_{E}, E ∈ C_{b}^{∞}(IR^{N}),  (1.2)

ISSN 0944-6532 / $ 2.50 ◯^{c} Heldermann Verlag

92 G. Bellettini / Variational approximation of some functionals ...

and, if v ∈ C^{∞}(IR^{N}) has compact support,

F_{h}(v) = ∫_{IR^{n}\\{|∇v|=0\}} [1 + f(x, \frac{∇v}{|∇v|}, \frac{P_{∇v}∇^{2}vP_{∇v}}{|∇v|} )] dμ_{h}(x).  (1.3)

Here H^{N−1} is the (N − 1)-dimensional Hausdorff measure, f is a non-negative continuous function, 1_{E} is the characteristic function of E, ∇^{2}v is the Hessian of v, P_{∇v} = Id −|∇v|^{−2}∇v ⊗ ∇v, and μ_{h} (which depend on v) are as in (1.1). Let us extend F and F_{h} with value +∞ on the whole L^{1}(IR^{N}). Then, if F is L^{1}(IR^{N})-lower semicontinuous on C_{b}^{∞}(IR^{N}), we prove that

Γ − lim_{h→+∞} F_{h} = 2c_{0}\overline{F}  on L^{1}(IR^{N}),  (1.4)

where c_{0} = ∫_{0}^{1} \sqrt{W (t)} dt, \overline{F} denotes the L^{1}(IR^{N})-lower semicontinuous envelope of F, and the Γ-limit is computed with respect to the L^{1}(IR^{N})-topology. More in general, we prove that (1.4) holds if the measure H^{N−1} is replaced by a suitable anisotropic surface measure (see (4.1) and Section (2.1)), provided that the measures μ_{h} in (1.3) are consequently modified (see (4.2)). Our results apply, in particular, to the elastica functional F (see [14] and [20]) defined, in dimension N = 2, by

F (1_{E}) = ∫_{∂E}[1 + |κ|^{2}] dH^{1},  (1.5)

where κ(x) is the curvature of ∂E at the point x. Since F is L^{1}(IR^{2})-lower semicontinuous on C_{b}^{∞}(IR^{2}) [2], by (1.4) it can be approximated by the sequence of functionals

F_{h}(v) = ∫_{IR^{2}\{|∇v|=0}} [1 + ||∇v|^{−1}Δv − |∇v|^{−3} ∑_{i,j=1}^{2} ∇_{i}v∇_{j}v∇_{ij}^{2}v|^{2} ] dμ_{h}(x),

where ∇v = (∇_{1}v, ∇_{2}v) and (∇^{2}v)_{ij} = ∇_{ij}^{2}v. Thanks to (1.4) the approximation problem is reduced to study the lower semicontinuity of F. Conditions on f ensuring that F = \overline{F} on C_{b}^{∞}(IR^{N}) are partially investigated in the last part of the paper, in which the cases N = 2 and N ≥ 3 are separately considered. In particular, if N = 3, using a slicing argument [9] we prove the lower semicontinuity of F when f is a suitable quadratic form of the principal curvatures (Proposition 6.1). Moreover, if N ≥ 2, we prove that F = \overline{F} on C_{b}^{∞}(IR^{N}) provided that f is a convex function of the length of the second fundamental form (Theorem 6.3). The outline of the paper is the following. In Section 2 we give some notation. In Section 3 we give some preliminaries concerning the signed distance function and we recall the notion of geometric integrand f̃ (next used in the definition of the approximating functionals F_{h}), and we show some examples. In Section 4 we define the functionals F and F_{h}. In Theorems 4.2 and 4.3 we show the relations between F and the Γ-limit of the sequence {F_{h}}. In Section 5 we find, in dimension N = 2, some necessary conditions on f that guarantee the lower semicontinuity of F (Theorem 5.2). Finally, in Section 6 we exhibit some examples of lower semicontinuous functionals in dimension N ≥ 3.

2. Notation

Let E be a subset of IR^{N}, N ≥ 2; we denote by 1_{E} the characteristic function of E, i.e., 1_{E}(x) = 1 if x ∈ E, and 1_{E}(x) = 0 if x ∈ IR^{N} \ E. We write E ∈ C_{b}^{∞}(IR^{N}) if E is bounded, open, and near each point x ∈ ∂E the set E can be written as the subgraph of a function of class C^{∞}. If E ∈ C_{b}^{∞}(IR^{N}) its boundary ∂E is oriented by the inner unit normal vector field ν_{E}; moreover, we say that u : IR^{N} → IR is a defining function of ∂E if u is of class C^{∞} in a neighbourhood of ∂E, ∂E = {u = 0}, E = {u > 0}, and |∇u| ≠ 0 on ∂E. Given p, q ∈ IR^{N}, by p ⊗ q we mean the matrix whose entries are (p ⊗ q)_{ij} = p_{i}q_{j}, for i, j = 1, …, N. We set also p ⊙ q = (p ⊗ q + q ⊗ p)/2. If p ∈ IR^{N} \ {0}, by P_{p} : IR^{N} → IR^{N} we mean the projection matrix on the orthogonal subspace to p, i.e.,

P_{p} = Id −|p|^{−2}p ⊗ p.  (2.1)

Note that if q ∈ IR^{N} we have

P_{p} p ⊙ q P_{p} = 0.  (2.2)

We denote by Sym(N) the space of the real N×N symmetric matrices. For notational simplicity we set

T = IR^{N} × (IR^{N} \ {0}) × Sym(N).  (2.3)

2.1. Definitions of ϕ, ϕ^{o}, δ, δ_{E}, d_{E}, m, and P^{N−1}

We denote by | · | and d the norm and the distance in IR^{N} (in the euclidean sense), respectively. In what follows ϕ : IR_{x}^{N} × IR_{ξ}^{N} → [0, +∞[ is a symmetric convex Finsler metric, that is a continuous function which is locally equivalent (uniformly with respect to x) to the euclidean norm and such that ϕ(x, ·) is positively one-homogeneous and convex on IR^{N}. We denote by ϕ^{o} : IR_{x}^{N} × IR_{ξ^{∗}}^{N} → [0, +∞[ the dual of ϕ with respect to ξ [27, p. 128], defined as ϕ^{o}(x, ξ^{∗}) = sup{ξ · ξ^{∗} : ϕ(x, ξ) ≤ 1}. One can show that ϕ^{o} is a symmetric convex Finsler metric. We indicate by δ the distance on IR^{N} induced by the metric ϕ, and by δ_{E} the δ signed distance function from ∂E positive inside E ⊆ IR^{N}, i.e.,

δ_{E}(x) = δ(x, IR^{N} \ E) − δ(x, E).  (2.4)

By d_{E} we mean the euclidean signed distance function from ∂E positive inside E. In the sequel m : IR^{N} → ]0, +∞[ is a function of class C^{1}(IR^{N}) such that 0 < inf m ≤ sup m < +∞.

We denote by H^{k} the k-dimensional Hausdorff measure, for k ∈ IN, 0 ≤ k ≤ N, and by P^{N−1} the (N − 1)-dimensional measure associated to ϕ and m, defined, on a smooth

94 G. Bellettini / Variational approximation of some functionals ...

compact boundary ∂E, as the restriction to ∂E of H^{N−1} with density ϕ^{o}(x, ν_{E})m(x), see [5]. 2.2. Definitions of Γ-convergence and lower semicontinuous envelope Let F_{h} : L^{1}(IR^{N}) → [0, +∞] be a sequence of functionals. For any v ∈ L^{1}(IR^{N}) set

(Γ − lim inf_{h→+∞} F_{h})(v)=inf {lim inf_{h→+∞} F_{h}(v_{h}): {v_{h}} ⊆ L^{1}(IR^{N}), v_{h} ⟶^{h→+∞} v in L^{1}(IR^{N})}

(Γ − lim sup_{h→+∞}F_{h})(v)=inf {lim sup_{h→+∞} F_{h}(v_{h}):{v_{h}}⊆L^{1}(IR^{N}), v_{h} ⟶^{h→+∞} v in L^{1}(IR^{N})}.

