The BV -energy of maps into a manifold: relaxation and density results

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1 The BV -energy of maps into a manifold: relaxation and density results Mariano Giaquinta and Domenico Mucci Abstract. Let Y be a smooth compact oriented Riemannian manifold without boundary, and assume that its 1- homology group has no torsion. Wea limits of graphs of smooth maps u : B n Y with equibounded total variation give rise to equivalence classes of Cartesian currents in cart 1,1 (B n Y) for which we introduce a natural BV -energy. Assume moreover that the first homotopy group of Y is commutative. In any dimension n we prove that every element T in cart 1,1 (B n Y) can be approximated wealy in the sense of currents by a sequence of graphs of smooth maps u : B n Y with total variation converging to the BV -energy of T. As a consequence, we characterize the lower semicontinuous envelope of functions of bounded variations from B n into Y. In this paper we deal with sequences of smooth maps u : B n Y with equibounded total variation sup E 1,1 (u ) <, E 1,1 (u ) := Du dx B n and their limit points. Here B n is the unit ball in R n and Y is a smooth oriented Riemannian manifold of dimension M 1, isometrically embedded in R N for some N 2. We shall assume that Y is compact, connected, without boundary. In addition, we assume that the integral 1-homology group H 1 (Y) := H 1 (Y; Z) has no torsion. Modulo passing to a subsequence the (n, 1)-currents G u, integration over the graphs of u of n-forms with at most one vertical differential, converge to a current T cart 1,1 (B n Y), see Sec. 2 below. To every T cart 1,1 (B n Y) it corresponds a function u T BV (B n, Y), i.e., u T BV (B n, R N ) such that u T (x) Y for L n -a.e. x B n, compare [14, Vol. I, Sec. 4.2] [14, Vol. II, Sec. 5.4]. Also, the wea convergence G u T yields the convergence u u T wealy in the BV -sense. In order to analyze the wea limit currents, it is relevant first to consider the case n = 1. Therefore in Sec. 1 we study some of the structure properties of 1-dimensional Cartesian currents in B 1 Y, i.e., of currents in cart(b 1 R N ) with support spt T B 1 Y, compare [14, Vol. I]. In the simple case Y = S 1, the unit circle in R 2, and in any dimension n, for any current T cart(b n S 1 ) we can find a sequence of smooth maps {u } C 1 (B n, S 1 ) such that G u wealy converges to T and the area of the graph of the u s converges to the mass of T, i.e., M(G u ) M(T ), see [13] and [14, Vol. II, Sec ]. However, in case of general target manifolds, and even in dimension n = 1, a gap phenomenon occurs. More precisely, setting M(T ) := inf{lim inf M(G u ) {u } C 1 (B 1, Y), G u T wealy in D 1 (B 1 Y)}, there exist currents T cart(b 1 Y) for which M(T ) < M(T ), i.e., for every smooth sequence {u } C 1 (B 1, Y) such that G u T wealy in D 1 (B 1 Y) we have that lim inf M(G u ) M(T ) + C, where C > 0 is an absolute constant and, we recall, the mass of G u is the area of the graph of u M(G u ) = A(u ) := 1 + Du 2 dx. B 1 1

2 In order to deal with this gap phenomenon, we introduce the class cart 1,1 (B 1 Y) of equivalence classes of currents in cart(b 1 Y), where the equivalence relation is given by T T T (ω) = T (ω) ω Z 1,1 (B 1 Y), see Definition 1.6. Here Z 1,1 (B 1 Y) denotes the class of smooth forms ω D 1 (B 1 Y) such that d y ω (1) = 0, where d = d x + d y denotes the splitting into a horizontal and a vertical differential, and ω (1) is the component of ω with exactly one vertical differential. In other words cart 1,1 (B 1 Y) is the class of vertical homological representatives of the elements of cart(b 1 Y). Notice that if Y = S 1, actually cart 1,1 (B 1 S 1 ) agrees with the class cart(b 1 S 1 ). We then introduce on cart 1,1 (B 1 Y) the following energy A(T ) := 1 + ut (x) 2 dx + D C u T (B 1 ) + L T (x), B 1 x J c (T ) where u T and D C u T are respectively the absolutely continuous and the Cantor part of the distributional derivative of the underlying function u T BV (B 1, Y), and the countable set J c (T ) is the union J c (T ) := J ut {x i : i = 1,..., I} of the discontinuity set J ut of u T and of the finite set of points x i where the mass of T concentrates. In the above formula, L T (x) denotes the minimal length L(γ) among all Lipschitz curves γ : [0, 1] Y, with end points equal to the one-sided approximate limits of u T on x J c (T ), such that their image current γ # [[ (0, 1) ]] is equal to the 1-dimensional restriction π # (T {x} Y) of T over the point x. In the case Y = S 1, it turns out that A(T ) agrees with the mass of T, compare [13] and [14, Vol. II, Sec ]. We will show that the functional T A(T ) is lower semicontinuous in cart 1,1 (B 1 Y), Theorem 1.7, and that for every T there exists a sequence of smooth maps {u } C 1 (B 1, Y) such that G u T and M(G u ) A(T ) as, Theorem 1.8. As a consequence, we conclude that A(T ) coincides with the relaxed area functional Ã(T ) := inf{lim inf A(u ) {u } C 1 (B 1, Y), G u T }. In Sec. 2, we deal with the n-dimensional case, n 2, introducing the class cart 1,1 (B n Y) of vertical homological representatives. The BV -energy of a current T cart 1,1 (B n Y) is then defined by E 1,1 (T ) := u T (x) dx + D C u T (B n ) + L T (x) dh n 1 (x), B n J c (T ) see Definition 2.10, where J c (T ) is the countably H n 1 -rectifiable subset of B n given by the union of the Jump set J ut of u T and of the (n 1)-rectifiable set of mass-concentration of T. Finally, the integrand L T (x) is defined as above, by taing into account that the 1-dimensional restriction π # (T {x} Y) of T is well-defined for H n 1 -a.e. point x J c (T ). Notice that, if T = G u, where u : B n Y is smooth or at least in W 1,1, then E 1,1 (G u ) = E 1,1 (u). Moreover, in the case Y = S 1, we have cart 1,1 (B n S 1 ) = cart(b n S 1 ) and, due to the absence of gap phenomenon, the functional E 1,1 (T ) agrees with the parametric variational integral associated to the total variation integral, see Definition 2.5, and can be dealt with as in [13], see also [14, Vol. II, Sec. 6.2], [8], [19]. The functional T E 1,1 (T ) turns out to be lower semicontinuous in cart 1,1 (B n Y), see Theorem 2.12 and Sec. 3. Moreover, assuming in addition that the first homotopy group π 1 (Y) is commutative, in Sec. 4 and Sec. 5 we will prove in any dimension n 2 that for every T cart 1,1 (B n Y) there exists a sequence of smooth maps {u } C 1 (B n, Y) such that G u T and E 1,1 (u ) E 1,1 (T ) as, Theorem Consequently, we show that a closure-compactness property holds in cart 1,1 (B n Y), Theorem We stress that the commutativity hypothesis on π 1 (Y) cannot be removed, see Remar 5.2. In Sec. 6, extending the classical notion of total variation of vector-valued maps, compare e.g. [1], we introduce in a natural way the total variation of functions u BV (B n, Y), given by E T V (u) := u(x) dx + D C u (B n ) + H 1 (l x ) dh n 1 (x), B n J u 2

