On the definitions of Sobolev and BV spaces into singular spaces and the trace problem

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1 On the definitions of Sobolev and BV spaces into singular spaces and the trace problem David CHIRON Laboratoire J.A. DIEUDONNE, Université de Nice - Sophia Antipolis, Parc Valrose, 68 Nice Cedex 2, France chiron@math.unice.fr Abstract The purpose of this paper is to relate two notions of Sobolev and BV spaces into metric spaces, due to N. Korevaar and R. Schoen on the one hand, and J. Jost on the other hand. We prove that these two notions coincide and define the same p-energies. We review also other definitions, due to L. Ambrosio (for BV maps into metric spaces), Y.G. Reshetnyak and finally to the notion of Newtonian-Sobolev spaces. These last approaches define the same Sobolev (or BV) spaces, but with a different energy, which does not extend the standard Dirichlet energy. We also prove a characterization of Sobolev spaces in the spirit of J. Bourgain, H. Brezis and P. Mironescu in terms of limit of the space W s,p as s, < s <, and finally following the approach proposed by H.M. Nguyen. We also establish the W s p,p regularity of traces of maps in W s,p ( < s < sp). Keywords: Sobolev spaces; BV maps; metric spaces; traces. AMS Classification : 46E35. Introduction The aim of this paper is to relate various definitions of Sobolev and BV spaces into metric spaces. The first one is due to N. Korevaar and R. Schoen (see [2]), which follows the pioneering work of M. Gromov and R. Schoen [9]. The second one is the approach of J. Jost ([]). We will next focus on other approaches. For the domain, we restrict ourselves to an open bounded and smooth subset of R N, N N. Many results extend straightforwardly when is a smooth riemannian manifold. Extensions are also possible when is a measured metric space (see [], [2], []). However, we do not aim such a generality. For the target space, we will work with a complete metric space (X, d). First, we recall the definition of L p (, X), for p. A measurable map with separable essential range u : X is said to be in L p (, X) if there exists z X such that d(u( ), z) L p (, R). If is bounded (and more generally if < ), then the point z is not relevant: if u L p (, X), then for any z X, d(u(.), z) L p (, R). The space L p (, X) is complete for the distance ( d L p(u, v) ) d(u(x), v(x)) p p dx

2 if p < and for p =, is complete for the distance d L (u, v) supess x d(u(x), v(x)). We recall the definitions of Sobolev spaces W,p (, X), < p < and BV (, X) spaces proposed by these authors. These spaces naturally appeared in the theory of harmonic functions with values into spaces coming from complex group actions, or into infinite dimensional spaces. This is the reason why [2] developped in the context of metric spaces targets the well-known theory of Sobolev maps: compact embeddings, Poincaré inequality, traces, regularity results for minimizing maps for nonpositively curved metric spaces (even though X is only a metric space, the notion of non-positively curved metric space does make sense, and we refer to [2] for instance for the definition)... Since we allow the target space X to be singular, one can not reasonably define Sobolev spaces of higher order. Our interest for these spaces was motivated by the study of topological defects in ordered media, such as liquid crystal, where an energy of the type Dirichlet integral plus a potential term appears naturally for maps with values into cones (see [6]). It is then important that this Sobolev theory extends naturally the usual Dirichlet integral.. Definition of W,p (, X) and BV (, X). Let p <. We recall in this subsection the definition of Sobolev spaces of N. Korevaar and R. Schoen given in [2], naturally based on limits of finite differences in L p. For ε >, we set For u L p (, X) and ε >, we introduce e ε (u)(x) ε {x, d(x, ) > ε}. d(u(x), u(y)) p S N S ε(x) ε N+p dh N (y) if x ε, otherwise. Here, S R (x) is the sphere in of center x and radius R >. For u L p (, X), e ε (u) has a meanning as an L (, R) function. We then consider the linear functional Eε u on C c () defined for f C c () by Eε u (f) f(x)e ε (u)(x) dx. Next, we set E p (u) ( sup f C c(), f lim sup ε ) Eε u (f) R + { }. Definition ([2]) Let p <. A map u L p (, X) is said to be in W,p (, X) for < p < or in BV (, X) for p = if and only if E p (u) <. We summarize now some of the main properties of the spaces W,p (, X) and BV (, X), which come from the theory developped in [2] (section there). Notice that we have chosen to divide the original density measure e ε(u) of [2] by the factor S N so that all approximate derivatives are based on averages. 2

3 Proposition ([2]) Let p <. Then, for every u L p (, X) such that E p (u) <, there exists a non-negative Radon measure u p in such that e ε (u) u p weakly as measures as ε (hence E p (u) = u p ()). Moreover, if < p <, u p L (, R + ). Remark We lay the emphasis on the fact that the weak convergence of e ε (u) to u p as ε holds for the real parameter ε and not for a sequence ε n. This is due to a monotonicity property (see Lemma 5 below). The last statement in Proposition is valid only for p >, which motivates the definition of the space W, (, X). Definition 2 ([2]) We define W, (, X) {u BV (, X), u L (, R + )}. Let us define for p <, the constants < K p,n by K p,n S N ω e p dh N (ω), () S N e denoting any unit vector in R N. Definitions and 2 extend the classical Sobolev spaces W,p (, R) and BV (, R). Theorem ([2]) Assume X = R is endowed with the standard distance. Then, for p <, W,p (, X) = W,p (, R) and BV (, X) = BV (, R). Moreover, if u W,p (, R) for p < or u BV (, R), then E p (u) = K p,n u p or E (u) = K,N u (), Remark 2 If p = 2 and X = R n (euclidean), n N arbitrary, then, by Theorem, we have E 2 (u) = K 2,N u 2, hence E 2 coincides, up to a constant factor K 2,N with the usual Dirichlet energy. This remains true if X is a smooth complete riemannian manifold. However, for X = R n but p 2, u p or u (in the sense of measures) may not be equal (even up to a constant factor) to the standard quantities u p or u (in the sense of measures). The second result is the lower-semicontinuity of E p for the L p (, X) topology. Theorem 2 ([2]) Let p <. Then, the p-energy is lower semicontinuous for the strong L p (, X) topology. In other words, if u n u in L p (, X) as n +, then E p (u) lim inf n + Ep (u n ) R + {+ }. 3