We say that the sequence {F_{h}} is Γ-convergent with respect to the L^{1}(IR^{N})-topology if Γ − lim inf_{h→+∞} F_{h} = Γ − lim sup_{h→+∞} F_{h} =: Γ − lim_{h→+∞} F_{h} [13,11]. If F_{h} = F for any h ∈ IN, the Γ − lim_{h→+∞} F_{h} is denoted by \overline{F} and is called the L^{1}(IR^{N})- lower semicontinuous envelope of F [8].

3. Preliminaries We recall that if E ∈ C_{b}^{∞}(IR^{N}) then there exists a neighbourhood U of ∂E such that d_{E} ∈ C^{∞}(U) and |∇d_{E}| = 1 in U; moreover, on ∂E, ∇d_{E} = ν_{E} and −(N − 1)^{−1}Δd_{E} is the scalar mean curvature of ∂E (see [18, 14.6]). Since ∇d_{E} ∈ Ker(∇^{2}d_{E}) in U, we have

P_{∇d_{E}}∇^{2}d_{E}P_{∇d_{E}} = ∇^{2}d_{E}  on ∂E.  (3.1)

The following lemma, which is useful for the definition of f̃ in (3.3), shows the relations between ∇^{2}d_{E} and ∇^{2}u, where u is any defining function of ∂E. Lemma 3.1. Let E ∈ C_{b}^{∞}(IR^{N}) and suppose that u, v : IR^{N} → IR are two defining functions of ∂E. Then

|∇u|^{−1}P_{∇u}∇^{2}uP_{∇u} = |∇v|^{−1}P_{∇v}∇^{2}vP_{∇v} = ∇^{2}d_{E}  on ∂E.  (3.2)

Proof. Fix x ∈ ∂E. By [4, Lemma 3.2] we have

|∇u(x)|^{−1}∇^{2}u(x) = |∇v(x)|^{−1}∇^{2}v(x) + ∇u(x) ⊙ q,

for a suitable q ∈ IR^{N}. Then (3.2) follows from (2.2) and (3.1). Let us recall the definition of the geometric integrand f̃ associated to f [17]. Definition 3.2. Let T be as in (2.3) and let f : T → IR be a continuous function. We define the function f̃ : T → IR as follows: if (x, p, X) ∈ T we set

f̃(x, p, X) = f (x, \frac{p}{|p|}, \frac{P_{p}XP_{p}}{|p|} ) ,  (3.3)

where P_{p} is defined in (2.1).

The following properties are consequences of (3.3), (2.2) and (3.2).

Lemma 3.3. The function f̃ is continuous, ˜ f̃= f̃ and

(i) for any (x, p, X) ∈ T, any λ > 0, and any q ∈ IR^{N} we have

f̃(x, λp, λX + p ⊙ q) = f̃(x, p, X);  (3.4)

(ii) if E ∈ C_{b}^{∞}(IR^{N}) and u is any defining function of ∂E then

f̃(x, ∇u, ∇^{2}u) = f̃(x, ∇d_{E}, ∇^{2}d_{E}) = f(x, ∇d_{E}, ∇^{2}d_{E})  on ∂E.  (3.5)

Let E ∈ C_{b}^{∞}(IR^{N}) and let u be any defining function of ∂E.

Example 3.4. Let α ∈ [1, +∞[ and let f_{1}, f_{2} : T → [0, +∞[ be defined as

\begin{matrix}f_{1}(x, p, X) = f_{1}(X) = ((N − 1))^{−α} |tr (X)|^{α} , \\ f_{2}(x, p, X) = f_{2}(X) = ((N − 1))^{−1}[ tr (X^{2})]^{1/2}. \end{matrix}

Then f̃_{1}(∇u, ∇^{2}u) equals |H|^{α} on ∂E, where H is the scalar mean curvature of ∂E, while ((N − 1)f̃_{2}(∇u, ∇^{2}u))^{2} equals, on ∂E, the sum of the squares of the principal curvatures

of ∂E, and (f_{2}(∇^{2}d_{E}))^{2} = (N − 1)^{−2} ∑_{i,j=1}^{N} |∇_{ij}^{2}d_{E}|^{2}.

Example 3.5. Assume that ϕ^{2}, (ϕ^{o})^{2} are of class C^{2} and that (ϕ(x, ·))^{2}, (ϕ^{o}(x, ·))^{2} are strictly convex. Let f : T → [0, +∞[ be defined as

f(x, p, X) = |∑_{i=1}^{N} \frac{∂^{2}(ϕ^{o})}{∂x_{i}∂ξ_{i}}(x, p) + ∇_{ξ}^{2}(ϕ^{o})(x, p)X + ∇_{ξ}(ϕ^{o})(x, p) · ∇(log m)| .

Then one can show that f = f̃; f(x, ∇u, ∇^{2}u) represents the absolute value of the anisotropic scalar mean curvature of ∂E with respect to ϕ and m [5,4].

4. Definitions of the functionals. Γ-convergence result

Given the continuous function f : T → [0, +∞[ we define the map F : L^{1}(IR^{N}) → [0, +∞] as

F (u) = ∫_{∂E} [1 + f(x, ∇d_{E}, ∇^{2}d_{E})] dP^{N−1}  (4.1)

if u is the characteristic function of a set E ∈ C_{b}^{∞}(IR^{N}), and F = +∞ elsewhere. The integral in (4.1) will be also denoted by F (E), thus emphasizing the dependence of F on the set E rather than on its boundary ∂E. Note that by (3.5) we can replace f by f̃ in (4.1) without affecting the result.

96 G. Bellettini / Variational approximation of some functionals ...

4.1. The approximating functionals F_{h} For any v ∈ C_{c}^{∞}(IR^{N}) (the space of functions of class C^{∞}(IR^{N}) with compact support) and h ∈ IN we define the non-negative Radon measures μ_{h} on IR^{N} by

dμ_{h}(x) = [h^{−1}(ϕ^{o}(x, ∇v))^{2} + hW (v)] m(x) dx,  (4.2)

where W : IR → [0, +∞[ is defined by W (t) = t^{2}(1 − t)^{2} and dx stands for the Lebesgue measure. For simplicity of notation we omit the dependence on v in the notation of the measures μ_{h}. We set c_{0} = ∫_{0}^{1} \sqrt{W (t)} dt. Let f : T → [0, +∞[ be a continuous function; for any h ∈ IN we define the map F_{h} : L^{1}(IR^{N}) → [0, +∞] as

F_{h}(v) = ∫_{IR^{N} \{|∇v|=0}} [1 + f̃(x, ∇v, ∇^{2}v)] dμ_{h}(x) if v ∈ C_{c}^{∞}(IR^{N}),  (4.3)

and F_{h} = +∞ elsewhere. Note the presence of f̃ in (4.3). The following remark is a consequence of (3.5) and the definition of \overline{F}, and will be useful in the proof of Theorem 4.2. Remark 4.1. Let {E_{k}} be a sequence in C_{b}^{∞}(IR^{N}) converging in L^{1}(IR^{N}) to E with 1_{E} ∈ L^{1}(IR^{N}). For any k ∈ IN let u_{k} be a defining function of ∂E_{k}. Then

lim inf_{k→+∞} ∫_{∂E_{k}}[1 + f̃(x, ∇u_{k}, ∇^{2}u_{k})] dP^{N−1} ≥ \overline{F} (E).

