Curvature-dependent energies: The elastic case

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1 Nonlinear Analysis 153 (2017) 7 34 Contents lists available at ScienceDirect Nonlinear Analysis Curvature-dependent energies: The elastic case Emilio Acerbi, Domenico Mucci University of Parma, Department of Mathematics and Computer Science, Parco Area delle Scienze 53/A, Parma, taly a r t i c l e i n f o a b s t r a c t Article history: Received 22 March 2016 Accepted 14 May 2016 Communicated by Enzo Mitidieri Dedicated to Prof. Nicola Fusco on his 60th birthday Keywords: Curvature Cartesian currents mage restoration We continue our analysis on functionals depending on the curvature of graphs of curves in high codimension Euclidean space. We deal with the elastic case, corresponding to a superlinear dependence on the pointwise curvature. We introduce the corresponding relaxed energy functional and prove an explicit representation formula. Different phenomena w.r.t. the plastic case, i.e. to the relaxation of the total curvature functional, are observed. A p-curvature functional is well-defined on continuous curves with finite relaxed energy, and the relaxed energy is given by the length plus the p-curvature. The wider class of graphs of one-dimensional BV-functions is treated Elsevier Ltd. All rights reserved. 1. ntroduction n this paper we continue our analysis begun in [1] concerning the curvature functional for non-smooth Cartesian curves in codimension higher than one. n the mathematical literature, energies depending on second order derivatives have recently been applied e.g. in image restoration processes, in order to overcome some drawbacks typical of approaches based on first order functionals, as the total variation. One instance is the approach by Chan Marquina Mulet [4] who proposed to consider regularizing terms given by second order functionals of the type u dx + ψ( u ) h( u) dx Ω Ω for scalar-valued functions u defined in two-dimensional domains, where the function ψ satisfies suitable conditions at infinity in order to allow jumps. The downscaled one-dimensional version of the above functional is given by b a u dt + b a ψ( u ) ü p dt, p 1, u : [a, b] R (1.1) Corresponding author. addresses: emilio.acerbi@unipr.it (E. Acerbi), domenico.mucci@unipr.it (D. Mucci) X/ 2016 Elsevier Ltd. All rights reserved.

2 8 E. Acerbi, D. Mucci / Nonlinear Analysis 153 (2017) 7 34 and it has been thoroughly studied in [5], where Dal Maso Fonseca Leoni Morini proved an explicit formula for the relaxed energy, under suitable assumptions on the function ψ. The prototypical example is the curvature energy functional, given by choosing ψ p (t) := n this case, in fact, the above functional takes the form E p (u) := b a u dt + b a 1 (1 + t 2 ) (3p 1)/ u(t)2 k u (t) p dt, k u (t) := ü(t) (1 + u(t) 2 ) 3/2. Therefore, in the smooth case, considering the Cartesian curve c u (t) := (t, u(t)), since k u (t) is the curvature at the point c u (t), and replacing the first term with the integral of, 1 + u 2, by the area formula one obtains an intrinsic formulation on the graph curve c u as E p (u) = L(c u ) + ku p dh 1, (1.2) c u where L is the length. Formula (1.2) can be taken in higher codimension N 2 as the definition of our energy functional. n fact, the curvature of a smooth Cartesian curve c u (t) = (t, u 1 (t),..., u N (t)) is given at the point c u (t) by the formula k u = k cu = ċ u c u u ü (1 + u 2 ) ü 2 ( u ü) 2 1/2 ċ u 3 = ċ u 3 =. (1 + u 2 ) 3/2 The relaxed energy. n this paper, we shall prove in high codimension a complete explicit formula for the relaxed energy. More precisely, we define E p (u) := inf lim inf E p(u h ) {u h } C 2 (, R N ), u h u in L 1 h for any summable function u L 1 (, R N ), where = [a, b]. n our recent paper [1] we treated the linear case p = 1, which corresponds to plasticity. We analyze here the elastic case p > 1, where different phenomena appear, as we now briefly illustrate. The superlinear growth of the curvature term implies that in the relaxation process the energy does not remain bounded if the curvature radius goes to zero at some point, see Example 3.7, as it instead happens in the plastic case. n codimension N = 1, one then obtains that a Cartesian curve with finite relaxed energy cannot have creases or edges, compare [5]. However, in the higher codimension case, corner points (only of a special type, say vertical ) are allowed. This phenomenon is illustrated in Example 6.2. Roughly speaking, there is sufficient freedom in the vertical directions in order that a sequence of smooth Cartesian curves approaches a curve with a corner point, by producing a twist in order to keep the curvature radius greater than some positive threshold, depending on p > 1. Finding the optimal twist at some corner point seems to be a great challenge. This is due to the difficulties in solving the Euler equation satisfied by energy minimizing loops, compare (5.8). Notwithstanding, differently to the case p = 1, the set of corner points is always finite. This follows from the fact that the contribution of the relaxed energy at any corner point is at least π/2, as soon as the exponent p > 1, compare Theorem 6.3. The Gauss map. We shall take advantage of several results that we proved in [1]. n fact, it turns out that a function u with finite relaxed energy E p (u) for some p > 1 also satisfies E 1 (u) <.