4 .2 Alternative definition (J. Jost) We recall now the alternate definition proposed by J. Jost ([]). We mention that this work considers a metric measured space as a domain instead of an open subset of R N or a riemannian manifold as in [2]. We will however not consider this general setting. Let, for x R N and ε >, µ ε x(y) dy B ε (x), (2) and consider the functionals for p < Jε p (u) d(u(x), u(y))p dµ ε x(y) x y p dµ ε x (y) dx. Note that the initial point of view of J. Jost was to deal with harmonic maps, and thus he only considered the case p = 2. The definition relies on Γ-convergence (see [7]), and is as follows. First, we define J p : L p (, X) R + { } to be the Γ-limit of Jε p as ε, or for a subsequence ε n as n +, for the L p (, X) topology. We recall that this means that for any sequence u ε u as ε (or u n u as n + ) in L p (, X), lim inf ε J p ε (u ε ) J p (u) ( ) or lim inf J p n + ε n (u n ) J p (u), and there exists a sequence u ε u as ε (or u n u as n + ) in L p (, X) such that ( ) lim J ε p (u ε ) = J p (u) or lim J p ε n + ε n (u n ) = J p (u). The existence of this Γ-limit for some subsequence ε n as n + is guaranteed if L p (, X) satisfies the second axiom of countability (see [7]). Definition 3 ([]) Let be given p < and u L p (, X). Then, u is said to be in W,p (, X) for < p < or in BV(, X) for p = if and only if J p (u) <. We denote for the moment W,p (, X) and BV(, X) since we do not know yet that these spaces are actually W,p (, X) and BV (, X). Notice that the approach of J. Jost does not allow to define directly W, (, X). Let us point out that the notion of Γ-convergence is in general stated for a countable sequence ε n, and not for a real parameter ε. One main problem with Definition 3 is that we do not know at this stage if the Γ-limit has to be taken for the full family ε, or for a subsequence ε n. In particular, it is not shown in [] that the functional J does not depend on the choice of the subsequence ε n. This will be however a consequence of our result, and in fact that the Γ-convergence holds for the full family ε. Theorem 3 Let p <. As ε (resp. for any sequence ε n as n + ), the functional E p is the pointwise and the Γ-limit, in the L p (, X) topology, of the functionals J p ε (resp. J p ε n ). The functional J p is now well-defined and J p = E p. In particular, for < p <, W,p (, X) = W,p (, X) and BV (, X) = BV(, X). 4

5 This theorem clarifies the notion of W,p (, X) (for < p < ) and BV (, X) (for p = ) of J. Jost: J p does not depend on the choice of some subsequence. Although natural, we have not been able to find in the literature a result concerning the fact that these two definitions coincide. If the definition of J. Jost allows to derive existence results more general than in [2] for harmonic maps into homotopy classes in non-positively curved metric spaces, [2] gives regularity results (namely lipschitzian). Fortunately, these two notions coincide for the Sobolev space H (, X) W,2 (, X), with the same energy thus, one can apply, for harmonic maps, the existence results of [] and the regularity results in [2]. Notice however that [3] establishes existence results similar to those of [] in a slightly different context. Our second result concerns the pointwise and Γ-limit of the functional E ε : L p (, R) R +, defined as E ε (u) e e (u)(x) dx = Eε u () = sup Eε u (f), f C c(), f as ε in the L p (, X) topology. Proposition 2 Let p < and ε n be a positive sequence, ε n as n +. Then, the functional E p is the pointwise and the Γ-limit for the L p (, X)-topology of the functionals E ε (resp. E εn ) as ε (resp. n + ). We turn now to other approaches to the definition of Sobolev spaces into metric spaces, based on a characterization through post-composition with lipschitz maps..3 Characterizations of Sobolev and BV maps by post-composition In this Section, we are interested in characterizations of BV and Sobolev maps with values in X by composition with -lipschitzian maps ϕ : X R..3. BV maps into metric spaces In [], L. Ambrosio has proposed another approach to define the space BV (, X) when X is a locally compact and separable metric space. First, let us define for u L (, X) the measure (of possibly infinite mass) Du on to be the least measure such that for every borelian set B and every -lipschitzian map ϕ : X R, ϕ u BV (,R) (B) Du (B). Here, ϕ u BV (,R) is the usual measure in the BV sense. From [], such a measure exists. Definition 4 ([]) We define BV(, X) {u L (, X), Du () < }. Here, we do not know at this stage that BV(, X) = BV (, X) in the sense of Definition, which justifies that we denote BV(, X) this space to avoid confusions. Proposition 3 Assume X is locally compact and separable. Then, BV(, X) = BV (, X). Moreover, for u BV (, X), we have in the sense of measures K,N Du u Du. 5

6 Remark 3 In the case N = (since then K,N = ) or X = R, then E (u) = u () and the two BV energies are equal. However, in general, the two constants K,N and are optimal in the sense that for X the euclidean space R N and with v, v 2 : R N defined by v (x) (x,,..., ) and v 2 (x) x, then K,N Dv = K,N dx = v > and v 2 = dx = Dv 2 >. Therefore, the two BV energies are only equivalent and not equal up to a constant factor even in the usual euclidean (vectorial) case: the minimizers of Du () or E (u) (in some subset of BV (, X)) may then be different. To overcome this difficulty, when X has dimension n, we should choose maps ϕ with values into R n instead of R. L. Ambrosio has developped in this framework the well-known results concerning BV maps, in particular the definition of the regular part of the measure Du, the definition of the jump set of of a BV map as a rectifiable set of locally finite H N measure and the notion of approximate limit along the orthogonal direction to this jump set. By Proposition 3, one can use equivalently any of these two definitions, when X is locally compact and separable..3.2 Reshetnyak s characterization of Sobolev spaces In [9], Y.G. Reshetnyak proposed a similar approach to define the Sobolev maps for a metric space target. Let p < and u L p (, X). Set } R(u) inf { w plp(,r), z X, (d(u( ), z)) w a.e. in R + { }. Definition 5 ([9]) Let p < be given. The Reshetnyak-Sobolev space is defined to be the set R,p (, X) {u L p (, X), R(u) < }. Paralleling the proof of Proposition 3, we have the following result (see also [] for a similar result when X is a Banach space). Proposition 4 Let p <. Then, we have Morever, for u W,p (, X), we have R,p (, X) = W,p (, X). K p,n R(u) E p (u) R(u). Remark 4 When X is locally compact and separable, we could also have defined the Reshetnyak- Sobolev space as the set of maps u BV(, X) in the sense of L. Ambrosio such that the measure Du belongs to L p (, R), with p-energy Du p L p (). This gives, in this case, the existence of a minimizer w for R(u) for any p < (this is not obvious for p = ). Remark 5 As in Remark 3, if N = or X = R, then E p = R. Moreover, the constants K p,n and are optimal (consider the maps v, v 2 : R N defined in Remark 3). Therefore, if we are interested in the minimization of the Dirichlet energy into a riemannian manifold with singularities, then R does not extend the standard Dirichlet integral (up to a constant factor). 6