The main result of this section is contained in the following two theorems. Theorem 4.2. Let T be as in (2.3), let f : T → [0, +∞[ be a continuous function and let f̃ be defined as in (3.3). Let F, F_{h} be the functionals defined in (4.1) and (4.3), respectively. Then

Γ − lim inf_{h→+∞} F_{h} ≥ 2c_{0}\overline{F} on L^{1}(IR^{N}).

Proof. Let u ∈ L^{1}(IR^{N}) and let {v_{h}} be a sequence in C_{c}^{∞}(IR^{N}) converging to u in L^{1}(IR^{N}) such that lim inf_{h→+∞} F_{h}(v_{h}) = (Γ − lim inf_{h→+∞} F_{h})(u) < +∞. Passing to a subsequence, we may assume that the lim inf_{h→+∞} is a lim_{h→+∞} and that {v_{h}} converges to u almost everywhere. We have ∫_{IR}_{N} W (v_{h}) dx ≤ O(h^{−1}), so that u is the characteristic function of some measurable set E ⊆ IR^{N}. As h^{−1}(ϕ^{o}(x, ∇v_{h}))^{2} + hW (v_{h}) ≥ 2ϕ^{o}(x, ∇v_{h})\sqrt{W (v_{h})}, using the fact that ϕ^{o}(x, ·) is one-homogeneous and the coarea formula [16] we have

\begin{aligned}F_{h}(v_{h}) & ≥ 2 ∫_{\{|∇v_{h}|≠0\}} |∇v_{h}|\sqrt{W (v_{h})} [1 + f̃(x, ∇v_{h}, ∇^{2}v_{h})] ϕ^{o}(x, \frac{∇v_{h}}{|∇v_{h}|}) mdx \\ & ≥ 2 ∫_{0}^{1} \sqrt{W (t)} ∫_{\{v_{h}=t\}∩\{|∇v_{h}|≠0\}} [1 + f̃(x, ∇v_{h}, ∇^{2}v_{h})] dP^{N−1}dt. \end{aligned}

Observe that, given h ∈ IN, ∂{v_{h} > t} is compact for every t ∈ ]0, 1[. Moreover, Sard’s theorem [16, 3.4.3] implies that |∇v_{h}| ≠ 0 on {v_{h} = t} for almost every t ∈ ]0, 1[, hence {v_{h} = t} = ∂{v_{h} > t} and {v_{h} > t} ∈ C_{b}^{∞}(IR^{N}). Let I_{h} ⊆ ]0, 1[ be such that the Lebesgue measure |I_{h}| of I_{h} is zero and {v_{h} > t} verifies the properties listed above for any t ∈ ]0, 1[ \I_{h}. Letting I := ⋃_{h∈IN} I_{h} we have |I| = 0 and, for any h ∈ IN,

F_{h}(v_{h}) ≥ 2 ∫_{]0,1[\I} \sqrt{W (t)} ∫_{∂\{v_{h}>t\}} [1 + f̃ (x, ∇v_{h}, ∇^{2}v_{h})] dP^{N−1}dt.  (4.4)

Using the Cavalieri formula there exist a subsequence {v_{k}} of {v_{h}} and a set J ⊆]0, 1[ with |J| = 0 such that 1_{{v_{k}>t}} → 1_{{u>t}} = E in L^{1}(IR^{N}) as k → +∞ for any t ∈ ]0, 1[\(I ∪ J). Fix t ∈ ]0, 1[\(I ∪ J) and set u_{k} = v_{k} − t. Using (4.4), Fatou’s lemma, applying Remark 4.1 with E_{k} replaced by {u_{k} > 0} and observing that ∇u_{k} = ∇v_{k}, ∇^{2}u_{k} = ∇^{2}v_{k}, we obtain

(Γ − lim inf_{h→+∞} F_{h})(u) = lim_{h→+∞} F_{h}(v_{h}) = lim_{k→+∞} F_{k}(v_{k})

≥ 2 ∫_{]0,1[\I∪J} \sqrt{W (t)} lim inf_{k→+∞} ∫_{∂\{u_{k}>0\}} [1 + f̃(x, ∇u_{k}, ∇^{2}u_{k})] dP^{N−1}dt

≥ 2\overline{F} (E) ∫_{]0,1[\(I∪J)} \sqrt{W (t)} dt = 2c_{0}\overline{F} (E).

This concludes the proof.

Theorem 4.3. Assume that ϕ does not depend on x and that ϕ^{2}, (ϕ^{o})^{2} are of class C^{∞} and strictly convex. Let F and F_{h} be as in the statement of Theorem 4.2. Suppose that F is L^{1}(IR^{N})-lower semicontinuous on C_{b}^{∞}(IR^{N}). Then

Γ − lim sup_{h→+∞} F_{h} ≤ 2c_{0}\overline{F}  on L^{1}(IR^{N}).  (4.5)

Proof. Let u ∈ L^{1}(IR^{N}); we can assume that \overline{F} (u) < +∞, so that u = 1_{E} for some measurable set E ⊆ IR^{N}. Let us first suppose that E ∈ C_{b}^{∞}(IR^{N}), so that \overline{F} (E) = F (E). Denote by γ the minimizer of the one-dimensional functional ∫_{IR}(|ζ′|^{2} + W (ζ)) dt under the constraints ζ ∈ H_{loc}^{1}(IR), lim_{t→+∞} ζ(t) = 1, lim_{t→−∞} ζ(t) = 0, ζ(0) = 1/2. With our choice

of W it turns out that γ(t) = (tgh(t/2) + 1) /2. For any h ∈ IN let t_{h} = log h. Denote by χ_{h} : [0, +∞[→ [0, 1] a function of class C^{∞} with the following properties: χ_{h} = 1 on [0, t_{h}], χ_{h} = 0 on [2t_{h}, +∞[, χ′_{h} < 0 in ]t_{h}, 2t_{h}[ and ∥χ′_{h}∥_{L^{∞}([t_{h},2t_{h}])} = O(1/ log h). Let

γ̂_{h}(t) = \{ \begin{matrix}γ(t)χ_{h}(t) + 1 − χ_{h}(t) & if t ≥ 0, \\ 1 − γ̂_{h}(−t) & if t < 0, \end{matrix} γ_{h}(t) = γ̂_{h}(ht),

and

u_{h}(x) = γ_{h}(δ_{E}(x)) ∀x ∈ IR^{N},  (4.6)

where δ_{E} is defined in (2.4). One can show that u_{h} → 1_{E} in L^{1}(IR^{N}) as h → +∞ and that, under our assumptions on ϕ and ϕ^{o}, the function δ_{E} is of class C^{∞} in a suitable neighbourhood of ∂E. Hence u_{h} ∈ C_{c}^{∞}(IR^{N}) for any h large enough. Set

A_{h} = {x ∈ IR^{N} : |δ_{E}(x)| < h^{−1}t_{h}}, B_{h} = {x ∈ IR^{N} : h^{−1}t_{h} < |δ_{E}(x)| < 2h^{−1}t_{h}}.