3 E. Acerbi, D. Mucci / Nonlinear Analysis 153 (2017) As a consequence, the function u has bounded variation, with distributional derivative decomposed as usual by Du = u L 1 + D C u + D J u, see [2]. A crucial role is played by the Gauss map τ u : S N that is defined a.e. in by means of the approximate gradient u, namely τ u = ċu ċ u, ċ u = (1, u 1,..., u N ). (1.3) Using that E 1 (u) <, in [1, Thm. 4.7] we proved that also the Gauss map τ u is a function with bounded variation. n Theorem 8.1, we shall see that for p > 1 actually τ u is a special function with bounded variation, i.e., its distributional derivative has no Cantor part: D C τ u = 0. Therefore, in the particular case of continuous functions u with finite relaxed energy and no corner points, we readily infer that τ u is a Sobolev function in W 1,1 (, S N ). Notice moreover that Theorem 8.1 is false for p = 1, even in codimension N = 1. A counterexample is given by the primitive u(t) := t v(s) ds of the classical Cantor Vitali function v : [0, 1] R associated to the 0 middle thirds Cantor set. We in fact have E 1 (u) < and D C τ u = f(v) D C v, where f(t) = (1, t)/ 1 + t 2, but E p (u) = for every exponent p > 1, see Remark 8.3. The total curvature. For our purposes, we recall that the total curvature TC(c) of a curve c has been defined by Milnor [10] as the supremum of the total curvature (i.e. the sum of the turning angles) of the polygons inscribed in c, see also [11]. A curve with finite total curvature is rectifiable, and hence it admits a Lipschitz parameterization. Therefore, it is well defined the tantrix (or tangent indicatrix), that assigns to a.e. point the oriented unit tangent vector t c. Moreover, the total curvature agrees with the essential total variation of the tantrix. For smooth Cartesian curves c u the tantrix t cu agrees with the Gauss map τ u, whence ċ u k u = τ u and the total curvature TC(c u ) is equal to the total variation of τ u, namely: TC(c u ) = k cu dh 1 = τ u dt. c u For continuous functions such that E 1 (u) <, in [1] we proved that the relaxed energy is the sum of the length and of the total curvature of the Cartesian curve c u, together with its representation as the total variation of the distributional derivative of the Gauss map: E 1 (u) = L(c u ) + TC(c u ), TC(c u ) = Dτ u (). The p-curvature functional. n this paper, we shall define the p-curvature functional of smooth Cartesian curves as TC p (c u ) := kc p u dh 1, p > 1. (1.4) c u Actually, using that ċ u k u = τ u, by the area formula we get TC p (c u ) = ċ u 1 p τ u p dt, p > 1. (1.5) We wish to extend our definition to the non-smooth case of continuous functions u with relaxed energy and with no corner points. To this purpose, in the proof of Theorem 7.1 we shall see that the arc-length parameterization L s γ(s) of the curve c u is a Sobolev function in W 2,p, where L := [0, L] and L = L(c u ). We can thus define the p-curvature functional by means of (1.4), where the curvature of c u at the point γ(s) is given a.e. by k cu (γ(s)) := γ(s) γ(s) γ(s) 3, s L.