3 where, for any x J u, we let H 1 (l x ) denote the length of a geodesic arc l x in Y with initial and final points u (x) and u + (x). Extending the density result of Bethuel [5], in Theorem 6.5 we will show that for every u BV (B n, Y) we can find a sequence of maps {u } R1 (B n, Y) such that u u as wealy in the BV -sense and lim Du dx = E T V (u). B n If n = 1, the class R1 (B n, Y) agrees with C 1 (B n, Y). If n 2, it is given by all the maps u W 1,1 (B n, Y) which are smooth except on a singular set which is discrete, if n = 2, and is the finite union of smooth (n 2)-dimensional subsets of B n with smooth boundary, if n 3. Therefore, if π 1 (Y) = 0, we obtain that smooth maps in C 1 (B n, Y) are dense in BV (B n, Y) in the strong sense above mentioned. However, in Sec. 7 we will show that E T V (u) does not agree with the relaxed of the total variation { } Ẽ T V (u) := inf lim inf Du dx {u } C 1 (B n, Y), u u wealy in the BV -sense B n if n 2, and we have ẼT V (u) <, Theorem 7.3, and that Ẽ T V (u) = inf{e 1,1 (T ) T T u }, Theorem 7.4, where T u is the class of Cartesian current T in cart 1,1 (B n Y) with underlying BV -function u T equal to u, this way obtaining the representation formula { } Ẽ T V (u) = u(x) dx + D C u (B n ) + inf L T (x) dh n 1 (x) T T u. B n J c(t ) We finally specify the above relaxation results to u W 1,1 (B n, Y) and/or Y = S 1, recovering in particular previous results in [13], [8], and [19]. 1 Cartesian currents in dimension one In this section we discuss some features of 1-dimensional Cartesian currents in B 1 Y and, in particular, we discuss a gap phenomenon and the relaxed area functional. First let us introduce a few notation about BV -functions and Cartesian currents in the general context B n Y. Vector valued BV -functions. Let u : B n R N be a function in BV (B n, R N ), i.e., u = (u 1,... u N ) with all components u BV (B n ). The Jump set of u is the countably H n 1 -rectifiable set J u in B n given by the union of the complements of the Lebesgue sets of the u s. Let ν = ν u (x) be a unit vector in R n orthogonal to J u at H n 1 -a.e. point x J u. Let u ± (x) denote the one-sided approximate limits of u on J u, so that for H n 1 -a.e. point x J u lim ρ 0 + ρ n B ± ρ (x) u(x) u ± (x) dx = 0, where B ± ρ (x) := {y B ρ (x) : ± y x, ν(x) 0}. Note that a change of sign of ν induces a permutation of u + and u and that only for scalar functions there is a canonical choice of the sign of ν which ensures that u + (x) > u (x). The distributional derivative of u is the sum of a gradient measure, which is absolutely continuous with respect to the Lebesgue measure, of a ump measure, concentrated on a set that is σ-finite with respect to the H n 1 -measure, and of a Cantor-type measure. More precisely, Du = D a u + D J u + D C u, where D a u = u dx, D J u = (u + (x) u (x)) ν(x) H n 1 J u, 3

4 u := ( 1 u,..., n u) being the approximate gradient of u, compare e.g. [2] or [14, Vol. I]. We also recall that {u } is said to converge to u wealy in the BV -sense, u u, if u u strongly in L 1 (B n, R N ) and Du Du wealy in the sense of (vector-valued) measures. We will finally denote BV (B n, Y) := {u BV (B n, R N ) u(x) Y for L n -a.e. x B n }. Cartesian currents. The class of Cartesian currents cart(b n R N ), compare [14, Vol. I], is defined as the class of integer multiplicity (say i.m.) rectifiable currents T in R n (B n R N ) which have no inner boundary, T B n R N = 0, have finite mass, M(T ) <, and are such that where T 1 <, π # (T ) = [[ B n ]] and T 00 0, T 1 := sup{t (ϕ(x, y) y dx) ϕ C 0 c (B n R N ) and ϕ 1} and T 00 is the Radon measure in B n R N given by T 00 (ϕ(x, y)) = T (ϕ(x, y) dx) ϕ C 0 c (B n R N ). Finally, here and in the sequel π : R n+n R n and π : R n+n R N denote the proections onto the first n and the last N coordinates, respectively. It is shown in [14, Vol. I] that for every T cart(b n R N ) there exists a function u T BV (B n, R N ) such that T (φ(x, y) dx) = φ(x, u T (x)) dx B n (1.1) for all φ C 0 (B n R N ) such that φ(x, y) C (1 + y ), and ( 1) n i T (ϕ(x) dx i dy ) = D i u T, ϕ := B n u T (x) D iϕ(x) dx for all ϕ C 1 c (B n ), where dx i := dx 1 dx i 1 dx i 1 dx n. In particular, we have T 1 = u T L 1 (B n,r N ). Definition 1.1 If n = 1 we set cart(b 1 Y) := {T cart(b 1 R N ) spt T B 1 Y}. Notice that the class cart(b 1 Y) contains the wea limits of sequences of graphs of smooth maps u : B 1 Y with equibounded W 1,1 -energies. Moreover, it is closed under wea convergence in D 1 (B 1 Y) with equibounded masses. Finally, the BV -function u T associated to currents T in cart(b 1 Y) clearly belongs to BV (B 1, Y). Restriction over one point. Let T cart(b 1 Y). Since T has finite mass, η T (χ Br(x) η), where x B 1 and 0 < r < 1 x, defines a current in D 1 (Y). The 1-dimensional restriction of T over the point x π # (T {x} Y) D 1 (Y) is the limit π # (T {x} Y)(η) := lim r 0 + T (χ B r (x) η), η D 1 (Y). Canonical decomposition. There is a canonical way to decompose a current T cart(b 1 Y). We first observe that the 1-dimensional restriction of T over any point x in the ump set J ut of u T is given by π # (T {x} Y) = Γ x, Γ x being a 1-dimensional integral chain on Y such that Γ x = δ u + T (x) δ u (x), where u+ T (x) and u T (x) T here and in the sequel denote the right and left limits of u T at x, respectively. Therefore, by applying 4

5 Federer s decomposition theorem [9], we find an indecomposable 1-dimensional integral chain γ x on Y, satisfying γ x = δ u + T (x) δ u (x), and an integral 1-cycle C x in Y, satisfying C x = 0, such that T Γ x = γ x + C x and M(Γ x ) = M(γ x ) + M(C x ). (1.2) Currents associated to graphs of BV -functions. Next we associate to any T cart(b 1 Y) a current G T D 1 (B 1 Y) carried by the graph of the function u T BV (B 1, Y) corresponding to T, and acting in a linear way on forms ω in D 1 (B 1 Y) as follows. We first split ω = ω (0) + ω (1) according to the number of vertical differentials, so that ω (0) = φ(x, y) dx and ω (1) = N φ (x, y) dy for some φ, φ C0 (B 1 Y). We then decompose G T into its absolutely continuous, Cantor, and Jump parts G T := T a + T C + T J and define T C (ω (0) ) = T J (ω (0) ) = 0 and T a (ω (0) ) := φ(x, u T (x)) dx B N 1 T a (ω (1) ) := φ (x, u T (x)) u T (x) dx B 1 T C (ω (1) ) := T J (ω (1) ) := =1 N D C u T, φ (, u T ( )) =1 N =1 J ut ( γ x φ (x, y) dy =1 ) ν(x) dh 0 (x). Here, γ x is the indecomposable 1-dimensional integral chain defined by means of the 1-dimensional restriction of T over the point x J ut, see (1.2). Notice that the definition of G T obviously depends on γ x and hence, in conclusion, on the current T cart(b 1 Y). Moreover, we readily infer that the mass of G T is given by M(G T ) = M(T a ) + M(T C ) + M(T J ), where M(T a ) = 1 + ut (x) 2 dx, M(T C ) = D C u T (B 1 ), M(T J ) = H 1 (γ x ) dh 0 (x). B 1 J ut A density result. We recall from [14] that if u : B 1 Y is smooth, or at least e.g. u W 1,1 (B 1, Y), the current G u integration of 1-forms in D 1 (B 1 Y) over the rectifiable graph of u is defined in a wea sense by G u := (Id u) # [ B 1 ]], i.e., by letting G u (ω) = (Id u) # (ω) for every ω D 1 (B 1 Y), where (Id u)(x) := (x, u(x)). Moreover, the mass of G u agrees with the area A(u) of the graph of u M(G u ) = A(u) := 1 + Du(x) 2 dx. B 1 By a straightforward adaptation of the proof of Theorem 1.8 below, we readily obtain the following strong density result for the mass of G T. Proposition 1.2 For every T cart(b 1 Y) there exists a sequence of smooth maps {u } C 1 (B 1, Y) such that u u T wealy in the BV -sense, G u G T wealy in D 1 (B 1 Y) and M(G u ) M(G T ) as. 5