Using the definition of F_{h} we have

F_{h}(u_{h}) = ∫_{A_{h}} dμ_{h}(x) + ∫_{A_{h}} f̃(x, ∇u_{h}, ∇^{2}u_{h}) dμ_{h}(x)
+ ∫_{B_{h}} dμ_{h}(x) + ∫_{B_{h}} f̃(x, ∇u_{h}, ∇^{2}u_{h}) dμ_{h}(x)
=: F_{h}^{(1)}(u_{h}) + F_{h}^{(2)}(u_{h}) + F_{h}^{(3)}(u_{h}) + F_{h}^{(4)}(u_{h}).

We shall show that

lim_{h→+∞} (F_{h}^{(1)}(u_{h}) + F_{h}^{(2)}(u_{h})) = 2c_{0}F (E),  (4.7)
lim_{h→+∞} (F_{h}^{(3)}(u_{h}) + F_{h}^{(4)}(u_{h})) = 0,  (4.8)

and this will conclude the proof of (4.5) when E ∈ C_{b}^{∞}(IR^{N}). We recall a basic property of δ_{E} [5, Theorem 3.2]: if E ∈ C_{b}^{∞}(IR^{N}) then

ϕ^{o}(∇δ_{E}) = 1  (4.9)

in a suitable neighbourhood of ∂E, which implies, in particular, that the infimum of |∇δ_{E}| in this neighbourhood is strictly positive. Let us prove (4.7). First it is well known [5,6,26] that

lim_{h→+∞} F_{h}^{(1)}(u_{h}) = 2c_{0} ∫_{∂E} dP^{N−1}.  (4.10)

By (4.6), for h large enough we have

∇u_{h} = γ′_{h}(δ_{E})∇δ_{E},  ∇^{2}u_{h} = γ′′_{h}(δ_{E})∇δ_{E} ⊗ ∇δ_{E} + γ′_{h}(δ_{E})∇^{2}δ_{E}.  (4.11)

Hence by (4.9) and since γ′ = \sqrt{W (γ)}, on A_{h} we have

\begin{matrix}h^{−1}(ϕ^{o}(∇u_{h}))^{2} + hW (u_{h}) = h^{−1} (γ′_{h}(δ_{E}))^{2} + hW (γ_{h}(δ_{E})) \\ = 2γ′_{h}(δ_{E})\sqrt{W (γ_{h}(δ_{E}))}. \end{matrix}  (4.12)

In addition, using (4.11) and (3.4) (for x ∈ A_{h}) applied with λ = γ′_{h}(δ_{E}(x)) > 0, p = ∇δ_{E}(x) ∈ IR^{N} \ {0}, and q = γ′′_{h}(δ_{E}(x))∇δ_{E}(x)/2, we obtain that

f̃(x, ∇u_{h}, ∇^{2}u_{h}) = f̃(x, ∇δ_{E}, ∇^{2}δ_{E}) on A_{h}.  (4.13)

By (4.12) and (4.13), the fact that |∇δ_{E}|^{−1} = ϕ^{o}(\frac{∇δ_{E}}{|∇δ_{E}|}) in a neighbourhood of ∂E (see

(4.9)) and the coarea formula we get, for h large enough,

\begin{aligned}F_{h}^{(2)}(u_{h}) & = 2 ∫_{A_{h}}|∇δ_{E}|γ′_{h}(δ_{E})\sqrt{W (γ_{h}(δ_{E}))} f̃(x, ∇δ_{E}, ∇^{2}δ_{E})ϕ^{o}(\frac{∇δ_{E}}{|∇δ_{E}|})m dx \\ & = 2 ∫_{−h^{−1}t_{h}}^{h^{−1}t_{h}} γ′_{h}(t)\sqrt{W (γ_{h}(t))} ∫_{\{δ_{E}=t\}} f̃(x, ∇δ_{E}, ∇^{2}δ_{E}) dP^{N−1}dt. \end{aligned}

We observe now that for any h large enough

\begin{matrix}∫_{\{δ_{E}=t\}} f̃(x, ∇δ_{E}, ∇^{2}δ_{E}) dP^{N−1} \\ = ∫_{∂E} f̃(x, ∇δ_{E}, ∇^{2}δ_{E}) dP^{N−1} + o(h^{−1}t_{h}) \end{matrix}  (4.14)

for any t ∈ [−h^{−1}t_{h}, h^{−1}t_{h}]. Indeed, if t ∈ [−h^{−1}t_{h}, h^{−1}t_{h}], x ∈ {δ_{E} = t} and if π(x) is the (unique) projection of x onto ∂E with respect to the metric ϕ, we have x = π(x) + t(∇ϕ^{o})(ν_{E}(π(x))) [4], ∇_{i}δ_{E}(x) = ∇_{i}δ_{E}(π(x)) + O(h^{−1}t_{h}), ∇_{ij}^{2}δ_{E}(x) = ∇_{ij}^{2}δ_{E}(π(x)) +

O(h^{−1}t_{h}), and similar relations hold for ϕ^{o} and m. Hence, as all \overline{A_{h}} are contained (for h large enough) in a fixed neighbourhood of ∂E where δ_{E} is of class C^{∞} and inf |∇δ_{E}| > 0, and since f̃, ϕ^{o} and m are continuous, we have

\begin{matrix}f̃(x, ∇δ_{E}(x), ∇^{2}δ_{E}(x))ϕ^{o}(\frac{∇δ_{E}(x)}{|∇δ_{E}(x)|})m(x) \\ = f̃(π(x), ∇δ_{E}(π(x)), ∇^{2}δ_{E}(π(x)))ϕ^{o}(\frac{∇δ_{E}(π(x))}{|∇δ_{E}(π(x))|})m(π(x)) + o(h^{−1}t_{h}) \end{matrix}

for any x ∈ A_{h}, and this, together with a change of variables, implies (4.14). We then get

\begin{aligned}F_{h}^{(2)}(u_{h}) & = 2 ∫_{∂E} f̃(x, ∇δ_{E}, ∇^{2}δ_{E}) dP^{N−1} ∫_{−h^{−1}t_{h}}^{h^{−1}t_{h}} γ′_{h}(t)\sqrt{W (γ_{h}(t))} dt \\ & + o(h^{−1}t_{h}) ∫_{−h^{−1}t_{h}}^{h^{−1}t_{h}} γ′_{h}(t)\sqrt{W (γ_{h}(t))} dt =: I_{h} + II_{h}. \end{aligned}

Now

∫_{−h^{−1}t_{h}}^{h^{−1}t_{h}} γ′_{h}(t)\sqrt{W (γ_{h}(t))} dt = ∫_{γ(−t_{h})}^{γ(t_{h})} \sqrt{W (τ)} dτ → c_{0} as h → +∞.

Therefore lim_{h→+∞} II_{h} = 0, and using (3.5)

lim_{h→+∞} I_{h} = 2c_{0} ∫_{∂E} f̃(x, ∇δ_{E}, ∇^{2}δ_{E}) dP^{N−1} = 2c_{0} ∫_{∂E} f(x, ∇d_{E}, ∇^{2}d_{E}) dP^{N−1}.

Then (4.7) follows recalling (4.10).