7 .4 Link with the Newtonian-Sobolev and Cheeger-Sobolev spaces The paper [] generalizes a definition of Sobolev spaces introduced in [2] as Newtonian-Sobolev spaces, for an arbitrary metric space as target space. Definition 6 ([]) Let p < be given and u L p (, X). Then, u is said to be in the Newtonian-Sobolev space N,p (, X) if there exist ρ, w L p (, R) such that for any -lipschitzian curve γ : [, l], d(u γ(l), u γ()) l ρ γ or l w γ =. The Newtonian p-energy of u is then the infimum of ρ p L p () for all the possible ρ s: N(u) inf ρ p. ρ The maps ρ are called upper gradients for u. Theorem 3.7 in [] establishes that the Newtonian- Sobolev energy is the same as the Reshetnyak-Sobolev energy. In particular, for p = 2, this energy R(u) is not equal to the standard Dirichlet energy u 2 dx (up to a constant factor). Proposition 5 ([]) Let p <. Then, in L p (, X), we have R = N. In particular, N,p (, X) = R,p (, X) = W,p (, X). We emphasize that neither [9] nor [] (extending [2]) allow to define the space BV (, X). Moreover, neither compactness result (as Theorem.3 in [2], when X is locally compact) nor lower-semicontinuity (see Theorem 2) result for the energy are given. In fact, R = N is not lower semi-continuous for p = and for the L (, X) topology. Finally, [6] extends the notion of Cheeger type Sobolev spaces (see [5]) to a metric space target. Let p < be given and u L p (, X). We set { } H(u) inf lim inf g p j + j R + { }, where the infimum is computed over all sequences (u j ) L p (, X) and (g j ) L p (, R) such that u j u in L p (, X) as j + and for any j N and any -lipschitzian curve γ : [, l], d(u j γ(l), u j γ()) l g j γ. (3) Definition 7 ([6]) Let p <. The Cheeger type Sobolev space is defined to be the set H,p (, X) {u L p (, X), H(u) < }. Remark 6 We would like to emphasize the role played by the target space for all these definitions of Sobolev and BV maps. Let ( X, d) be a complete metric space and let X be a closed subset of X endowed with the induced distance, so that X is also a complete metric space. Then, if we view u as a map ū in L p (, X) instead of L p (, X) the quantities E p (u), J p (u), Du (), R(u), and N(u) do not change 2. However, it is not clear whether H(u) = H(ū) or not since the computation of H(ū) uses sequences (u j ) X-valued instead of X-valued. In view of Proposition 6 below, we have at least H(u) C H(ū), for some constant C, when X is a length space. 2 This follows from the definitions. For Du, we use the well-known fact that any -lipschitzian map ϕ : X R can be extended in a -lipschitzian map ϕ : X R, for instance by the formula ϕ(x) supξ X (ϕ(ξ) d(x, ξ)). 7

8 Actually, Proposition 5 is a direct consequence of Remark 6 and Theorem 3.7 in [], using an embedding of X into l (X). In the statement of the following Proposition, we will need the definition of a length space. Definition 8 Let (X, d) be a complete metric space. Then, (X, d) is said to be a length space if, for every points x, y X, there exists a -lipschitzian map γ : [, d(x, y)] X such that γ() = x and γ(d(x, y)) = y. The result in the next Proposition is already known for < p < and X a Banach space in [2] (Theorem ). However, for sake of completeness and in view of Remark 6, we have included a proof here when X is an arbitrary metric space. Furthermore, the following Proposition takes into account the case p = when X is a length space. Proposition 6 Assume < p <. Then, in L p (, X), we have H = N = R. In particular, H,p (, X) = N,p (, X) = R,p (, X) = W,p (, X). Assume p = and, moreover, that X is a length space. Then H, (, X) = BV (, X), and there exists a constant C = C(, N) such that for every u BV (, X), K,N E (u) H(u) C(, N)E (u). Remark 7 We emphasize that for p =, the result may be false if X is not a length space. Indeed (see [6]), if X = {, } R, then u : (, ) X defined by u(x) = if x > and if x belongs to BV ((, ), R) = H, ((, ), R) (by Proposition 6) but not to H, ((, ), X), since X is not path-connected (see also Remark 6). Remark 8 Actually, the -energy in the space H, (, X) is close to the BV energy in the sense of L. Ambrosio (see Lemma 8 in section 2.7)..5 Other characterizations of Sobolev spaces In [2], J. Bourgain, H. Brezis and P. Mironescu have introduced a new characterization of the usual Sobolev spaces W,p (, R) of real-valued maps, viewed as the limit of the spaces W s,p, < s <, as s. This does not require a notion of gradient, and is very close to the two previous definitions ([2] and []). We generalize in the following Theorem the results of [2] and [8] in the case of a metric space target. We consider a family of radial mollifiers (ρ ε ) ε> (or a sequence (ρ n )). We recall that this means that ρ ε is in L (R N ), radial, ρ ε, R ρ N ε = and for all δ >, ρ ε (x) dx as ε. x >δ Then, given p <, we define, for u L p (, X), d p (u(x), u(y)) F ε (u) x y p ρ ε (x y) dxdy R +. Theorem 4 Let p < and (ε n ) be any positive sequence, ε n as n +. Then, for every u L p (, X), lim ε F ε (u) and lim n + F εn (u) exist in R + {+ } and lim F d p (u(x), u(y)) ε(u) = lim ε ε x y p ρ ε ( x y ) dxdy = lim F ε n + n (u) = E p (u). Moreover, the functionals (F ε ) (and (F εn )) Γ-converge as ε (and as n + ), in the L p (, X) topology, to E p. 8

9 Theorem 4 combined with Theorem extend the results of [2] and [8] which correspond to the case of real-valued maps. Indeed, from [2] and [8], given p < and f L p (, R), then f(x) f(y) p lim inf n + x y p ρ n ( x y ) dxdy = K p,n f p dx R + { }. (4) The quantity (4) is if f W,p (, R) (if < p < ) or f BV (, R) (if p = ). Moreover, the right-hand of (4) has to be understood as the mass of the measure u in the BV case. This approach defines the same p-energy as [2] (thus extends the classical Dirichlet energy), but does not allow to define W, (, X). Remark 9 Theorem 4 emphasizes that the result of J. Bourgain, H. Brezis and P. Mironescu (and also J. Dávila) does not require a linear structure, neither usual Sobolev tools, such that density of smooth functions or integration by parts. Remark It is a natural question to wonder what may occur if one chooses non-radial functions ρ n. The problem is that in the non-radial case, there can be privileged directions for [ ρ n, for ] instance, ] in R 2, ρ n can be the characteristic function of the thin rectangle in the x direction n, n [ n, 2 n 2 suitably normalized, and then we have R2 f(x) f(y) p lim inf n + x y p ρ n ( x y ) dxdy = f p dx. x R 2 However, for ρ n s which do not privilege any direction, A. C. Ponce (see [7]) proves the existence of a non-negative measure µ on the sphere such that the results of [2] still hold with the right-hand side of (4) replaced by ω f p dµ(ω)dx. S N Therefore, it is reasonable to take radial distributions of masses to define W,p (, X) and BV (, X). Remark Finally, we mention that if N 2, then A.C. Ponce in [8] (theorem.2) showed that for a family u ε L p (, R) (or for a sequence ε n ), the inequality lim inf ε F ε (u ε ) < yields precompactness in L p (, R) for (u ε ) as ε (if is bounded). This result is false if N = (see [2], counterexample ). It is plausible that these result extend to the case where R is replaced by a complete metric space X, at least locally compact. We conclude this section with a generalization of the paper [5] of H-M. Nguyen. This article gives a generalization of the previous work [4] of H-M. Nguyen and the paper of J. Bourgain and H-M. Nguyen [3]. For this approach, we consider a sequence of functions g n : R + R + such that: (a) for every n N, g n is non-decreasing; (b) sup n N + g n (t) dt < and for every n N, tp+ (c) (g n ) n N converges locally uniformly in (, + ) to. R 2 g n (t) dt = ; tp+ 9