6 Vertical Homology. Let now Z 1,1 (B 1 Y) denote the class of vertically closed forms Z 1,1 (B 1 Y) := {ω D 1 (B 1 Y) d y ω (1) = 0}, where d = d x + d y denotes the splitting of the exterior differential d into a horizontal and a vertical differential. We say that T T wealy in Z 1,1 (B 1 Y) if T (ω) T (ω) for every ω Z 1,1 (B 1 Y). Homological vertical part. By Proposition 1.2, since by Stoes theorem G u G u G T, we obtain that G T B 1 Y = 0. B 1 Y = 0, whereas Remar 1.3 In higher dimension n 2 in general G T has a non-zero boundary, i.e., G T B n Y = 0, see Remar 2.2. Setting then S T := T G T, by (1.1) we infer that S T (φ(x, y) dx) = 0 and S T (dφ) = 0 for every φ C 0 (B 1 Y). Therefore, by homological reasons, since similarly to [14, Vol. II, Sec ] we infer that inf{m(c) C Z 1 (Y), C is non trivial in Y} > 0, S T = I δ xi C i on Z 1,1 (B 1 Y), i=1 where {x i : i = 1,..., I} is a finite disoint set of points in B 1, possibly intersecting the Jump set J ut, and C i is a non-trivial homological integral 1-cycle in Y. Notice that the integral 1-homology group H 1 (Y) is finitely generated. Remar 1.4 Setting S T,sing := T G T I δ xi C i, it turns out that S T,sing is nonzero only possibly on forms ω with non-zero vertical component, ω (1) 0, and such that d y ω (1) 0. Therefore, S T,sing is a homologically trivial i.m. rectifiable current in R 1 (B 1 Y). Consequently, setting for T cart(b 1 Y) T H := i=1 I δ xi C i, (1.3) T decomposes into the absolutely continuous, Cantor, Jump, Homological, and Singular parts, i=1 T = T a + T C + T J + T H + S T,sing. Gap phenomenon. However, a gap phenomenon occurs in cart(b 1 Y). More precisely, if we set M(T ) := inf{lim inf M(G u ) {u } C 1 (B 1, Y), G u T wealy in D 1 (B 1 Y)}, we see that there exist Cartesian currents T cart(b 1 Y) for which M(T ) < M(T ). For example, as in [14, Vol. I, Sec ], if T = G u + δ 0 C, where u P Y is a constant map and C Z 1 (Y) is a 1-cycle in Y, it readily follows that for every smooth sequence {u } C 1 (B 1, Y) such that G u T wealy in D 1 (B 1 Y) we have that where dist Y denotes the geodesic distance in Y. lim inf M(G u ) M(T ) + 2d, d := dist Y (P, spt C), 6

7 Remar 1.5 This gap phenomenon is due to the structure of the area integrand u 1 + Du 2, and it is typical of integrands with linear growth of the gradient, e.g., the total variation integrand u Du, since the images of smooth approximating sequences may have to connect the point P to the cycle C, this way paying a cost in term of the distance d. This does not happen e.g. for the Dirichlet integrand u 1 2 Du 2 in dimension 2, compare [15]. In this case, in fact, the connection from one point P to any 2-cycle C Z 2 (Y) can be obtained by means of cylinders of small 2-dimensional mapping area and, therefore, of small Dirichlet integral, on account of Morrey s ε-conformality theorem. Homological theory. In order to study the currents which arise as wea limits of graphs of smooth maps u : B 1 Y with equibounded total variations, sup Du L 1 <, the previous facts lead us to consider vertical homology equivalence classes of currents in cart(b 1 Y). More precisely, we give the following Definition 1.6 We denote by cart 1,1 (B 1 Y) the set of equivalence classes of currents in cart(b 1 Y), where T T T (ω) = T (ω) ω Z 1,1 (B 1 Y). If T T, then the underlying BV -functions coincide, i.e., u T = ue T. Therefore, we have T a = T a T C = T C, whereas in general T J T J. However, we have that and T J + T H = T J + T H on Z 1,1 (B 1 Y). Jump-concentration points. For future use, we let J c (T ) := J ut {x i : i = 1,..., I} (1.4) denote the set of points of ump and concentration, where the x i s are given by (1.3). We infer that J c (T ) is an at most countable set which does not depend on the representative T, i.e., J c (T ) = J c ( T ) if T T. By extending the notion of 1-dimensional restriction π # (T {x} Y) to equivalence classes, we infer that π # (T {x} Y) = 0 if x / J c (T ). As to ump-concentration points, letting if x J ut, with x x i, we infer that Z 1 (Y) := {η D 1 (Y) d y η = 0}, π # (T {x} Y) = γ x on Z 1 (Y), where γ x is the indecomposable 1-dimensional integral chain defined by (1.2), and if x = x i, see (1.4), π # (T {x} Y) = γ xi + C i on Z 1 (Y), where C i Z 1 (Y) is the non-trivial 1-cycle defined by (1.3), and γ xi = 0 if x i / J ut. Vertical minimal connection. For every Cartesian current T cart 1,1 (B 1 Y) and every point x J c (T ) we will denote by Γ T (x) := {γ Lip([0, 1], Y) γ(0) = u T (x), γ(1) = u+ T (x), γ # [[ (0, 1) ]](η) = π # (T {x} Y)(η) η Z 1 (Y)} (1.5) the family of all smooth curves γ in Y, with end points u ± T (x), such that their image current γ #[[ (0, 1) ]] agrees with the 1-dimensional restriction π # (T {x} Y) on closed 1-forms in Z 1 (Y). Moreover, we denote by L T (x) := inf{l(γ) γ Γ T (x)}, x J c (T ), (1.6) the minimal length of curves γ connecting the vertical part of T over x to the graph of u T. For future use, we remar that the infimum in (1.6) is attained, i.e., x J c (T ), γ Γ T (x) : L(γ) = L T (x). (1.7) 7