100 G. Bellettini / Variational approximation of some functionals ...

Let us prove (4.8). One can check that

∥γ̂_{h} − 1∥_{L^{∞}([t_{h},2t_{h}])} = O(h^{−1}), ∥γ̂′_{h}∥_{L^{∞}([t_{h},2t_{h}])} = o(h^{−1}). (4.15)

Applying (4.9), the coarea formula and (4.15) we have, for h large enough,

\begin{aligned}F_{h}^{(3)}(u_{h}) & = ∫_{B_{h}} |∇δ_{E}|ϕ^{o}(\frac{∇δ_{E}}{|∇δ_{E}|}) dμ_{h}(x) \\ & ≤ O(1) ∫_{h^{−1}t_{h}}^{2h^{−1}t_{h}} h^{−1}|γ′_{h}(t)|^{2} + hW (γ_{h}(t)) dt = o(1). \end{aligned}

It remains to show that lim_{h→+∞} F_{h}^{(4)}(u_{h}) = 0. Reasoning as in (4.13) we have f̃(x, ∇u_{h}, ∇^{2}u_{h}) = f̃(x, ∇δ_{E}, ∇^{2}δ_{E}) on B_{h}. As δ_{E} ∈ C^{∞}(B_{h}) and inf_{B_{h}} |∇δ_{E}| > 0 for h large enough and since f̃ is continuous, we then have

F_{h}^{(4)}(u_{h}) ≤ ∫_{B_{h}} f̃(x, ∇δ_{E}, ∇^{2}δ_{E}) dμ_{h}(x) ≤ O(1) ∫_{B_{h}} dμ_{h}(x) = o(1).

The proof of (4.8) is concluded. We have showed that (4.5) holds for E ∈ C_{b}^{∞}(IR^{N}). If u = 1_{E} ∈ L^{1}(IR^{N}) we can find a sequence \{E_{h}\} in C_{b}^{∞}(IR^{N}) converging to u in L^{1}(IR^{N}) such that F (E_{h}) → \overline{F} (E) < +∞ as h → +∞. Then (4.5) follows from the previous case and a diagonal procedure. Using Theorems 4.2 and 4.3 we have the following result, which shows that the Γ-approxi-mability of the functional F is reduced to study its lower semicontinuous envelope \overline{F}. Corollary 4.4. Under the assumptions of Theorem 4.3 we have

Γ − lim_{h→+∞} F_{h} = \{ \begin{matrix}2c_{0}F & on C_{b}^{∞}(IR^{N}), \\ 2c_{0}\overline{F} & elsewhere on L^{1}(IR^{N}). \end{matrix}

Remark 4.5. Let c ∈ [0, +∞[ and assume that {v_{h}} is a sequence in L^{1}(IR^{N}) such that F_{h}(v_{h}) ≤ c for any h ∈ IN. Then {v_{h}} admits a subsequence converging in L^{1}(IR^{N}) to a characteristic function of a set E of finite perimeter such that \overline{F} (E) < +∞. Indeed, as c ≥ F_{h}(v_{h}) ≥ O(1) ∫_{IR}_{N} h^{−1}|∇v_{h}|^{2} + hW (v_{h}) dx, we have [22] that {v_{h}} has a subsequence converging in L^{1}(IR^{N}) to a characteristic function of a set E of finite perimeter. Reasoning as in the proof of Theorem 4.2, one then obtains \overline{F} (E) < +∞.

5. Lower semicontinuity of F in two dimensions In this section we find some necessary conditions on f which ensure the L^{1}-lower semicontinuity of F in two dimensions. In view of Corollary 4.4, F can be approximated via Γ-convergence. A particular case has been illustrated in Section 1, see (1.5).

Let N = 2; if p = (p_{1}, p_{2}) ∈ IR^{2} we set p^{⊥} = (−p_{2}, p_{1}).

Remark 5.1. If N = 2 we have

∇^{2}d_{E} = Δd_{E}(∇d_{E})^{⊥} ⊗ (∇d_{E})^{⊥} on ∂E.  (5.1)

Proof. It is enough to observe that if p ∈ IR^{2}, |p| = 1, and X ∈ Sym(2) is such that P_{p}XP_{p} = X, then X = tr(X)p^{⊥} ⊗ p^{⊥}. Letting x ∈ ∂E, p = ∇d_{E}(x), X = ∇^{2}d_{E}(x) and using (3.1), the assertion follows.

Let E ∈ C_{b}^{∞}(IR^{2}) and let γ : [0, l(γ)] → IR^{2} be an oriented parametrization by arc length of a connected component of ∂E (here l(γ) denotes the length of γ). Then at x = γ(s) ∈ ∂E, we have

∇d_{E} = (γ̇)^{⊥}, Δd_{E} = −κ, ∇^{2}d_{E} = −κγ̇ ⊗ γ̇, γ̈ = κ∇d_{E},

where κ is the curvature (with sign) of ∂E. If f : T = IR^{2} × (IR^{2} \ {0}) × Sym(2) → [0, +∞[ is a continuous function, using also (5.1) it follows that at x = γ(s) ∈ ∂E we have

f (x, ∇d_{E}, ∇^{2}d_{E}) = g(γ, γ̇, γ̈),  (5.2)

where g : IR^{2} × (IR^{2} \ {0}) × IR^{2} → [0, +∞[ is the continuous function defined by

g(x, ξ, η) = f(x, ξ^{⊥}, ξ ⊗ η^{⊥}).  (5.3)

Theorem 5.2. Let f : T → [0, +∞[ be a continuous function such that f = f̃ (see (3.3)) and satisfying the following properties:

(i) there exist constants c > 0 and α > 1 such that f(x, p, X) ≥ c |\frac{tr(P_{p}XP_{p})}{|p|} |^{α} for any

(x, p, X) ∈ T; (ii) for any x ∈ IR^{2} and any p ∈ IR^{2} \ {0} the function q → f(x, p^{⊥}, p ⊗ q^{⊥}) is convex on

IR^{2}. Then the map F defined in (1.2) (with N = 2) is L^{1}(IR^{2})-lower semicontinuous on C_{b}^{∞}(IR^{2}).

Proof. Let E ∈ C_{b}^{∞}(IR^{2}) and let {E_{h}} be a sequence in C_{b}^{∞}(IR^{2}) converging to E in L^{1}(IR^{2}) as h → +∞. We must prove that

F (E) ≤ lim inf_{h→+∞} F (E_{h}).  (5.4)

We can suppose, possibly passing to a subsequence, that the right hand side of (5.4) is a finite limit. For simplicity, this subsequence (and any further subsequence) will be still denoted by {E_{h}}. Using assumption (i) we then have

sup_{h} ∫_{∂E_{h}} |Δd_{E_{h}}|^{α} dH^{1} ≤ c^{−1} sup_{h} ∫_{∂E_{h}} f(x, ∇d_{E_{h}}, ∇^{2}d_{E_{h}}) dH^{1} < +∞,

so that the L_{H^{1}}^{α}-norm of the curvature of ∂E_{h} is uniformly bounded with respect to h. Since also {H^{1}(∂E_{h})} is bounded, possibly passing to a subsequence, for any h the set ∂E_{h} has a finite number m̄ of connected components [2], and m̄ is independent of h. Let Δ_{h} be an oriented parametrization of ∂E_{h} [2, Definition 3.2]; then Δ_{h} = {γ_{h}^{1}, …, γm̄ ; h}, where γ_{h}^{i} are closed regular simple smooth disjoint curves parametrized with constant velocity on [0, 1]. Let {γ_{h}^{i_{1}}, …, γ_{h}^{i_{n}}} be the H^{2,α}-equibounded sequence and let {γ^{i_{1}}, …, γ^{i_{n}}} be the limit system of curves of class H^{2,α} constructed (starting from Δ_{h}) in the proof of Lemma 3.3 of [2]. Then m̄ ≥ n,

⋃_{j=1}^{n} γ^{i_{j}}([0, 1]) ⊇ ∂E,  (5.5)

and for any j = 1, …, n we have that γ_{h}^{i_{j}} ⇀ γ^{i_{j}} weakly in H^{2,α}([0, 1]) as h → +∞ (in

particular lim_{h→+∞} l(γ_{h}^{i_{j}}) = l(γ^{i_{j}})). Recalling m̄ ≥ n, (5.5) and (5.2), we obtain

lim inf_{h→+∞} F (E_{h}) ≥ lim inf_{h→+∞} [ ∑_{j=1}^{n} l(γ_{h}^{i_{j}}) + ∑_{j=1}^{n} ∫_{0}^{l(γ_{h}^{i_{j}})} g(γ_{h}^{i_{j}}, γ̇_{h}^{i_{j}}, γ̈_{h}^{i_{j}}) ds]

≥ H^{1}(∂E) + ∑_{j=1}^{n} lim inf_{h→+∞} ∫_{0}^{l(γ_{h}^{i_{j}})} g(γ_{h}^{i_{j}}, γ̇_{h}^{i_{j}}, γ̈_{h}^{i_{j}}) ds.