4 10 E. Acerbi, D. Mucci / Nonlinear Analysis 153 (2017) 7 34 n fact, see Proposition 9.6, formula (1.5) continues to hold, and actually the p-curvature functional agrees with the integral of the p-power of the second derivative of the arc length parameterization: TC p (c u ) = γ(s) p ds <. L Main results. For continuous functions with finite relaxed energy and no corner points, in Corollary 9.7 we shall obtain a nice geometric formula for the relaxed energy: E p (u) = L(c u ) + TC p (c u ). (1.6) The above expression extends to the wider class of continuous functions with finite relaxed energy. n fact, using that the set J u of corner points is finite we shall introduce, Definition 9.8, the generalized p-curvature functional by setting TC p (c u ) := ċ u 1 p τ u p dt + Ep 0 (Γ p t ). t J u n the above formula, roughly speaking, the energy contribution Ep 0 (Γ p t ) is the integral of the p-power of the curvature of the optimal vertical curve that allows to smoothly connect the two edges on the graph of u at the corner c u (t), i.e., the optimal twist that we previously described. The more general expression of the relaxed energy (without assuming continuity of u) is: E p (u) = ċ u 1 p ċ u p + τ u p dt + D C u () + Ep 0 (Γ p t ). t J Φu This formula reduces to (1.6) in case of continuous functions. Here, we denote by J Φu the discontinuity set of the map Φ u (t) := (c u (t), τ u (t)), and Ep 0 (Γ p t ) is the energy of a minimizer among all vertical curves in {t} R N S N with end points given by the right and left limits Φ u (t ± ). For this reason, using tools from geometric measure theory, we need to extend our energy functional to 1-dimensional currents with vertical parts, i.e., to Cartesian currents. Plan of the paper. n Section 2, we introduce the energy functional and recall from [1] the definition of Gauss graph of Cartesian curves, whereas in Section 3, we deal with the energy functional E 0 p on currents, following the approach by Giaquinta Modica Souček [9]. n Section 4, we report from [1] the structure properties of the class of currents that naturally arise in the relaxation process. n Section 5, we then introduce a class of minimal currents associated to our relaxation problem. The following energy lower bound holds: the relaxed energy of u is greater than the energy of the corresponding minimal current with underlying function u. n Section 6, we shall outline some features concerning functions with finite relaxed energy. n particular, we shall prove that (in high codimension) the set of corner points is always finite, Theorem 6.3. The energy upper bound is then obtained in Section 7, by means of a suitable approximation result, Theorem 7.1. As a consequence, in Section 8 we shall prove that the Gauss map τ u of a function with finite relaxed energy is a BV-function with no Cantor part, Theorem 8.1. Therefore, if u is continuous and with no corner points, then τ u is a Sobolev function. Finally, Section 9 collects the main results that we previously described. 2. Notation and preliminary results n this section we introduce the energy functional and recall from [1] the definition of Gauss graph of Cartesian maps. Consider a C 2 -function u : R N where = [a, b] and N 1, so that the corresponding Cartesian curve is c u : R N+1 defined by c u (t) := (t, u(t)), where u = (u 1,..., u N ) in components. Any smooth

5 E. Acerbi, D. Mucci / Nonlinear Analysis 153 (2017) Cartesian curve is automatically regular, as ċ u (t) = (1, u(t)) for each t. For simplicity of notation we shall correspondingly denote by τ u and k u the tantrix and curvature of c u, respectively. We thus have τ u = ċu ċ u, ċ u = (1, u 1,..., u N ), ċ u = 1 + u 2. n codimension N = 1, i.e. for c u (t) = (t, u(t)) : R 2, the curvature of c u at the point c u (t) is k u (t) = ü(t) 1 + u(t) 2 3/2 (2.1) so that ċ u (t) k u (t) = v(t), where v(t) := arctan u(t). n general, denoting by v 1 v 2 and v 1 v 2 the wedge and scalar product of vectors in R n, respectively, the area of the parallelogram generated by v 1 and v 2 is area[v 1, v 2 ] = v 1 v 2 = { v 1 2 v 2 2 (v 1 v 2 ) 2 } 1/2. Therefore, the curvature of a smooth regular curve c in R n is k c := area[ċ, c] ċ 3 = ċ c ċ 3 = { ċ 2 c 2 (ċ c) 2 } 1/2 ċ 3. n particular, for a smooth Cartesian curve c u, where c u (t) = (t, u(t)) : R N+1, the curvature at the point c u (t) is k u := k cu = ċ u c u u ü (1 + u 2 ) ü 2 ( u ü) 2 1/2 ċ u 3 = ċ u 3 =. (2.2) (1 + u 2 ) 3/2 Energy functional. Let p 1 a real exponent. For u C 2 (, R N ) we denote E p (u) := L(c u ) + ku p dh 1. c u Using that τ u = u ü ċ u 2, u ü = ċ u c u = (1 + u 2 ) ü 2 ( u ü) 2 1/2, (2.3) we have ċ u k u = τ u. Whence by the area formula we get E p (u) = ċ u 1 + ku p dt = ċ u 1 p ċ u p + τ u p dt. We define E p (u) := inf{lim inf h E p(u h ) {u h } C 2 (, R N ), u h u in L 1 } for any summable function u L 1 (, R N ), and correspondingly E p (, R N ) := {u L 1 (, R N ) E p (u) < }. (2.4) Currents. Let U := R N. We deal with currents Σ D 1 (U S N ) of the type Σ = [[M, θ, ξ ]. This means that Σ acts on compactly supported and smooth 1-forms ω D 1 (U S N ) as Σ, ω = ω, ξ θ dh 1 ω D 1 (U S N ) M