8 Relaxed area functional. We finally introduce the functional A(T, B) := 1 + ut (x) 2 dx + D C u T (B) + for every Borel set B B 1, and we let Notice that for every T cart 1,1 (B 1 Y) we have B A(T ) := A(T, B 1 ). J c (T ) B L T (x) dh 0 (x) min{m( T ) : T T } A(T ). (1.8) Main results. We first prove the following lower semicontinuity property. Theorem 1.7 Let T cart 1,1 (B 1 Y). For every sequence of smooth maps {u } C 1 (B 1, Y) such that G u T wealy in Z 1,1 (B 1 Y), we have Then we prove the following density result. lim inf M(G u ) A(T ). Theorem 1.8 Let T cart 1,1 (B 1 Y). There exists a sequence of smooth maps {u } C 1 (B 1, Y) such that G u T wealy in Z 1,1 (B 1 Y) and M(G u ) A(T ) as. As a consequence, if we denote, in the same spirit as Lebesgue s relaxed area, Ã(T ) := inf{lim inf A(u ) {u } C 1 (B 1, Y), G u T wealy in Z 1,1 (B 1 Y)}, by Theorems 1.7 and 1.8 we readily conclude that A(T ) = Ã(T ) T cart1,1 (B 1 Y). Properties. From Theorems 1.7 and 1.8, (1.8) and the closure of the class cart(b 1 Y) we infer: (i) the functional T A(T ) is lower semicontinuous in cart 1,1 (B 1 Y) w.r.t. the wea convergence in Z 1,1 (B 1 Y); (ii) the class cart 1,1 (B 1 Y) is closed and compact under wea convergence in Z 1,1 (B 1 Y) with equibounded A-energies. We finally notice that similar properties hold if one considers the total variation integrand u Du instead of the area integrand u 1 + Du 2. In particular, setting E 1,1 (T ) := u T (x) dx + D C u T (B 1 ) + L T (x) dh 0 (x), B 1 J c (T ) for every T cart 1,1 (B 1 Y) we have { } E 1,1 (T ) = inf lim inf Du dx {u } C 1 (B 1, Y), B 1 G u T wealy in Z 1,1 (B 1 Y). Remar 1.9 For future use, we denote Y ε := {y R N dist(y, Y) ε} the ε-neighborhood of Y and we observe that, since Y is smooth, there exists ε 0 > 0 such that for 0 < ε ε 0 the nearest point proection Π ε of Y ε onto Y is a well defined Lipschitz map with Lipschitz constant L ε 1 + as ε 0 +. Note that for 0 < ε ε 0 the set Y ε is equivalent to Y in the sense of the algebraic topology. In particular, we have π 1 (Y ε ) = π 1 (Y). 8

9 Proof of Theorem 1.7: Let {x i } i>i B 1 be the at most countable set of discontinuity points in J ut \ {x i : i = 1,..., I}, see (1.4). By the properties of Y we have L T (x i ) C u + T (x i) u T (x i) i > I, where C = C(Y) > 0 is an absolute constant, see (1.6). Therefore, since D J u T (B 1 ) = for every ε > 0 we find l(ε) > I such that u + T (x i) u T (x i) <, i=1 i=l(ε)+1 L T (x i ) < ε. (1.9) After rearranging in an increasing way the set {x i : i l(ε)}, and setting x 0 = 1, x l(ε)+1 = 1, we let 2δ = 2δ(ε) := min{ x i x i+1 : i = 0,..., l(ε)} > 0. For i {1,..., l(ε)}, due to the wea convergence u u T in the BV -sense, possibly passing to a subsequence, we find the existence of sequences of points a i ]x i δ/, x i [ and b i ]x i, x i + δ/[ such that dist Y ( u (a i ), u T (x i) ) < 1 and dist Y ( u (b i ), u + T (x i) ) < 1 (1.10) for every, where dist Y denotes the geodesic distance in Y. Let γ i : [0, 1] Y be the Lipschitz reparametrization with constant velocity of the smooth curve u [a i,b i ]. From the wea convergence G u T we infer that γ i #[[ (0, 1) ]](η) π # (T {x} Y)(η) η Z 1 (Y) (1.11) as, where π # (T {x} Y) is the previously defined restriction of T over x. Moreover, by connecting the end points u (a i ) and u (b i ) with u T (x i) and u + T (x i), respectively, due to (1.10) we find a sequence of Lipschitz arcs γ i : [0, 1] Y, with end points γi (0) = u T (x i) and γ i (1) = u+ T (x i), such that ( γ i # [ (0, 1) ]] γ i # [[ (0, 1) ]]) (η) 0 for every η Z 1 (Y) as and L( γ i ) L(γ i ) + 2. By the construction we also infer that { γ i } is a sequence of equibounded and equicontinuous maps. Therefore, by Ascoli s theorem, possibly passing to a subsequence, we find that γ i converges uniformly to a Lipschitz arc γ i : [0, 1] Y, with end points u T (x i), satisfying by (1.11) γ i #[[ (0, 1) ]](η) = π # (T {x} Y)(η) η Z 1 (Y). We then obtain that γ i Γ T (x i ), according to the definition (1.5). Moreover, by the lower semicontinuity of the length functional w.r.t. the uniform convergence, we have L( γ i ) lim inf L( γi ). By (1.6) and by the above estimates we conclude that Now, since by the wea BV -convergence of u u T L T (x i ) lim inf L(γi ) i = 1,..., l(ε). (1.12) we have B ut (x) 2 dx + D C u T (B 1 ) lim inf A(u ), 9

10 by the previous argument, taing into account (1.9) and (1.12), we readily infer that and hence the assertion, by letting ε 0. A(T ) ε lim inf A(u ) Proof of Theorem 1.8: Let {x i } i>i, l(ε) and δ = δ(ε) be defined as in the proof of Theorem 1.7, so that (1.9) holds true. Let γ i Γ T (x i ) be such that L(γ i ) L T (x i ) + ε 2 i, see (1.5) and (1.6). For fixed δ (0, δ(ε)), and for every i = 1,..., l(ε), we first define u ε δ : [x i δ, x i + δ] Y by reparametrising with the same orientation the arc γ i, i.e., ( 1 u ε δ(x) := γ i ) 2δ (x x i). Setting I i :=]x i + δ, x i+1 δ[ if i = 1,..., l(ε) 1, and I 1 :=] 1, x 1 δ[, I l(ε) :=]x l(ε) + δ, 1[, we then extend u ε δ to the whole of B1 by letting u ε δ (x) := u T (Ψ i (x)) if x I i for some i = 0,..., l(ε), where Ψ i is the biective and increasing affine map between the intervals I i and ]x i, x i+1 [. We then apply a mollification procedure to the function u ε δ, defining this way a smooth map vε δ : B1 R N such that vδ ε u ε δ L 1 (B 1 ) δ and Dvδ ε dx Du ε δ (B 1 ) + δ. B 1 Since u T is continuous outside the Jump set J ut and (1.9) holds true, for every σ > 0 we find η = η(σ, δ, ε) > 0 such that, in the a.e. sense, As a consequence, we may and do define v ε δ x, y B 1, x y < η = u ε δ(x) u ε δ(y) < σ + ε. in such a way that in particular dist(v ε δ(x), Y) < ε x B 1. Setting now wδ ε := Π ε vδ ε : B1 Y, compare Remar 1.9, taing first δ small w.r.t. ε, and letting then ε 0, by a diagonal procedure we find a smooth approximating sequence. 2 Cartesian currents, BV -energy and wea limits In this section we deal with the wea limits of graphs of smooth maps u : B n Y with equibounded W 1,1 -energies. We first state a few preliminary results. Homological facts. Since H 1 (Y) has no torsion, there are generators [γ 1 ],..., [γ s ], i.e. integral 1-cycles in Z 1 (Y), such that { s } H 1 (Y) = n s [γ s ] n s Z, s=1 see e.g. [14], Vol. I, Sec By de Rham s theorem the first real homology group is in duality with the first cohomology group HdR 1 (Y), the duality being given by the natural pairing [γ], [ω] := γ(ω) = ω, [γ] H 1 (Y; R), [ω] HdR(Y) 1. γ We will then denote by [ω 1 ],..., [ω s ] a dual basis in HdR 1 (Y) so that γ s(ω r ) = δ sr, where δ sr Kronecer symbols. denotes the D n,1 -currents. For p = 1,..., n, every differential p-form ω D p (B n Y) splits as a sum ω = p =0 ω(), where p := min(p, M), M = dim(y), and the ω () s are the p-forms that contain exactly differentials in 10