(5.6)

By assumption (ii), for any x ∈ IR^{2} and ξ ∈ IR^{2} \ {0}, the function η → g(x, ξ, η) is convex on IR^{2}. Consequently the functional γ → ∫_{I} g(γ, γ̇, γ̈) ds is weakly H^{2,α}(I)-lower semicontinuous (see, for instance, [8, Section 2.3]), where I ⊆ [0, l(γ)] is an open interval.

For any j = 1, …, n let 0 < λ_{j} < l(γ^{i_{j}}). We have l(γ_{h}^{i_{j}}) > λ_{j} for any j = 1, …, n provided that h is large enough. Therefore by (5.6)

lim inf_{h→+∞} F (E_{h}) ≥ H^{1}(∂E) + ∑_{j=1}^{n} ∫_{0}^{λ_{j}} g (γ^{i_{j}}, γ̇^{i_{j}}, γ̈^{i_{j}}) ds.

Since this inequality holds for any λ_{j} < l(γ^{i_{j}}) we have

lim inf_{h→+∞} F (E_{h}) ≥ H^{1}(∂E) + ∑_{j=1}^{n} ∫_{0}^{l(γ^{i_{j}})} g (γ^{i_{j}}, γ̇^{i_{j}}, γ̈^{i_{j}}) ds.  (5.7)

Using (5.5) and reasoning as in [2, Lemma 3.4] we also have

∑_{j=1}^{n} ∫_{0}^{l(γ^{i_{j}})} g (γ^{i_{j}}, γ̇^{i_{j}}, γ̈^{i_{j}}) ds ≥ ∫_{∂E} f(x, ∇d_{E}, ∇^{2}d_{E}) dH^{1}.  (5.8)

Then (5.4) follows from (5.7) and (5.8).

Example 5.3. The function f(p, X) = |p|^{−α}| tr(P_{p}XP_{p})|^{α} for α > 1 (which corresponds

to f̃_{1} of Example 3.4) verifies the assumptions of Theorem 5.2. Indeed given p = (p_{1}, p_{2}) ∈ IR^{2} \ {0}, we have, for q = (q_{1}, q_{2}) ∈ IR^{2},

f(p^{⊥}, p ⊗ q^{⊥}) = |p|^{−α}| tr(p ⊗ q^{⊥}P_{p^{⊥}})|^{α} = |p|^{−α}|p_{2}q_{1} − p_{1}q_{2}|^{α},

so that the map q → f(p^{⊥}, p ⊗ q^{⊥}) is convex on IR^{2}.

The case in which the integrand f has linear growth at infinity is useful for some applications, see [25]. In particular the functional F (E) = ∫_{∂E}[1 + ζ(κ)] dH^{1}, where ζ(t) = c_{1}|t|^{2}1_{{|t|≤τ}}+ [c_{2}|t| − c_{3}]1_{{|t|>τ}}, for τ > 0, c_{1} > 0, c_{1}τ^{2} = c_{2}τ − c_{3}, c_{2} = 2c_{1}τ, is lower semicontinuous on C_{b}^{∞}(IR^{2}), see [3, Theorem 6.1]. Hence by Corollary 4.4 F is the Γ-limit of the sequence in (4.3) where m ≡ 1, ϕ(x, ξ) = |ξ|, and f(X) = c_{1} |tr(X)|^{2} 1_{{X∈Sym(2):|tr(X)|≤τ}} +(c_{2}| tr(X)| − c_{3}) 1_{{X∈Sym(2):|tr(X)|>τ}}.

6. Lower semicontinuity of F in dimension N ≥ 3

In this section we show some examples of functionals F in dimension N ≥ 3 depending on curvatures and which are L^{1}(IR^{N})-lower semicontinuous on smooth compact boundaries. In the proof of the following proposition we use a slicing argument [9].

Proposition 6.1. Let N = 3, let f_{1}, f_{2} be as in Example 3.4 with α = 2, let a, b ∈ ]0, +∞[, and set

f(p, X) = af̃_{1}(p, X) + b(f̃_{2}(p, X))^{2}  ∀p ∈ IR^{3} \ {0}, ∀X ∈ Sym(3).  (6.1)

Then if b ≥ 2a the map F defined in (1.2) (with N = 3) is L^{1}(IR^{3})-lower semicontinuous on C_{b}^{∞}(IR^{3}).

Note that, if κ_{1}, κ_{2} are the principal curvatures of ∂E, letting H = (κ_{1} + κ_{2})/2 and K = κ_{1}κ_{2}, then the functional F in (1.2) corresponding to f in (6.1) can be written as

\begin{aligned}F (E) & = ∫_{∂E} [1 + \frac{a}{4}|Δd_{E}|^{2} + \frac{b}{4} ∑_{i,j=1}^{3} |∇_{ij}^{2}d_{E}|^{2}]dH^{2} \\ & = ∫_{∂E} [1 + \frac{a}{4}|κ_{1} + κ_{2}|^{2} + \frac{b}{4}(κ_{1}^{2} + κ_{2}^{2})]dH^{2} = ∫_{∂E} [1 + (a + b)|H|^{2} − \frac{b}{2}K] dH^{2}. \end{aligned}

Proof of Proposition 6.1. Let E ∈ C_{b}^{∞}(IR^{3}) and let {E_{h}} be a sequence in C_{b}^{∞}(IR^{3}) converging to E in L^{1}(IR^{3}) as h → +∞. We must prove inequality (5.4). We can suppose, possibly passing to a subsequence, that the right hand side of (5.4) is a finite limit. Hence, in view of [7, Remark 29.4.9 with N − 1 = m = 2], possibly passing to a further subsequence, we can assume that each ∂E_{h} has a finite number of connected components independent of h.