10 Define then for n N and u L p (, X) (comparing with the quantity used in [5], we have just introduced the factor S N ) G n (u) g n (d(u(x), u(y))) S N x y N+p dxdy. Theorem 5 Let < p <, and (g n ) be a sequence verifying (a), (b) and (c). Then, for every u L p (, X), lim n + G n (u) exists in R + {+ } and lim G n(u) = E p (u). n + The key point in the proof of Theorem 5 is the generalization of the result by J. Bourgain and H-M. Nguyen in [3] (Lemma ) given in theorem 4 in [5]. Remark 2 The above theorem remains true when we replace the sequence g n by a familly g δ, < δ < satisfying the corresponding hypothesis to (a), (b) and (c). It would be interesting to know whether the functionals G n Γ-converge as n +, in the L p (, X) topology, to E p. When X = R, see the partial result Theorem 5 in [5]. Remark 3 As shown by an example due to A. Ponce, these results do not hold for p =. Namely, there exists a map f W, (, ), R) such that G δ (f) + as δ, with g δ (t) = pδ p χ t δ. However, the inequality E (u) lim inf n + G n (u) for u L (, X) remains true..6 Metric topology for W,p (, X) and BV (, X) It is a natural question to wonder whether the spaces W,p (, X), p <, and BV (, X) can be endowed with a metric topology. Actually, one may propose (at least) two distances on these spaces. Then, we define: Dp (u, v) d L p (,X)(u, v) + E p (u) p E p (v) p if E p (u) + E p (v) <, DW 2 (u, v) d,p L p (,X)(u, v) + u p v p L p (,R) if u, v W,p (, R), p < and DBV 2 (u, v) d [C L (,X)(u, v) + u v (,R)] if u, v BV (, R). Note that these quantities are well-defined on the spaces W,p (, X) for p <, and on the space BV (, X). Moreover, it is easily checked that they define distances on these spaces. The first natural question is to ask if Dp, D2 W and D 2,p BV induce, for X = R, the same topology as the classical one on W,p (, R) and BV (, R), induced by the norm f W,p f L p (,R) + f L p (,R) if f W,p (, R), p <, f BV f L (,R) + f [C (,R)] if f BV (, R). Given u, v W,p (, R) or BV (, R), there holds (by Theorem and note that K p,n ) D p(u, v) D 2 W,p (u, v) u v W,p and D (u, v) D 2 BV (u, v) u v BV. (5)

11 Lemma Assume X = R. If < p <, then Dp and D2 W induce the same topology as the,p classical one on W,p (, R). Proof. Assume first < p <. If u n u in W,p (, R) in the topology of the norm, then (5) implies that Dp(u n, u) and DW 2 (u,p n, u). Assume then that Dp(u n, u) as n +. Then, u n u in L p (, R) and ( u n ) is bounded in L p (, R N ), hence ( < p < ) u n u in L p (, R N ). Furthermore, since Dp(u n, u) and using Theorem, K p,n u n p dx = E p (u n ) E p (u) = K p,n u p dx, thus by strict convexity of L p (, R N ) (since < p < ), we have u n u in L p (, R N ), so that u n u W,p. We have nevertheless some negative results for these metrics. Lemma 2 (i) The distance D does not induce the classical topology on W, (, R) and BV (, R). (ii) For p <, the space W,p ((, ), R) (resp. BV ((, ), R)), endowed with the metrics Dp or DW 2 (resp. D,p or D2 BV ) is not complete. Proof. We consider the sequence v n : (, ) R defined in the following way. Let, for k < n, I + k [ k n, k n + 2n [, and I k [ k n + 2n, k n + n [. Then, let, for k < n, v n(x) x k n if x I+ k and v n (x) k+ n x if x I k. The map v n is Lipschitz continuous, and we have (χ is the characteristic function) v n L ((,)) = n 2n, v n = χ I + χ k I. k As a consequence, v n in L ((, )) as n +, but v n = a.e. in (, ). Therefore, (v n ) is not convergent in BV ((, ), R), but (v n ) is a Cauchy sequence for Dp, DW 2 and D 2,p BV. This proves (ii). Consider now u n (x) x + v n (x) in (, ). We have then u n u(x) x in L ((, ), R) but (u n ) does not converge in BV ((, ), R). However, we have u n(x) = + v n(x) (since v n = ± a.e.) and u (x) =, hence E (u n ) = + v n(x) dx = and E (u) =, so that, as n +, k= D (u n, u) = u n u L = v n L. Hence D (u n, u) but u n u BV = u n u W, = v n L =. This proves (i)..7 The trace problem In [2] (Section.2), the trace of a map in W,p (, X) is defined, namely, for p < and a smooth R N, there exists a map tr : W,p (, X) L p (, X) such that tr(u) = u for u W,p (, X) C(, X). Moreover, for < p <, the map tr is continuous in the sense that if u n W,p (, X) is such that (E p (u n )) n N is bounded and u n u in L p (, X), then tr(u n ) tr(u) in L p (, X). In view of (5), this implies that the trace map is continuous from W,p (, X) endowed with the distances D or D 2 into L p (, X), < p <. We complete this

12 notion of trace by proving the W p,p regularity of the trace. First, given a compact submanifold M of R N of dimension M (such as or ), we define, for < s < and p <, W s,p (M, X) {u L p (M, X), u pw s,p(m,x) d(u(x), u(y)) p } dxdy <. x y sp+m M M Proposition 7 Let p < and < s such that sp > or s =. Then the trace map tr : W s,p (, X) L p (, X) is well-defined and has values into W s p,p (, X). Moreover, there exists C, depending on, s and p such that for every ξ X, tr(u) W s p,p (,X) C( d(u, ξ) L p (,R) + u W s,p (,X)). Similarly to W,p (, X), we may endow W s,p (M, X) ( < s < ) with the distances defined by Ds,p(u, v) d L p (M,X)(u, v) + u W s,p v W s,p, Ds,p 2 (u, v) d L p (M,X)(u, v) + d(u(x), u(y)) d(v(x), v(y)) L p( The following Lemma summarizes the properties of these distances on W s,p. ). dxdy x y sp+m Lemma 3 Assume < s <. (i) If X = R and < p <, Ds,p and Ds,p 2 induce the usual topology on W s,p (, R). (ii) For p <, the space W s,p ((, ), R) endowed with the metric Ds,p is not complete. (iii) If p <, W s,p (, X) endowed with the metric Ds,p 2 is complete. Proof. Assertion (i) follows as in Lemma since, for < p <, W s,p (, R) is uniformly convex. To prove (ii), let w : R R be defined by: w(x) x if x, w(x) otherwise. Consider then w n : (, ) R defined by: w n (x) n s w(nx). Then, w n L = n s as n +. Moreover, simple scaling yields w n W s,p = w W s,p = Cte >. Hence, (w n ) is Cauchy for Ds,p but does not converge in W s,p ((, ), R). Assertion (iv) is straightforward: if (u n ) is Cauchy for Ds,p, 2 then (u n ) and ( d(un(x),un(y)) ) are Cauchy sequences in L p (, X) and L p (, R), hence converge. Thus, u x y s+ M n u p in L p (, X) and d(un(x),un(y)) g in L p (, R). Clearly, g = d(u(x),u(y)), so Ds,p(u 2 n, u). x y s+ M p x y s+ M p We have not investigated whether the trace map is continuous or not, from W s,p (, X) into W s p,p (, X) for the distances we have defined. The plan of the paper is the following. Section 2 contains the detailed proofs of Proposition 2 (subsection 2.2) Proposition 3 (subsection 2.4), Proposition 4 (subsection 2.5), Proposition 6 (subsection 2.7). For the main results, Theorem 3 is proved in subsection 2.3, then Theorem 4 in subsection 2.8 and finally Theorem 5 in subsection 2.9. We conclude with the proof of Proposition 7 about traces, given in Section 3. 2 Proofs of the main results 2. An extension Lemma Here is a standard extension lemma for smooth domains (a lipschitz boundary is enough). 2