11 the vertical Y variables. We denote by D p,1 (B n Y) the subspace of D p (B n Y) of p-forms of the type ω = ω (0) +ω (1), and by D p,1 (B n Y) the dual space of D p,1 (B n Y). Every (p, 1)-current T D p,1 (B n Y) splits as T = T (0) + T (1), where T () (ω) := T (ω () ). For example, if u W 1,1 (B n, Y), then G u is an (n, 1)- current in D n,1 (B n Y) defined in an approximate sense by where (Id u)(x) := (x, u(x)), compare [14], see also [4]. G u := (Id u) # [[ B n ]], (2.1) Wea D n,1 -convergence. If {T } D n,1 (B n Y), we say that {T } converges wealy in D n,1 (B n Y), T T, if T (ω) T (ω) for every ω D n,1 (B n Y). Trivially, the class D n,1 (B n Y) is closed under wea convergence. E 1,1 -norm. For ω D n,1 (B n Y) and T D n,1 (B n Y) we set { ω (0) } (x, y) ω E1,1 := max sup, sup ω (1) (x, y) dx, { x,y 1 + y B n y } T E1,1 := sup T (ω) ω D n,1 (B n Y), ω E1,1 1. It is not difficult to show that T E1,1 is a norm on {T D n,1 (B n Y) : T E1,1 < }. Moreover, E1,1 is wealy lower semicontinuous in D n,1, so that {T D n,1 (B n Y) : T E1,1 < } is closed under wea D n,1 -convergence with equibounded E 1,1 -norms. Finally, if sup T E1,1 < there is a subsequence that wealy converges to some T D n,1 (B n Y) with T E1,1 <. Boundaries. The exterior differential d splits into a horizontal and a vertical differential d = d x + d y. Of course x T (ω) := T (d x ω) defines a boundary operator x : D n,1 (B n Y) D n 1,1 (B n Y). Now, for any ω D n 1,1 (B n Y), d y ω belongs to D n,1 (B n Y) if and only if d y ω (1) = 0. Then y T maes sense only as an element of the dual space of Z n 1,1 (B n Y), where Z p,1 (B n Y) := {ω D p,1 (B n Y) d y ω (1) = 0}. Graphs of BV -maps. We introduce a class of D n,1 -currents associated to the graphs of BV -functions. To this aim, we observe that any form ω = ω (1) D n,1 (B n Y) can be written as ω (1) = n i=1 =1 for some φ i C 0 (B n Y), and we will set φ := (φ 1,..., φ n). N ( 1) n i φ i (x, y) dx i dy (2.2) Definition 2.1 We say that a current G D n,1 (B n Y) is in BV graph(b n Y) if it decomposes into its absolutely continuous, Cantor, and Jump parts G := G a + G C + G J, where G C (0) = GJ (0) = 0, and its action on forms in Dn,1 (B n Y) is given for any φ Cc (B n Y) by G(φ(x, y) dx) = G a (φ(x, y) dx) := φ(x, u(x)) dx B n for some function u = u(g) BV (B n, Y) and, on forms ω = ω (1) G a (ω (1) ) := G C (ω (1) ) := G J (ω (1) ) := N u, φ (x, u(x)) dx =1 B n N φ (x, u(x)) dd C u =1 B n N n ( φ i (x, y) dy γ x =1 i=1 J u satisfying (2.2), by ) ν i dh n 1 (x), 11

12 where γ x is a 1-dimensional integral chain in Y satisfying γ x = δ u + (x) δ u (x) and ν = (ν 1,..., ν n ) is the unit normal to J u at x, for H n 1 -a.e. x J u. Remar 2.2 If n 2 in general the current G has a non-zero boundary in B n Y, even if u W 1,1 (B n, Y), i.e., if G = G a. Tae for example n = 2, Y = S 1 R 2, and u(x) = x/ x, so that G = G u := (Id u) # [[ B 2 ]] and hence G B 2 S 1 = δ 0 [[ S 1 ]], where δ 0 is the unit Dirac mass at the origin. However, as we shall see in Remar 6.10 below, the boundary G is null on every (n 1)-form ω in B n Y which has no vertical differentials. Wea limits of smooth graphs. Let {u } C 1 (B n, Y) be a sequence of smooth maps with equibounded W 1,1 -energies, sup Du L 1 <. The currents G u carried by the graphs of the u s are well defined currents in D n,1 (B n Y) with equibounded E 1,1 -norms. Therefore, possibly passing to a subsequence, we infer that G u T wealy in D n,1 (B n Y) to some current T D n,1 (B n Y), and u u T wealy in the BV -sense to some function u T BV (B n, Y). Therefore, we clearly have that T (φ(x, y) dx) = φ(x, u T (x)) dx φ Cc (B n Y). (2.3) B n Moreover, by lower semicontinuity we have T E1,1 B n Y, by the wea convergence we also infer < whereas, since the G u s have no boundary in T = 0 on Z n 1,1 (B n Y). (2.4) Currents associated to graphs of BV -functions. Arguing as in Sec. 1, we associate to the wea limit current T a current G T BV graph(b n Y), see Definition 2.1, where the function u = u(g T ) BV (B n, Y) is given by u T and the γ x s in the definition of the ump part G J T are the indecomposable 1- dimensional integral chains defined as in the previous section, but for H n 1 -a.e. x J ut, since T E1,1 <, compare (1.2) and Definition 2.8 below. In general G T B n Y = 0. However, setting S T := T G T, we clearly have S T (φ(x, y) dx) = 0 for every φ C c (B n Y). Moreover, we also have: Proposition 2.3 S T (ω) = 0 for every form ω = ω (1) such that ω = d y ω for some ω D n 1,0 (B n Y). Proof: Write ω := ω ϕ η for some η C0 (Y) and ϕ = (ϕ 1,..., ϕ n ) C0 (B n, R n ), where n ω ϕ := ( 1) i 1 ϕ i (x) dx i. (2.5) i=1 Since d(ω ϕ η) = divϕ(x)η(y) dx + ( 1) n 1 ω ϕ d y η and T (d(ω ϕ η)) = T (ω ϕ η) = 0, we have ( 1) n T (divϕ(x)η(y) dx) = T (ω ϕ d y η), so that S T (ω ϕ d y η) = ( 1) n T (divϕ(x)η(y) dx) G T (ω ϕ d y η). Moreover, since T (0) = G T (0), by (2.3) we have T (divϕ(x)η(y) dx) = divϕ(x)η(u T (x)) dx = D(η u T ), ϕ B n 12

13 whereas, taing φ i = ϕi D y η in (2.2), by the definition of G T, since γ x = δ u + T (x) δ u T (x) ( 1) n 1 G T (ω ϕ d y η) = + + N =1 N =1 B n B n η y (u T (x)) u T (x), ϕ(x) dx η y (u T (x)) ϕ(x) dd C u T Finally, by the chain rule for the derivative D(η u T ) we obtain J ut ( η(u + T (x)) η(u T (x)) ϕ(x), ν(x) dh n 1. ( 1) n 1 G T (ω ϕ d y η) = D(η u T ), ϕ we infer and hence that S T (ω ϕ d y η) = 0. In conclusion, similarly to [14], Vol. II, Sec , we infer that the wea limit current T is given by T = G T + S T, S T = where Ls(T ) D n 1 (B n ) is defined by so that Notice that by (2.4) we have Finally, setting s Ls(T ) γ s on Z n,1 (B n Y), (2.6) s=1 Ls(T ) = ( 1) n 1 π # (S T π # ω s ), s = 1,..., s, (2.7) Ls(T )(φ) = S T (π # φ π # ω s ) φ D n 1 (B n ). Ls(T ) B n = ( 1) n 1 π # (( G T ) π # ω s ) s = 1,..., s. S T,sing := T G T s Ls(T ) γ s, (2.8) see Remar 1.4, it turns out that S T,sing is nonzero only possibly on forms ω with non-zero vertical component, ω (1) 0, and such that d y ω (1) 0. Parametric polyconvex l.s.c. extension of the total variation. Following [14], Vol. II, Sec. 1.2, we recall that the parametric polyconvex l.s.c. extension T V of the total variation integrand of mappings from B n to R N has the form s=1 ξ T V := ξ (1) ξ Λ n R n+n such that ξ 00 0, (2.9) where ξ 00 denotes the coefficient of the first component of any n-vector ξ Λ n R n+n and ξ (1) is the euclidean norm of the component ξ (1) of ξ in Λ n 1 R n Λ 1 R N. We have Proposition 2.4 The parametric polyconvex l.s.c. extension F (x, u, ξ) : B n R N Λ n R n+n R + total variation integrand of mappings from B n into any smooth manifold Y R N is given by { ξ T F (x, u, ξ) := V if u Y, ξ Λ n (R n T u Y) + otherwise, of the (2.10) where ξ T V is given by (2.9) and T u Y is the tangent space to Y at u. 13