We recall now a particular case of a kinematic formula due to C.-S. Chen [9, Section 5] : if π denotes a plane in IR^{3}, one has

∫_{∂E} [6|H|^{2} − 2K] dH^{2} = c_{2} ∫_{π} ∫_{π∩∂E} κ^{2} dH^{1} dπ,  (6.2)

where dπ denotes the density for the planes in IR^{3} [28, II.12.5], c_{2} > 0 depends only on the dimension, and κ denotes the curvature of the curve π ∩ ∂E (note that for dπ-almost every π the curve π ∩ ∂E is of class C^{∞} by Sard’s theorem, and π ∩ ∂E = ∂(π ∩ E)). Moreover

H^{2}(∂E) = c_{1} ∫_{π} H^{1}(π ∩ ∂E) dπ,  (6.3)

where c_{1} > 0 depends only on the dimension (see [10, formula (81) with e = 0]). Observe now that if E_{h} → E in L^{1}(IR^{3}), then by Fubini’s theorem for dπ-almost every π we have π ∩ E_{h} → π ∩ E in L^{1}(π). Let us first suppose that b = 2a. For any h denote by J_{h} the complement of all planes π so that π ∩∂E_{h} = ∂(π ∩E_{h}) and π ∩E_{h} is of class C^{∞}. Using the L^{1}(π)-lower semicontinuity on C_{b}^{∞}(π) of the map

π ∩ E → ∫_{∂(π∩E)}[c_{1} + c_{2}κ^{2}] dH^{1}

proved in [2, Theorem 3.2], using Fatou’s lemma, letting J = ∪_{h}J_{h} (which has zero dπ-measure), and denoting by κ_{h} the curvature of ∂(π ∩ E_{h}), by (6.2) and (6.3) we have

lim inf_{h→+∞} F (E_{h}) ≥ \frac{a}{2} ∫_{π\J} lim inf_{h→+∞} ∫_{∂(π∩E_{h})}[c_{1} + c_{2}κ_{h}^{2}] dH^{1} dπ

≥ \frac{a}{2} ∫_{π\J} ∫_{∂(π∩E)}[c_{1} + c_{2}κ^{2}] dH^{1} dπ = F (E).

If b ≥ 2a split F as

\begin{aligned}F (E) & = ∫_{∂E} [\frac{1}{2} + a\{f_{1}(∇^{2}d_{E}) + 2(f_{2}(∇^{2}d_{E}))^{2}\}] dH^{2} \\ & + ∫_{∂E} [\frac{1}{2} + (b − 2a)(f_{2}(∇^{2}d_{E}))^{2}] dH^{2} =: F_{1}(E) + F_{2}(E). \end{aligned}

Then F_{1} is lower semicontinuous by the previuos case and F_{2} is lower semicontinuous by Theorem 6.3 below. This concludes the proof.

The next remark shows the connection between the second fundamental form of ∂E and ∇^{2}d_{E}, and is useful in the proof of Theorem 6.3.

Remark 6.2. If B_{ij}^{k}, i, j, k ∈ {1, …, N}, denotes the second fundamental form of ∂E

with respect to the canonical basis of IR^{N}, then B_{ij}^{k} = −∇_{ij}^{2}d_{E}∇_{k}d_{E} on ∂E. In particular

∑_{i,j,k=1}^{N} |B_{ij}^{k}|^{2} = ∑_{i,j=1}^{N} |∇_{ij}^{2}d_{E}|^{2} on ∂E.

Indeed, following [19, Proposition 5.1.1 (i)], if δ_{ij} is the Kronecker delta, summing over repeated indices from 1 to N and using the fact that ∇d_{E} ∈ Ker(∇^{2}d_{E}) on ∂E, we get, at x ∈ ∂E,

\begin{aligned}B_{ij}^{k} & = −(δ_{lj} − ∇_{l}d_{E}∇_{j}d_{E})(δ_{ih} − ∇_{i}d_{E}∇_{h}d_{E})(∇_{kh}^{2}d_{E}∇_{l}d_{E} + ∇_{lh}^{2}d_{E}∇_{k}d_{E}) \\ & = −(δ_{lj} − ∇_{l}d_{E}∇_{j}d_{E})(∇_{ki}^{2}d_{E}∇_{l}d_{E} + ∇_{li}^{2}d_{E}∇_{k}d_{E}) = −∇_{ij}^{2}d_{E}∇_{k}d_{E}. \end{aligned}

Theorem 6.3. Let α > 1 and let ζ : [0, +∞[→ [0, +∞[ be a strictly convex function such that ct^{α} ≤ ζ(t) for any t ∈ [0, +∞[, for some constant c > 0. Let F : C_{b}^{∞}(IR^{N}) → [0, +∞] be the map defined by

F (E) = ∫_{∂E} [1 + ζ([ ∑_{i=1}^{N−1} κ_{i}^{2}]^{1/2})] dH^{N−1},  (6.4)

where κ_{1}, …, κ_{N−1} are the principal curvatures of ∂E. Then F is L^{1}(IR^{N})-lower semicontinuous on C_{b}^{∞}(IR^{N}).

Proof. Let E ∈ C_{b}^{∞}(IR^{N}) and let {E_{h}} be a sequence in C_{b}^{∞}(IR^{N}) converging to E in L^{1}(IR^{N}) as h → +∞. We must prove inequality (5.4). We can suppose, possibly passing to a subsequence, that the right hand side of (5.4) is a finite limit. Set

L(∂E_{h}) := ∫_{∂E_{h}} ζ([ ∑_{i=1}^{N−1}(κ_{i}^{h})^{2}]^{1/2}) dH^{N−1},

where κ_{1}^{h}, …, κ_{N−1}^{h} are the principal curvatures of ∂E_{h}. We have

sup_{h} H^{N−1}(∂E_{h}) < +∞,  sup_{h} L(∂E_{h}) < +∞,  (6.5)

hence, if H_{h} denotes the scalar mean curvature of ∂E_{h}, by the Hölder inequality and the properties of ζ we deduce that

∫_{∂E_{h}} |H_{h}| dH^{N−1} ≤ O(1) (H^{N−1}(∂E_{h}))^{1/α′} [L(∂E_{h})]^{1/α} = O(1),  (6.6)

where α^{−1} + (α′)^{−1} = 1. We need now some tools of geometric measure theory and we refer to the book of Simon [28]. We associate to each ∂E_{h} the integer (N −1)-dimensional rectifiable varifold (∂E_{h}, 1_{∂E_{h}}), which can be seen as a measure V_{h} on IR^{N} times the Grassmannian of the (N − 1)-dimensional linear subspaces of IR^{N}. The weight λ_{h} of V_{h} is the measure defined as the restriction of H^{N−1} to ∂E_{h}. Using (6.6) and [28, Theorem 42.7 and Remark 42.8] there exists an integer (N − 1)-dimensional rectifiable varifold V = (M, θ) with weight λ_{V} and a subsequence (still denoted by {V_{h}}) such that V_{h} ⇀ V in the sense of varifolds. We claim that

∂E ⊆ M.  (6.7)

Indeed, let x ∈ ∂E and let B_{r}(x) be the open ball centered at x with radius r > 0. Since V_{h} ⇀ V we have λ_{h} ⇀ λ_{V} weakly as measures. Using [15, Section 1.9], for any r > 0 such that λ_{V} (∂B_{r}(x)) = 0 (hence for all r > 0 up to a countable set) we have

λ_{V} (B_{r}(x)) = lim_{h→+∞} λ_{h}(B_{r}(x)) = lim_{h→+∞} H^{N−1}(B_{r}(x) ∩ ∂E_{h}) ≥ H^{N−1}(B_{r}(x) ∩ ∂E),

where the last inequality follows by using the L^{1}-lower semicontinuity of the perimeter of a set relatively to B_{r}(x) [15, Section 5.2.1]. Since M = spt(λ_{V}), we have x ∈ M, and this proves the claim. Observe also that by (6.5) and [19, Theorem 5.3.2] V is a varifold with generalised second fundamental form in L^{α} in the sense of Hutchinson, and

lim inf_{h→+∞} L(∂E_{h}) ≥ L(V),  (6.8)

where

L(V) := ∫_{M} ζ([ ∑_{i,j,k=1}^{N} |B_{ij}^{k}|^{2}]^{1/2})θ dH^{N−1},

and B_{ij}^{k} are defined in [19, Definition 5.2.5].