13 Lemma 4 Let p < and R N be smooth. Then, there exists δ >, depending only on the smoothness of, and C, depending on p and the smoothness of such that, for any u W,p (, X) or BV (, X), there exists U W,p ( δ, X) or BV ( δ, X) such that U = u in and E p (U) CE p (u). Proof. The extension is done by standard reflection across the boundary. Hence, by standard localization and using local charts, we may assume for simplicity that is locally the half-plane R N + RN R +. Then, we define U : RN X as U(x) u(x) if x N > and if x N <, U(x) u(x,..., x N, x N ). It then follows directly that e ε (U)(x) 2e ε (u)(x), which establishes the Lemma. 2.2 Proof of Proposition 2 A useful fact we will need in the sequel is to allow other approximate derivative than e ε (u), for instance ball averages instead of sphere averages. Therefore, we consider as in [2] νe ε (u)(x) + e λε (u)(x) dν(λ), where ν is a probability measure on (, + ), and we define ν E ε : L p (, X) R + {+ } by νe ε (u) = νe ε (u)(x) dx. Note that in [2], it is required that ν satisfies Supp ν (, 2) and 2 λ p dν(λ) <. (6) When ν = δ, ν E ε (u) is simply the norm of the linear functional E ε (u) : C c (, R) R. Therefore, Proposition 2 is a particular case of Theorem 6 below when ν = δ. Theorem 6 Let p < and ν be a probability measure on R +. Then, the functional E p is the pointwise and the Γ-limit for the L p (, X)-topology of the functionals ν E ε as ε (or for any sequence ε n as n + ). Moreover, the following inequality holds, for u L p (, X), νe p (u) C(, p)e p (u). Proof. Let us first assume that u W,p (, X) or BV (, X). Using Lemma 4, we first extend u, defined in, in a map U defined in the δ neighborhood δ of. The energy of the extension is finite and E p (U, δ ) C(, p)e p (u). Combining Theorem.8. and Lemma.9. in [2] about directional energies, we infer, for h < δ d p (U(x + h), U(x)) dx C(, p, N) h p E p (u). (7) Therefore, using Fubini s Theorem, νe ε (u) S N + = S N + S λε () d p (U(x), U(y)) dhn (y) dν(λ) dx (λε) N+p ( d(u(x + h), U(x)) ) p dh N (h) dx h h N dν(λ). (8) S λε (x) 3

14 By (7), we then deduce νe ε (u) C(, p) Ep (u) S N + S λε () dh N (h) h N dν(λ) = C(, p)e p (u), which concludes the proof of the last assertion. Proof of the liminf inequality. Let u, u ε L p (, X) be such that u ε u in L p (, X) (the case of a sequence is analogous). We wish to prove that E p (u) lim inf ε νe ε (u ε ), (9) and we may assume without loss of generality that the right-hand side is finite. decreasing sequence ε j such that There exists a lim inf ε νe ε (u ε ) = lim j + ν E εj (u εj ) <. We argue as in Lemma.4.2 in [2], letting, for ε >, n j be the integer part of ε/ε j. Therefore, n j ε j ε as j +. Since u εj u in L p (, X), then ν e nj ε j (u εj ) ν e ε (u) pointwise almost everywhere, thus by Fatou s lemma, νe ε (u) = νe ε (u) lim inf j + νe nj ε j (u εj ) = lim inf j + ν E nj ε j (u εj ). () We apply now the following Lemma, stating a result analogous to the fact that the approximate length of a lipschitz curve with respect to a partition is increased if the partition is refined. This Lemma comes from [2], Lemma.3., with the only change that we take f (the proof clearly works for this f). Lemma 5 Assume ν is a probability measure on (, + ). Let p <, and u L p (, X). Then, given n N and λ i >, i n such that n i= λ i =, we have for every ε > νe ε (u) p n λ i (νe λi ε(u) i= ) p. Applying Lemma 5 with u = u εj, n = n j, ε = n j ε j and λ i = n j > (thus λ i = ) we obtain νe nj ε j (u εj ) p ν E εj (u εj ) p ν E εj (u εj ) p. Passing to the limit as j + and using (), we infer νe ε (u) lim inf j + ν E nj ε j (u εj ) lim inf j + ν E εj (u εj ) = lim inf νe ɛ (u ɛ ). ɛ As a consequence, and (9) follows. lim sup νe ε (u) lim inf νe ɛ (u ɛ ) <, ε ɛ Proof of the pointwise limit. Let u L p (, X). We wish to prove lim ε ν E ε (u) = E p (u) R + {+ }. 4

15 We know from (9) (with u ε = u) that the equality is true if E p (u) = +, hence we may assume E p (u) < +, and, still with (9) (and u ε = u), it suffices to show lim sup νe ε (u) E p (u) < +. () ε First, by monotone convergence, ν((/r, R)) if R +, hence for R > large enough (so that ν((/r, R)) /2 for instance), we can define the measure ν R ν (/R, R) ν((/r, R)). This probability measure satisfies Supp ν R (, R) and λ p dν R (λ) R p R + R + dν R = R p <. This is not exactly ) (6), but this is sufficient however to apply Theorem.7 in [2], which asserts that lim ε (ν R E ε (u) exists and ( ) lim ν ε R E ε (u) = E p (u). (2) Therefore, using (7) and arguing as for (8), we deduce that, for R > large enough, ( ν E ε (u) ν((/r, R)) ν R E ε (u)) ( d(u(x + h), U(x)) ) p dh N (h) dx S N h h N dν(λ) R + \(/R,R) S λε () C(, p)ν(r + \ (/R, R))E p (u). Now, letting ε and using (2) gives lim sup νe ε (u) ν((/r, R))E p (u) + C(, p, N)ν(R + \ (/R, R))Ep (u). ε We may now let R go to + to deduce, since ν(r + \ (/R, R)) and ν((/r, R)), that lim sup νe ε (u) E p (u). ε This finishes the proof of (), and hence of the pointwise limit. Combining the liminf inequality together with the pointwise limit, we deduce Theorem Proof of Theorem 3 Proof of the liminf inequality. First, we have, for x ε and µ ε x defined by (2), x y p dµ ε x (y) = SN ε r p+n dr = SN N + p εn+p. Thus, denoting dν = (N + p)λ N+p dλ (, ) the measure giving ball averages, we infer that for every u L p (, X) J p ε (u) ε νe ε (u) dx = ν E ε (u). 5