14 Parametric total variation. If T D n,1 (B n Y) is such that T E1,1 <, we denote by T = T E1,1 T the Radon-Niodym decomposition of T with respect to the E 1,1 -norm, T being identified with the R 1+Nn - valued linear functional T := ( T 00, (T i ) R Nn), i = 1,... n, = 1,... N, where T 00 (φ) := T (φ dx), T i (φ) := T (φ dx i dy ), φ C 0 (B n Y). Definition 2.5 The parametric variational integral associated to the total variation integral is defined for every Borel set B B n by F 1,1 (T, B Y) := F ( π(z), π(z), T (z) ) d T E1,1 (z) B Y where F (x, u, ξ) is given by (2.10), and we let F 1,1 (T ) := F 1,1 (T, B n Y). Gap phenomenon. If T D n,1 (B n Y) is the wea limit of a sequence {G u } of graphs of smooth maps {u } C 1 (B n, Y) with equibounded W 1,1 -energies, since F 1,1 (G u ) = Du L 1, by the lower semicontinuity of F 1,1 with respect to the wea convergence in D n,1 we infer that F 1,1 (T ) <. Moreover, if T decomposes as in (2.6) on the whole of D n,1 (B n Y), i.e., the singular part S T,sing defined in (2.8) vanishes, and if the Ls(T ) s are i.m. rectifiable currents, an explicit formula can be obtained. However, similarly to the case of dimension n = 1, a gap phenomenon occurs. More precisely, in general for every smooth sequence {u } C 1 (B n, Y) such that G u T wealy in D n,1 (B n Y) we have that for some absolute constant C > 0, see Remar 1.5. lim inf F 1,1(G u ) F 1,1 (T ) + C Vertical homology classes. As in Definition 1.6, we are therefore led to consider vertical homology equivalence classes of currents satisfying the same structure properties as wea limits of graphs of smooth maps u : B n Y with equibounded total variation, sup Du L 1 <. More precisely, we say that T T T (ω) = T (ω) ω Z n,1 (B n Y). (2.11) Moreover, we will say that T T wealy in Z n,1 (B n Y) if T (ω) T (ω) for every ω Z n,1 (B n Y). Definition 2.6 We denote by E 1,1 graph(b n Y) the set of equivalence classes, in the sense of (2.11), of currents T in D n,1 (B n Y) which have no interior boundary, T = 0 on Z n 1,1 (B n Y), finite E 1,1 -norm, i.e. T E1,1 { } := sup T (ω) ω Z n,1 (B n Y), ω E1,1 1 <, and decompose as T = G T + S T, S T = s Ls(T ) γ s on Z n,1 (B n Y), s=1 where G T BV graph(b n Y), see Definition 2.1, and Ls(T ) is an i.m. rectifiable current in R n 1 (B n ) for every s. 14

15 Remar 2.7 If T T, in general GeT G T. However, the corresponding BV -functions coincide, i.e., u(g T ) = u(ge T ), see Definition 2.1. This yields that we may refer to the underlying functions u T BV (B n, Y) associated to currents T in E 1,1 graph(b n Y). Jump-concentration set. Moreover, if L(T ) denotes the (n 1)-rectifiable set given by the union of the sets of positive multiplicity of the Ls(T ) s, we infer that the union J c (T ) := J ut L(T ) (2.12) does not depend on the choice of the representative in T. As in dimension one, the countably H n 1 -rectifiable set J c (T ) is said to be the set of points of ump-concentration of T. Restriction over points of ump-concentration. Let T E 1,1 graph(b n Y) and let ν T : J c (T ) S n 1 denote an extension to J c (T ) of the unit normal ν ut to the Jump set J ut. For any = 1,..., n 1, let P be an oriented -dimensional subspace in R n and P λ := P + n i=1 λ iν i the family of oriented -planes parallel to P, where λ := (λ 1,..., λ n ) R n, span(ν 1,..., ν n ) being the orthogonal space to P. Since T has finite E 1,1 -norm, similarly to the case of normal currents, for L n -a.e. λ such that P λ B n, the slice T π 1 (P λ ) of T over π 1 (P λ ) is a well defined -dimensional current in E 1,1 graph((b n P λ ) Y) with finite E 1,1 -norm. Moreover, for any such λ we have J c (T π 1 (P λ )) = J c (T ) P λ in the H 1 -a.e. sense, whereas the BV -function associated to T π 1 (P λ ) is equal to the restriction u T Pλ of u T to P λ. Therefore, in the particular case = 1, as in Sec. 1 the 1-dimensional restriction π # ( (T π 1 (P λ )) {x} Y ) D 1 (Y) (2.13) of the 1-dimensional current T π 1 (P λ ) over any point x J c (T ) P λ such that ν T (x) does not belong to P is well defined. In this case, from the slicing properties of BV -functions, if x (J c (T ) \ J ut ) P λ we have u T Pλ (x) = u T (x). Moreover, if x J ut P λ, the one-sided approximate limits of u T are equal to the one-sided limits of the restriction u T Pλ, i.e. u + T P λ (x) = u + T (x) and u T P λ (x) = u T (x), provided that ν, ν ut (x) > 0, where ν is an orienting unit vector to P, compare Theorem 3.2. We finally infer that for H n 1 -a.e. point x J c (T ) the 1-dimensional restriction (2.13), up to the orientation, does not depend on the choice of the oriented 1-space P and on λ R n 1, provided that x P λ and ν T (x) does not belong to P. As a consequence we may and do give the following Definition 2.8 For H n 1 -a.e. point x J c (T ), the 1-dimensional restriction π # (T {x} Y) is welldefined by (2.13) for any oriented 1-space P and λ R n 1 such that x P λ and ν, ν T (x) > 0, where ν is the orienting unit vector to P. BV -energy. The gap phenomenon and the properties previously described lead us to define the BV -energy of a current T E 1,1 graph(b n Y) as follows. Definition 2.9 For H n 1 -a.e. point x J c (T ) we define Γ T (x) and L T (x) by (1.5) and (1.6), respectively, where this time π # (T {x} Y) is the 1-dimensional restriction given by Definition 2.8. Definition 2.10 The BV -energy of a current T E 1,1 graph(b n Y) is defined for every Borel set B B n by E 1,1 (T, B Y) := u T (x) dx + D C u T (B) + L T (x) dh n 1 (x). B J c(t ) B We also let E 1,1 (T ) := E 1,1 (T, B n Y). 15