Recall now that the L^{1}(IR^{N})-lower semicontinuity of the perimeter implies that

lim inf_{h→+∞} H^{N−1}(∂E_{h}) ≥ H^{N−1}(∂E);

hence, in view of (6.8), to prove (5.4) it is enough to show that

L(V) ≥ L(∂E) := ∫_{∂E} ζ([ ∑_{i,j,k=1}^{N} |B_{ij}^{k}|^{2}]^{1/2}) dH^{N−1}.  (6.9)

We firstly remark that M is a measurable countably (H^{N−1}, N − 1)-rectifiable set. Hence by (6.7) the tangent space T_{x}^{∂E} to ∂E at x ∈ ∂E coincides with the approximate tangent space T_{x}^{M} to M at x ∈ ∂E for H^{N−1}-almost every x ∈ ∂E. Therefore the orthogonal projection P_{∂E} on T_{x}^{∂E} coincides with the orthogonal projection P_{M} on T_{x}^{M} for H^{N−1}- almost every x ∈ ∂E. We recall now that each B_{ij}^{k}(x), x ∈ M (resp. x ∈ ∂E) can be obtained by tangentially differentiating in the λ_{V} -approximate sense (resp. in the classical sense) the function P_{M} (resp. P_{∂E}), see [21, Theorem 5.4] and [19, Proposition 5.1.1 (i)]. Since P_{∂E} and P_{M} coincide H^{N−1}-almost everywhere on ∂E ∩ {θ = n}, n ∈ IN, n ≥ 1, they must have on ∂E ∩ {θ = n} the same approximate tangential differential H^{N−1}-almost everywhere. Hence by (6.7) and Remark 6.2

L(V) ≥ ∫_{∂E} ζ([ ∑_{i,j,k=1}^{N} |B_{ij}^{k}|^{2}]^{1/2}) θ dH^{N−1}

≥ ∑_{n=1}^{∞} ∫_{∂E∩{θ=n}} ζ([ ∑_{i,j,k=1}^{N} |B_{ij}^{k}|^{2}]^{1/2}) dH^{N−1}

= ∑_{i=1}^{∞} ∫_{∂E∩{θ=n}} ζ([ ∑_{i,j=1}^{N} |∇_{ij}^{2}d_{E}|^{2}]^{1/2})dH^{N−1} =∫_{∂E} ζ([ ∑_{i,j=1}^{N} |∇_{ij}^{2}d_{E}|^{2}]^{1/2})dH^{N−1},

which yields (6.9) and concludes the proof.

Remark 6.4. In view of Corollary 4.4 we have the following approximation results. Assume that F is as in Theorem 5.2, (6.1) and (6.4) respectively. Then

(Γ − lim_{h→+∞} F_{h})(u) = \{ \begin{matrix}2c_{0}F (u) & if u = 1_{E}, E ∈ C_{b}^{∞}(IR^{N}), \\ 2c_{0}\overline{F} (u) & elsewhere in L^{1}(IR^{N}), \end{matrix}

where N = 2, N = 3, N ≥ 2 respectively, F_{h} is defined in (4.3) with m ≡ 1, ϕ(x, ξ) = |ξ| and, concerning (6.4), f̃ = ζ((N − 1)f̃_{2}), where f_{2} is defined in Example 3.4.

Acknowledgements. I am grateful to Ennio De Giorgi and Luigi Ambrosio for some useful discussions.

References

[1] A. C. Barroso, I. Fonseca: Anisotropic singular perturbations-the vectorial case, Proceedings of the Royal Society of Edinburgh 124 A (1994) 527–571.

[2] G. Bellettini, G. Dal Maso, and M. Paolini: Semicontinuity and relaxation properties of a curvature depending functional in 2D, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 20 (1993) 247–297.

[3] G. Bellettini, M. Paolini: Variational properties of an image segmentation functional depending on contours curvature, Adv. Math. Sc. Appl. 5 (1995) 681–715.

[4] G. Bellettini, M. Paolini: Anisotropic motion by mean curvature in the context of Finsler geometry, Hokkaido Mathematical Journal 25 (1996) 537–566.

[5] G. Bellettini, M. Paolini, and S. Venturini: Some results on surface measures in Calculus of Variations, Ann. Mat. Pura Appl. CLXX (1996) 329–359.

[6] G. Bouchitté: Singular perturbations of variational problems arising from a two-phase transition model, Appl. Math. and Opt. 21 (1990) 289–315.

[7] Y.D. Burago, V.A. Zalgaller: Geometric inequalities, Springer-Verlag, Berlin, 1988.

[8] G. Buttazzo: Semicontinuity, Relaxation and Integral Representation in the Calculus of Variation, Longman Scientific & Technical, Harlow, 1989.

108 G. Bellettini / Variational approximation of some functionals ...

[9] C. Chen: On the kinematic formula of square of mean curvature vector, Indiana Univ. Math. Journal 22 (1973) 1163–1169.

[10] S.S. Chern: On the kinematic formula in integral geometry, J. Math. Mech. 16 (1966) 101–118.

[11] G. Dal Maso: An Introduction to Γ-Convergence, Birkhäuser, Boston, 1993.

[12] E. De Giorgi: Some remarks on Γ-convergence and least squares method, Proc. Composite media and homogeneization theory (Trieste, 1990, G. Dal Maso, G.F. Dell’Antonio, eds.), Birkhäuser, Boston (1991) 135–142.

[13] E. De Giorgi, T. Franzoni: Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58 (1975) 842–850.

[14] L. Euler: Additamentum I de curvis elasticis, methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, Opera omnia I, 24, Lausanne (1744) 231–297.

[15] L.C. Evans, R.F. Gariepy: Measure theory and fine properties of functions, CRC Press, 1992.

[16] H. Federer: Geometric Measure Theory, Springer-Verlag, Berlin, 1968.

[17] Y. Giga, S. Goto: Motion of hypersurfaces and geometric equations, J. Math. Soc. Japan 44 (1992) 99–111.

[18] D. Gilbarg, N.S. Trudinger: Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.

[19] J.E. Hutchinson: Second fundamental form for varifolds and the existence of surfaces minimizing curvature, Indiana Univ. Math. J. 35, no.1 (1986) 45–71.

[20] A. E. H. Love: A Treatise on the Mathematical Theory of Elasticity, Dover, New York, 1944.

[21] C. Mantegazza: Curvature varifolds with boundary, Preprint Scuola Normale Superiore di Pisa, to appear on J. Diff. Geom.

[22] L. Modica: Gradient theory of phase transitions with boundary contact energy, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987) 487–512.

[23] L. Modica, S. Mortola: Un esempio di Γ-convergenza, Boll. Un. Mat. Ital. B (5) 14 (1977) 285–299.

[24] D. Mumford, M. Nitzberg: The 2.1-D sketch, Dec. 1990, International Conference on Computer Vision, Computer Society Press, IEEE.

[25] D. Mumford, M. Nitzberg, and T. Shiota: Filtering, Segmentation and Depth, Springer-Verlag, Berlin, 1993.

[26] N. Owen, P. Sternberg: Non convex variational problems with anisotropic perturbations, Nonlinear Anal. 16 (1991) 705–719.

[27] R.T. Rockafellar: Convex Analysis, Princeton University Press, Princeton, New Jersey, 1972.

[28] L.A. Santaló: Integral Geometry and Geometric Probability, Addison-Wesley Publishing Company, London, 1976.

[29] L. Simon: Lectures on Geometric Measure Theory, Proceedings of the Centre of Mathematical Analysis, Canberra, 3, 1984.

[30] T.J. Willmore: Riemannian Geometry, Clarendon Press, Oxford, 1993.