16 Therefore, for any family u ε u in L p (, X) as ε (or for a sequence u n u in L p (, X) with ε n as n + ), we deduce from Theorem 6 that lim inf ε J p ε (u ε) lim inf ε νe ε (u ε ) E p (u), or the corresponding statement for the sequence ε n. In particular, if J p (u) <, that is if u W,p (, X) (or u BV,p (, X)), then u W,p (, X) (or u BV,p (, X)). Proof of the pointwise limit. We now prove that there will be equality in the above inequality for the particular family u ε = u for all ε > (or u n = u for all n). First, we extend by Lemma 4 u in U of finite p-energy in the δ -neighborhood of. Furthermore, we have for any ε > and x \ ε, since is smooth, x y p dµ ε x(y) x y p dµ ε C() x(y) = S N B ε(x) C() N + p εn+p, where C() depends only on the smoothness of. As a consequence, fixing < η < δ, we have for every < ε < η/2 Jε p (u) ν E ε (u) = d(u(x), U(y))p dµ ε x (y) \ ε x y p dµ ε dx C() x(y) ν e ε (U)(x) dx, η\ η from which we deduce, using Theorem 6, E p (u) = lim inf ε νe ε (u) lim inf ε J p ε (u) lim sup Jε p (u) ε ( ) lim sup νe ε (u) + C() ε ν e ε (U)(x) dx η\ η = E p (u) + C() U p dx. η\ η Letting η, the last term tends to. Indeed, U p L if < p <, and if p =, U is a measure such that U ( ) = (since U is the extension by reflection across the boundary). Thus, we infer lim ε J p ε (u) = E p (u), which is the desired equality: Jε p Γ-converge at u to E p (u), which is finite if u W,p (, X) or u BV (, X) and infinite otherwise. 2.4 Proof of Proposition 3 In this subection, we prove the equivalence of the two definitions of BV maps into a locally compact metric space X, namely Definitions and 4. First, if u BV (, X), and ϕ : X R is a -lipschitzian map, then for every ε > and x ε, e ε (ϕ u)(x) = S N ε N S N ε N S ε(x) S ε(x) ϕ u(x) ϕ u(y) dh N (y) d(u(x), u(y)) dh N (y) = e ε (u)(x). 6

17 Passing to the limit as ε and using Proposition and Theorem, we obtain K,N D(ϕ u) BV u. Since this is true for any -lipschitzian map ϕ : X R, we deduce that u BV(, X) and K,N Du () u. Let now u BV(, X) be given. We claim that for every ε > and h ε, we have ε d(u(x), u(x + h)) dx h Du (). (3) Proof of (3). If N = and = (a, b), then (3) follows from the inequality for x (a + ε, b ε) and < h < ε d(u + (x), u + (x + h)) Du (]x, x + h]) valid for the right-hand side continuous representative u + of u. Integrating this inequality yields b ε a+ε d(u(x), u(x + h)) dx = b ε a+ε d(u + (x), u + (x + h)) dx h Du ([a, b]), which is the desired inequality for N =. The general case follows by slicing in the direction h and using (2.5) in Proposition 2. in [] (see the analogous proof of (ii) in Lemma 3.2 in []). We conclude now the proof of Proposition 3. For ε > and ν = (N + )λ N dλ (, ) giving ball averages, we have, using (3), νe ε (u)(x) dx = N + ε S N ε N d(u(x), u(x + h)) dx dh B ε() ε N + S N ε N B ε() From Definition, we infer u BV (, X) and h Du () dh = N + ε N Du () E (u) Du (). ε r N dr = Du () We have therefore K,N Du () E (u) Du () for any u L (, X), which implies that BV (, X) = BV(, X). Given now u BV (, X) and an open ball B, we may apply this to u BV (B, X) to infer K,N Du (B) u (B) Du (B), hence K,N Du u Du as borelian measures. This concludes the proof of Proposition Proof of Proposition 4 Given p < and u W,p (, X), we have K p,n R(u) E p (u). Indeed (as in section 2.4), for any z X, since d(, z) is -lipschitzian, K p,n (d(u( ), z)) p = (d(u( ), z)) p u p L (, R). Hence, one can take w (K p,n u p) p and infer K p,n R(u) E p (u) for u W,p (, X). The inequality E p (u) R(u) will follow as in section 2.4 from the inequality, for h < ε, ε d p (u(x), u(x + h)) dx h p R(u), (4) 7

18 which is the analogue of (3). Let us first consider some w L p (, R) such that for all z X, (d(u( ), z)) w a.e. in. Using Fubini s Theorem, for a.e. x ε and h < ε, [, ] t d(u(x + th), u(x)) W,p and d dt (d(u(x + th), u(x))) h w(x + th), thus by integration ( d p ) p (u(x), u(x + h)) h w(x + th) dt h p w p (x + th) dt. (5) Integrating over ε and using Fubini s Theorem, we obtain ε d p (u(x), u(x + h)) dx h p w p L p, and the conclusion follows by taking the infimum over all such w s. This proves (as in section 2.4) that E p R in L p (, X). This concludes the proof of the second assertion of Proposition 4, together with the fact that if p >, R,p (, X) = W,p (, X), and (if p = ) W, (, X) R, (, X) BV (, X). In order to finish the proof, we have then just to prove that R, (, X) W, (, X). Consider then u R, (, X), and a function w L (, R) associated, so that (5) holds (with p = ). Therefore, by integration and changing variables, one has for ε >, χ ε standing for the characteristic function of ε, χ ε e ε (u)(x) S N S N w(x + εty) dtdh N (y). We notice that the right-hand side is in L (, R) and converges to w in L (, R) by dominated convergence, thus, by Proposition, e ε (u) converges as measure to u w L (, R), thus u L (, R) and u W, (, X). This ends the proof. 2.6 An approximation result Before going to the proof of Proposition 6, we prove the following approximation result. Such a result is analogous to Theorem 3.3 in [], which is an approximation result in L (, X) of BV maps by maps constant on cubes, with a control on the energy. Theorem 3.3 in [] is stated for X a locally compact separable metric space, and we will state first this approximation result when X is just a complete metric space. Lemma 6 Let N. For n N, P n is the usual partition of ], [ N in n N cubes of edge n. There exists a constant C = C (N, X) such that, for every n N and every u BV (, X), there exists v n BV (, X) such that v n is constant on each cube of P n, d L (],[ N,X)(u, v n ) C n E (u) and E (v n ) C E (u). Proof. The proof is based on the following Poincaré type inequality (see Theorem.4. in [3] for a general version in W,p (, X), p < ). There exists a constant C, depending only on N such that, for every l > and every u BV (], l[ N, X), there holds inf ξ X ],l[ N d(u(x), ξ) dx C l u (], l[ N ). (6) Proof of (6). By simple scaling, it suffices to treat the case l =. For u BV (], [ N, X), we first extend u as a map U :] 2, 3[ N X by successive reflections across the boundary (as in Lemma 4). 8