16 Of course, if T = G u is the current integration of n-forms in D n,1 (B n Y) over the graph of a smooth W 1,1 -function u : B n Y, then E 1,1 (u) = E 1,1 (G u ) = Du L 1. Definition 2.11 We denote by cart 1,1 (B n Y) the class of currents T in E 1,1 graph(b n Y) such that E 1,1 (T ) <. Lower semicontinuity. Using the lower semicontinuity result in dimension n = 1, see Theorem 1.7, and applying arguments as for instance in [7], in Sec. 3 we will prove in any dimension Theorem 2.12 Let n 2 and T cart 1,1 (B n Y). For every sequence of smooth maps {u } C 1 (B n, Y) such that G u T wealy in Z n,1 (B n Y), we have lim inf E 1,1(u ) E 1,1 (T ). A strong density result. In all the results stated below, we shall always assume that the first homotopy group π 1 (Y) is commutative. We shall prove in any dimension n 2 Theorem 2.13 Let T cart 1,1 (B n Y). There exists a sequence of smooth maps {u } C 1 (B n, Y) such that G u T wealy in Z n,1 (B n Y) and E 1,1 (u ) E 1,1 (T ) as. More precisely, in Sec. 4 we will prove Theorem 2.14 Let T cart 1,1 (B n Y). We can find a sequence of currents {T } cart 1,1 (B n Y) such that T T wealy in Z n,1 (B n Y), E 1,1 (T ) E 1,1 (T ) and for all the corresponding function u := u T in BV (B n, Y) has no Cantor part, i.e, D C u = 0 for every. Moreover, u wealy converges to u T in the BV -sense and In Sec. 5 we will then prove lim Du (B n ) = Du T (B n ). Theorem 2.15 Let T cart 1,1 (B n Y) be such that the corresponding BV -function u T BV (B n, Y) has no Cantor part, i.e, D C u T = 0. There exists a sequence of smooth maps {u } C 1 (B n, Y) such that G u T wealy in Z n,1 (B n Y) and the energy E 1,1 (u ) E 1,1 (T ) as. By a diagonal argument we then clearly obtain Theorem Relaxed total variation functional. As a consequence, setting { } Ẽ 1,1 (T ) := inf lim inf Du dx : {u } C 1 (B n, Y), B n G u T wealy in Z n,1 (B n Y), by Theorems 2.12 and 2.13 we conclude that E 1,1 (T ) = Ẽ1,1(T ) T cart 1,1 (B n Y). Properties. By Theorems 2.12 and 2.13 we readily infer the following lower semicontinuity result. Proposition 2.16 Let {T } cart 1,1 (B n Y) converge wealy in Z n,1 (B n Y), T T, to some T cart 1,1 (B n Y). Then E 1,1 (T ) lim inf E 1,1(T ). As a consequence of Theorem 2.13, in the final part of this section we prove that the class of Cartesian currents cart 1,1 (B n Y) is closed under wea convergence with equibounded energies. Theorem 2.17 Let {T } cart 1,1 (B n Y) converge wealy in Z n,1 (B n Y), T T, to some T D n,1 (B n Y), and sup E 1,1 (T ) <. Then T cart 1,1 (B n Y). 16

17 By the relative compactness of E 1,1 -bounded sets in D n,1 (B n Y), we then readily infer the following compactness property. Proposition 2.18 Let {T } cart 1,1 (B n Y) be such that sup E 1,1 (T ) <. Then, possibly passing to a subsequence, T T wealy in Z n,1 (B n Y) to some T cart 1,1 (B n Y). Proof of Theorem 2.17: By Theorem 2.13, and by a diagonal procedure, we may and will assume that T = G u for some smooth sequence {u } C 1 (B n, Y). As a consequence, by the first part of this section we infer that T satisfies (2.4) and (2.6). It then remains to show that the Ls(T ) s in (2.6) are i.m. rectifiable current in R n 1 (B n ). In this case, in fact, since T E1,1 <, we obtain that T E 1,1 graph(b n Y), see Definition 2.6, and hence, by lower semicontinuity, Theorem 2.12, and the condition sup E 1,1 (G u ) <, we conclude that E 1,1 (T ) <, which yields T cart 1,1 (B n Y), according to Definition To prove that the Ls(T ) s are i.m. rectifiable currents we mae use of the following slicing argument. As before, let P be an oriented 1-space in R n and {P λ } λ R n 1 the family of oriented straight lines parallel to P. For H n 1 -a.e. λ the slice T π 1 (P λ ) of T over π 1 (P λ ) is well defined on Z 1,1 ((B n P λ ) Y) and G u π 1 (P λ ) belongs to cart 1,1 ((B n P λ ) Y) for every. Moreover, since G u T wealy in Z n,1, for H n 1 -a.e. λ, passing to a subsequence we have G u π 1 (P λ ) T π 1 (P λ ) wealy in Z 1,1 ((B n P λ ) Y), with sup M(G u π 1 (P λ )) <, so that by the closure-compactness of cart 1,1 on 1-dimensional domains, we infer that T π 1 (P λ ) cart 1,1 ((B n P λ ) Y). Therefore, the 0-dimensional slices Ls(T ) π 1 (P λ ) are rectifiable in R 0 (B n P λ ), as T π 1 (P λ ) belongs to cart 1,1 ((B n P λ ) Y) and Ls(T ) π 1 (P λ ) = Ls(T π 1 (P λ )). Since the Ls(T ) s are flat chains, see Lemma 2.19 below, arguing as in [12], by White s rectifiability criterion [23], see also [3], we infer that Ls(T ) is an i.m. rectifiable current in R n 1 (B n ) for every s, as required. Lemma 2.19 The Ls(T ) s are flat chains in B n. Proof: By Theorem 2.13, we may and will assume that T is the wea limit of G u for some smooth sequence {u } C 1 (B n, Y) such that sup u W 1,1 <. The proof follows the same lines as the proof of [17, Thm. 2.15]. Since u BV (B n, Y) is smooth, for all and s we infer that Ls(G u ) := π # (G u π # ω s ) is a flat chain with equibounded flat norms. Recall that the flat norm F ( Ls(G u ) ) of Ls(G u ) is given by where F ( Ls(G u ) ) := sup{ls(g u )(φ) φ D n 1 (B n ), F(φ) 1}, { F(φ) := max sup φ(x), x B n } sup dφ(x). x B n Next, since u u T wealy in the BV -sense, we deduce that {Ls(G u )(φ)} is a Cauchy sequence for every φ such that F(φ) 1. If F n 1 (B n ) denotes a countable dense subset of smooth forms φ in D n 1 (B n ) satisfying F(φ) 1, by a diagonal argument we infer that sup{ ( Ls(G u ) Ls(G uh ) ) (φ) φ F n 1 (B n )} is small for, h large. This yields that {Ls(G u )} is a Cauchy sequence w.r.t. the flat norm, i.e., that F ( Ls(G u ) Ls(G uh ) ) := sup{ ( Ls(G u ) Ls(G uh ) ) (φ) φ D n 1 (B n ), F(φ) 1} is small for, h large and therefore, due to wea convergence of G u to T, that R s := π # (T π # ω s ) is a flat chain. Similarly, by using a trivial extension of Theorem 6.7 below, we infer that D s := π # (G T π # ω s ) is a flat chain and hence, since ( 1) n 1 Ls(T ) = R s D s, compare (2.6) and (2.7), we conclude that Ls(T ) is a flat chain, too. 3 Lower semicontinuity In this section we prove Theorem 2.12, by recovering it from the one dimensional case. To this aim, we recall the following properties from BV -functions theory, compare [2, Sec. 3.11]. 17