19 Next, we may write, using (3), d(u(x), u(y)) dxdy ],[ N ],[ N ],[ N d(u(x), U(x + h)) dxdh ],[ N h U (] 2, 3[ N ) dh K N u (], [ N ). ],[ N As a consequence, by the mean value formula, there exists y ], [ N such that ξ u(y) verifies (6). We complete now easily the proof of Lemma 6. Let u BV (], [ N, X) and n N be given. For any cube Q P n, there exists by (6) some ξ n (Q) X such that d(u(x), ξ n (Q)) dx 2K N n u (Q). (7) Q Therefore, we may define v n :], [ N X, constant on each cube Q of P n, of value ξ n (Q). Summing (7) over all Q s in P n yields the first estimate for v n d L (],[ N,X)(u, v n ) 2K N n We turn now to the second estimate. We have easily E (v n ) = 2 E (u). Q Q P n n N d(ξ n(q), ξ n (Q )), where Q Q means Q Q and Q, Q have one face in common. Moreover, for Q Q P n, say Q = [, n ]N and Q = [ n, 2 n ] [, n ]N = Q + n e, with e = (,,..., ) R N for simplicity, we have by the triangle inequality n N d(ξ n(q), ξ n (Q )) Q d(u(x), ξ n (Q)) dx+ Q d(u(x), u(x+ n e )) dx+ d(u(x+ Q n e ), ξ n (Q )) dx. The last integral is simply Q d(u(x), ξ n (Q )) dx. Thus, using (3) and (7), n N d(ξ n(q), ξ n (Q )) 2K N n u (Q) + ( n u [, 2 n ] [, n ]N ) + 2K N n u (Q ). Summing this inequality for Q Q P n yields the desired estimate. This Lemma 6 will enable us to establish the following approximation result when X is a length space (the proof is not very different from that of Theorem 3.3 in []). Lemma 7 Let N and assume X is a length space. Then, there exists a constant C, depending only on N such that for every u BV (], [ N, X), there exists a sequence u n W, (], [ N, X) such that u n (], [ N ) C u (], [ N ) for every n N and, as n +, u n u in L (], [ N, X). Remark 4 The map u n W, (], [ N, X) we construct is actually in a smaller space. If N =, u n is by construction Lipschitz continuous. For N 2, u n is continuous except at Σ n, where Σ n is a finite union of subsets Σ i n, i q n, where each Σ i n is the image by a linear injective map R N 2 R N of [, ] N 2. If N = 2, Σ i n is by convention a point. 9

20 Proof. Let u BV (], [ N, X). For n N, we consider the approximation v n given by Lemma 6, constant on each cube of the partition P n. The map v n is not in W, (], [ N, X), thus we now approximate the v n s. We denote v = v n for simplicity, and let < ε < /(2n). Assume first N =. For j n, v = a j on Q j = ( j n, j n ). Let 2l j d(a j, a j ). We fix, for every j n, a unit speed geodesic γ j : [ l j, l j ] in X from a j to a j. These geodesics exist since X is assumed to be a length space. We define then naturally u ε (x) = a j if x ( j n + ε, j n ε) for some < j < n, u ε (x) = a if x (, n ε), u ε(x) = a n if x ( n n ε, ), and for < j < n and x ( j n ε, j n + ε) Iε j, u ε(x) = γ j ( l j ε (x j n )). We easily check that u ε v a.e. in [, ], hence in L (], [ N, X) by dominated convergence, and u ε = n l j χ j= ε Ij ε v = n j= l j in measure as ε, with u ε (], [ N ) = v (], [ N ) (here, χ I ε j stands for the characteristic function of Ij ε). Since u ε W, (], [ N, X), the proof is complete if N =. constant geodesic interpolation ε homogeneous extension a a 2 a a 2 a 4 a a a The map v (left) and its approximation u ε (right) We assume now N = 2. We follow the approximation scheme given in the figure above, where we assume for simplicity that n = 2. By geodesic interpolation, for the upper rectangle for instance, we mean that, as in the -dimensional case, u ε (x) = γ( l ε (x 2 )), where 2l d(a, a 2 ) and γ : [ l, l] X is a unit speed geodesic from a to a 2 (in this rectangle, we have x 2 ε). Finally, by homogeneous extension, we mean that in the square C ε a + [ ε, ε] 2, where a = ( 2, 2 ), we have u ε (x) = u ε (a + ε x x ). Notice indeed that at this stage, u ε is already defined on the boundary of the square. The map u ε is therefore continuous except at a (if we are not in the case a = a 2 = a 3 = a 4 ), and in W, (], [ N, X) since u ε is lipschitz outside C ε and a direct computation gives u ε dx = ε u ε dh (x). C ε C ε Since u ε Cε on C ε, where C = C(a, a 2, a 3, a 4 ), we infer C ε u ε dx Cε. As a consequence, from the computations in the -dimensional case, we have d(a u ε = χ, a 2 ) d(a R ε 2 + χ 2, a 3 ) d(a R ε ε 23 + χ 3, a 4 ) d(a R ε ε 34 + χ 4, a ) R ε ε 4 + O ε L (ε), where R2 ε is the rectangle between a and a 2, where u ε is a geodesic interpolation between a and a 2 (and similarly for the other rectangles). Then, as ε, u ε v in L (], [ N, X) and u ε v as measure. This concludes the proof in the case N = 2. Finally, one extends easily this construction to an arbitrary N 2. The map u ε we construct in this case clearly belongs to W, (], [ N, X), and this completes the proof of the lemma. 2

21 2.7 Proof of Proposition 6 First, given p < and u N,p (, X), consider some functions w, ρ L p (, R) as in Definition 6. With the sequences u j = u and g j = ρ + w j, (3) is always satisfied for any -lipschitzian curve γ : [, l] (even when l w γ ds = ) one has immediately u H,p (, X) and H(u) N(u). The case p >. Let us embed X isometrically into a Banach space, for instance into l (X) by the Kuratowski embedding ι : X l (X) defined by ι(x) (d(x, z) d(x, y )) z X. Here, y X is an arbitrary fixed point and isometric means that ι(x) ι(y) l (X) = d(x, y) for every x, y X. It is clear that H(ι(u)) H(u). Hence, by [6], Theorem 3.2, there exists (since p > ) a function g L p (, R) such that H(ι(u)) = gp dx and for any -lipschitzian curve γ : [, l], ι(u) γ(l) ι(u) γ() l (X) = d(u γ(l), u γ()) l g γ ds. Therefore, u N,p (, X) and N(u) gp dx = H(u). Thus, if < p <, then, in L p (, X), H = N. The case p =. We assume now p =. Let u H, (, X). Then, there exist sequences u j L (, X) and g j L (, R + ) such that u j u and g j dx H(u) as j +, and, for any -lipschitzian curve γ : [, l], d(u j γ(), u j γ(l)) l g j γ ds. In particular, we have u j N, (, X) (take w =, ρ = g j ) and N(u j ) g j dx. As a consequence of Propositions 4 and 5, we infer u j W, (, X) and K,N E (u j ) R(u j ) = N(u j ) g j dx. Taking the liminf as j + and using the lower semi-continuity of E in L (, X) (Theorem 2), we deduce K,N E (u) K,N lim inf j + E (u j ) H(u). Therefore, H, (, X) BV (, X), and K,N E H on H, (, X). Assume now u BV (, X). To prove that u H, (, X), we have to construct a sequence in W, (, X) approximating u in L (, X) with controled -energy. First, since is smooth, we may extend u by an arbitrary constant outside, so that the energy of the extension is C()E (u). We may now assume, without loss of generality, that is a cube, and by scaling, that this cube is ], [ N. Now, we apply Lemma 7: there exists a sequence u n W, (, X) such that u n u in L (, X) as n + and E (u n ) C E (u) for every n N. Therefore, u H, (, X) (take g n = K,N u n as in section 2.4) and we obtain the second inequality H(u) lim inf n + g n dx K,N lim inf n + E (u n ) C K,N E (u). In the case where X is locally compact and separable, the link between H(U) and the BV energy Du () in the sense of L. Ambrosio can actually be made more precise. 2

22 Lemma 8 Assume that X is locally compact and separable. Then, Du () H(u) if u BV (, X), Du () = H(u) if u W, (, X). If, furthermore, X is a length space, then there exists C (N, X, ) such that for every u BV (, X), H(u) C Du (). Proof. The last inequality follows from Propositions 3 and 6. Let us prove the first one. Let u L (, X) and g L (, R + ) be such that d(u γ(l), u γ()) l g γ (8) for any -lipschitzian curve γ : [, l]. Then, if ϕ : X R is -lipschitzian, v ϕ u : R satisfies v γ(l) v γ() l g γ. Hence, given ε >, ω S N and taking, for x ε, γ(s) = x + sω, s [, ε], one obtains v(x + εω) v(x) ε g(x + εtω) dt. Assume for the moment that v, g : R are smooth. Dividing by ε and letting ε yields for every ω S N, which implies ω v (x) g(x) v g. The general case follows easily by standard regularization. As a consequence, u W, (, X) and Du () g dx. (9) If u H, (, X), then there exist sequences u j L (, X) and g j L (, R) such that (8) holds for any j and H(u) = lim j + g j dx. Therefore, Du j () g j dx by (9) gives at the limit Du () lim inf Du j () lim inf g j dx = H(u). j + j + Here, we have used that u Du () is lower semi-continuous for the L (, X) topology (see [], section ). This finishes the proof of the first inequality. Assume now u W, (, X). Let γ : [, l] be a -lipschitzian curve, and consider the -lipschitzian function ϕ d(, u γ()) : X R. Then, v ϕ u W, (, R) and Dv Du. Moreover, since v is real valued, Dv = v L (, R) (in the classical sense, see [], Remark 2.2). This implies d(u γ(l), u γ()) = v γ(l) v γ() l γ (s) ( v) γ(t) dt l Du γ(t) dt. Since Du L (, R), we infer that u H, (, X) and H(u) Du dx = Du (). The second inequality is proved. 22

23 2.8 Proof of Theorem 4 Let (ρ ε ) be a family of mollifiers (or (ρ n ) be a sequence of mollifiers), and recall ( d(u(x), u(y)) ) pρε F ε (u) = (x y) dxdy. x y We first prove that for any u L p (, X), lim sup F ε (u) E p (u). (2) ε Proof of (2). We may assume E p (u) <, that is u BV (, X) (if p = ) or u W,p (, X) (if < p < ). First, we have ( ( d(u(x), u(x + h)) ) pρε F ε (u) = (h) dx) dh. h R N \B η() ( h) R N ( h) Let us fix ξ X and η >. Then, ( ( d(u(x), u(x + h)) ) pρε (h) dx) dh h ( 2dL p(u, ξ) δ ) p R N \B η() ρ ε (h) dh and the last integral tends to as ε. Therefore, for any η >, ( d(u(x), u(x + h)) ) pρε F ε (u) = (h) dx) dh + o ε (). (2) h B η() ( h) Moreover, letting U be the extension by reflection across the boundary of u on δ as in Lemma 4, and using Theorem.8. and Lemma.9 in [2], we infer, assuming h η < δ /2, d p (u(x), u(x + h)) dx C(p, ) h p U p dx. ( ( h))\ η 2η \ 2η As a consequence, B η() ( ( h))\ η ( d(u(x), u(x + h)) h Inserting this into (2) yields for η < δ /2 lim sup F ε (u) lim sup ε ε B η() + C(p, ) ) pρε (h) dx) dh C(p, ) U p dx. 2η \ 2η η ( h) ( d(u(x), u(x + h)) h ) p dx ) ρ ε (h)dh 2η \ 2η U p dx (22) We denote ρ ε L (R + ) defined by ρ ε(x) = ρ ε ( x ). Finally, ω E p (u) standing for the p-energy in the direction ω S N (see [2], section.8), we have, using polar coordinates and [2], ( d(u(x), u(x + h)) ) p ) dx ρ ε (h)dh B η() η ( h) h η ( d(u(x), u(x + rω)) ) p ρ ε (r)r N dxdh N (ω)dr S N η r η S N ρ ε (r)r N [ ] ω S N E p (u) dh N (ω) dr. S N 23

24 From [2], Theorem.8., the term between brackets is E p (u), and S N η ρ ε(r)r N dr = B η ρ ε as ε, hence ( d(u(x), u(x + h)) ) p ) lim sup dx ρ ε (h)dh E p (u). ε h B η() η ( h) Combining this with (22) gives, for η < δ /2, lim sup F ε (u) E p (u) + C(p, ) U p dx. ε 2η \ 2η As in the proof of Theorem 3, the last integral tends to as η. This finishes the proof of (2). We now prove that if u ε u (or u n u) in L p (, X), then E p (u) lim inf ε F ε (u ε ) or E p (u) lim inf n + F n(u n ). (23) This will conclude the proof of Theorem 4, since for the pointwise limit, it suffices to take u ε = u (or u n = u) to obtain by (2) and (23) lim ε F ε (u) = E p (u) (or lim n + F n (u) = E p (u)), and the proof of the Γ-limit follows from (23) and the pointwise limit. Proof of (23). We will prove (23) only in the case of the sequence ρ n δ, since the general case will then follow. We may also assume without loss of generality that the lim inf n + is actually a lim n +. We denote ρ n (r) L loc (R + ) defined by ρ n(x) = ρ n ( x ). First, for any u L p (, X), ( d(u(x), u(y)) ) pρn F n (u) = (x y) dxdy x y [ ( d(u(x), u(y)) ) p = ρ n (r) dh (y)] N drdx S r(x) r ( ) S N r N ρ n (r) e r (u)(x) dx dr = E r (u)µ n (r) dr, (24) r where µ n (r) S N r N ρ n (r) is a probability measure on R + such that µ n δ as n +, i.e. for all δ >, δ µ n (r) dr as n +. (25) Suppose for a while that u n = u for all n N. Then, since E r (u) E p (u) R + {+ } as r, one infers easily that since µ n δ as n +, then E r (u)µ n (r) dr E p (u) R + {+ } as n +. Hence, lim inf n + F n (u) E p (u) R + {+ } as desired. To prove the result in the general case, we will prove in fact that there exist a subsequence n j and a sequence r j > such that j N, r j j and E rj (u nj ) F nj (u nj )( + ). (26) j We proceed by induction on j, and for j =, there is nothing to prove. Assume now r j (, j ) and n j constructed (satisfying (26)), such that (n,...n j ) is increasing. We fix, by (25), n j large enough so that n j > n j and j µ nj (r) dr 24 ( + j ).