18 One-dimensional restrictions of BV -functions. Let Ω R n be an open set. Given ν S n 1 we denote by π ν the hyperplane in R n orthogonal to ν and by Ω ν the orthogonal proection of Ω on π ν. For any y Ω ν we let Ω ν y := {t R y + tν Ω} denote the (non-empty) section of Ω corresponding to y. Accordingly, for any function u : B Ω R N and any y B ν the function u ν y : B ν y R N is defined by u ν y(t) := u(y + tν). Proposition 3.1 Let u L 1 (Ω, R N ). Then u BV (Ω, R N ) if and only if there exist n linearly independent unit vectors ν i such that u ν i y BV (Ω ν i y, R N ) for L n 1 -a.e. y Ω νi and Du νi y (Ω νi y ) dl n 1 (y) < Ω νi i = 1,..., n. Theorem 3.2 If u BV (Ω, R N ) and ν S n 1, then Du, ν = L n 1 Ω ν Du ν y, D a u, ν = L n 1 Ω ν D a u ν y, D J u, ν = L n 1 Ω ν D J u ν y, D C u, ν = L n 1 Ω ν D C u ν y. In addition, for L n 1 -a.e. y Ω ν the precise representative u has classical directional derivatives along ν L 1 -a.e. in Ω ν y, the function (u ) ν y is a good representative in the equivalence class of u ν y, its Jump set is (J u ) ν y and u ν (y + tν) = u(y + tν), ν for L1 -a.e. t Ω ν y. Finally, σ(t) := ν, ν u (y + tν) 0 for L n 1 -a.e. y Ω ν lim u (y + sν) = u + (y + tν), s t lim u (y + sν) = u (y + tν), s t and L 1 -a.e. t Ω ν y, and lim u (y + sν) = u (y + tν) s t if σ(t) > 0 lim u (y + sν) = u + (y + tν) s t if σ(t) < 0. One-dimensional restrictions of Cartesian currents. If T cart 1,1 (B n, Y), taing Ω = B n, for any ν S n 1 the 1-dimensional slice Ty ν := T (B n ) ν y Y defines a Cartesian current Ty ν cart 1,1 ((B n ) ν y Y) for L n 1 -a.e. y (B n ) ν. Also, by Theorem 3.2 and by Definition 2.10, we infer that the BV -energy of Ty ν is given for L n 1 -a.e. y (B n ) ν by E 1,1 (Ty ν, A ν y Y) = for any open set A B n. A ν y u T (y + tν), ν dt + D C (u T ) ν y (A ν y) + t (J c(t ) A) ν y L T (y + tν) (3.1) Proof of Theorem 2.12: We follow [2, Thm. 5.4], [7]. Since {u } C 1 (B n, Y) is such that G u wealy in Z n,1 (B n Y), for L n 1 -a.e. y (B n ) ν we infer that T where (G u ) ν y T ν y wealy in Z 1,1 ((B n ) ν y Y), (G u ) ν y = G (u ) ν y, (u ) ν y(t) := u (y + tν) C 1 ((B n ) ν y, Y). Therefore, arguing as in the proof of Theorem 1.7, we readily infer that E 1,1 (T ν y, A ν y Y) lim inf E 1,1((u ) ν y, A ν y) (3.2) 18

19 for any open set A B n, where E 1,1 ((u ) ν y, A ν y) = E 1,1 (G (u ) ν, Aν y y Y) = A ν y u (y + tν), ν dt. We now denote by ν T an extension to the countably H n 1 -rectifiable set J c (T ) of the outward unit normal to the Jump set J ut. By the coarea formula, for any ν S n 1 and any open set A B n, we have ν T (x), ν f(x) dh n 1 (x) = f(y + tν) dl n 1 (y) J c (T ) A π ν t (J c(t ) A) ν y for any Borel function f : J c (T ) A [0, + ]. Moreover, Theorem 3.2 gives ( ) u T, ν dx = (u T ) ν y(t) dt L n 1 (y) A π ν A ν y D C u T, ν (A) = D C (u T ) ν y (A ν y) dl n 1 (y). π ν Therefore, setting for every open set A B n and ν S n 1 E 1,1 (T, A Y, ν) := u T, ν dx + D C u T, ν (A) + A J c (T ) A ν T (x), ν L T (x) dh n 1 (x), by (3.1) we obtain the identity E 1,1 (T, A Y, ν) = E 1,1 (Ty ν, A ν y Y) dl n 1 (y). π ν (3.3) Similarly, for every we obtain E 1,1 (u, A, ν) := We also notice that Since A u, ν dx = E 1,1 ((u ) ν y, A ν y) dl n 1 (y). (3.4) π ν E 1,1 (T, A Y, ν) E 1,1 (T, A Y) and E 1,1 (u, A, ν) E 1,1 (u, A). ( ) lim (u ) ν y (u T ) ν y dt dl n 1 (y) = lim u u T dx = 0, π ν A ν y A we can find a sequence {(h)} such that lim inf E 1,1(u, A, ν) = lim E 1,1(u (h), A, ν) h and (G u(h) ) ν y converges to T ν y wealy in Z 1,1 (A ν y Y) as h for L n 1 -a.e. y π ν. The lower semicontinuity property in dimension one, see (3.2), implies then lim inf h E 1,1((u (h) ) ν y, A ν y) E 1,1 (T ν y, A ν y Y) for L n 1 -a.e. y π ν. Integrating both sides on π ν, using Fatou s lemma and (3.3), (3.4), we get lim inf E 1,1(u, A, ν) = lim E 1,1(u (h), A, ν) E 1,1 (T, A Y, ν). h Let λ := L n + L T ( ) H n 1 J c (T ) + D C u T and let {ν i } S n 1 be a countable dense sequence. Choosing an L n -negligible set E B n \ J c (T ) on which D C u T is concentrated, we can define u T (x), ν i if x B n \ (E J c (T )) ν ϕ i (x) := T (x), ν i L T (x) if x J c (T ) D C u T, ν i D C (x) if x E u T 19

20 and obtain from (3.3) that lim inf E 1,1(u, A) lim inf E 1,1(u, A, ν i ) E 1,1 (T, A Y, ν i ) = for any i N and any open set A B n. By the superadditivity of the lim inf operator, we obtain that lim inf E 1,1(u, B n ) ϕ i dλ i A i A ϕ i dλ for any finite family of pairwise disoint open sets A i B n. We now recall that by [2, Lemma 2.35] { } sup ϕ i dλ = sup ϕ i dλ, B n i N A i where the supremum is taen over all finite sets I N and all families {A i } i I of pairwise disoint open sets with compact closure in B n. We then conclude that lim inf E 1,1(u, B n ) sup ϕ i dλ B n i N = u T (x) dx + D C u T (B n ) + L T (x) dh n 1 (x) B n J c(t ) = E 1,1 (T, B n Y). i I 4 The density theorem: part I In this section we prove Theorem To this aim we first recall that every T cart 1,1 (B n Y) decomposes as s T = G T + S T, S T = Ls(T ) γ s on Z n,1 (B n Y), s=1 see Definition Let u = u T BV (B n, Y) be the BV -function associated to T, according to Remar 2.7. For every Borel set B B n we have E 1,1 (T, B Y) = u(x) dx + D C u (B) + L T (x) dh n 1 (x), B J c (T ) B where J c (T ), Γ T (x), and L T (x) are given by (2.12), (1.5), and (1.6), respectively, compare Definition Slicing properties. Similarly to the case of normal currents, for every point x 0 B n and for a.e. radius r (0, r 0 ), where 2r 0 := dist(x 0, B n ), the slice T, d x0, r = G T, d x0, r + S T, d x0, r, where d x0 (x, y) := x x 0, is a well-defined Cartesian current in cart 1,1 ( B r (x 0 ) Y). More precisely, let u (r,x0) := u Br(x 0) be the restriction of u to B r (x 0 ), which is a function in BV ( B r (x 0 ), Y) with ump set satisfying J = J u(r,x0 ) u B r (x 0 ) in the H n 1 -a.e. sense. The slice G T, d x0, r is an (n 1)- dimensional current in BV graph( B r (x 0 ) Y) such that its action on forms in D n 1,1 ( B r (x 0 ) Y), according to a straightforward extension of Definition 2.1, depends on the restriction u (r,x0 ) and on the 1-dimensional integral chains γ x in Y associated to the current G T BV graph(b n Y), so that in particular γ x = δ u + (r,x 0 ) (x) δ u (r,x 0 ) (x) for Hn 1 -a.e. x J. Also, u(r,x0 ) S T, d x0, r = s Ls(T ), δ x0, r γ s on Z n 1,1 ( B r (x 0 ) Y), s=1 20