6 12 E. Acerbi, D. Mucci / Nonlinear Analysis 153 (2017) 7 34 where M is a countably 1-rectifiable set, ξ : M R N+1 x R N+1 y is an H 1 M-measurable unit vector field and the multiplicity θ : M R is a non-negative H 1 M-measurable function. Whence the mass 1 of Σ is given by the formula M(Σ) = M θ dh1. n particular, the current Σ = [[M, θ, ξ ] is said to be an integer multiplicity (say i.m.) rectifiable current in R 1 (U S N ) if it has finite mass, the multiplicity function θ is integer-valued, and ξ is orienting the approximate tangent 1-space to M at H 1 M-a.e. point. n this case, H 1 (M) M(Σ) < and hence M is 1-rectifiable. Example 2.1. For smooth functions u C 2 (, R N ), in [1] we defined the Gauss-graph by the current GG u := Φ u# [[ ], where Φ u (t) := (c u (t), τ u (t)). Therefore, GG u is a 1-current in U S N, and setting ξ u (t) := Φ u (t) Φ u (t), where Φ u (t) = ċ u 1 + k 2 u (2.5) by the area formula for every ω D 1 (U S N ) we have GG u, ω := Φ u # ω = ξ u (t) ω(φ u (t), ξ u (t)) dt = ω, ξ u dh 1 GG u where GG u := {Φ u (t) t } is the Gauss graph of u. Whence GG u = [[GG u, 1, ξ u ] is an i.m. rectifiable current in R 1 (U S N ) with finite mass M(GG u ) = Φ u (t) dt <. More precisely, we shall denote by (dx 0, dx 1,..., dx N ) and (dy 0, dy 1,..., dy N ) the canonical bases of 1-forms in R N+1 x and R N+1 y, respectively. For any g Cc (U S N ) we thus obtain: GG u, g(x, y) dx 0 = g(φ u (t)) dt, GG u, g(x, y) dx j = g(φ u (t)) u j (t) dt, j = 1,..., N, (2.6) GG u, g(x, y) dy j = g(φ u (t)) τ u(t) j dt, j = 0, 1,..., N. Furthermore the current GG u has null-boundary 2 (in U S N ) as by Stokes theorem GG u, f := GG u, df = df = f = 0 f Cc (U S N ). GG u GG u 3. Energy functional on currents n this section we extend the energy functional to currents, following the approach by Giaquinta Modica Souček [9]. We prove a regularity property: if a current has finite p-energy, then it has finite mass, too. We then show that the converse implication is false, in general. Finally, we see that a sequential weak closure property holds. 1 The mass M(Σ) of a current Σ D1 (U S N ) is defined by M(Σ) := sup{ Σ, ω ω D 1 (U S N ), ω 1}. 2 The boundary Σ of a current Σ D1 (U S N ) is the 0-current (or distribution) in D 0 (U S N ) defined by Σ, f := Σ, df f C c (U SN ).

7 E. Acerbi, D. Mucci / Nonlinear Analysis 153 (2017) We shall denote by Π x and Π y the canonical projections of R N+1 x R N+1 y onto the first and the second factor, respectively. For any ξ R N+1 x R N+1 y, we let ξ (x) := Π x (ξ) and ξ (y) := Π x (ξ), so that ξ = (ξ (x), ξ (y) ). f u C 2 (, R N ), setting Ep 0 (GG u ) := ξ u (x) GG u 1 p ξ u (x) p + ξ u (y) p dh 1, ξ u (x) = ċ u Φ u, ξ(y) u = τ u Φ u where τ u = ċ u k u, by the area formula we have Ep 0 (GG u ) = Φ ċu u Φ u + ċ u ku p Φ dt = u ċ u 1 + ku p dt = Ep (u). (3.1) This suggests to introduce for smooth functions (u, v) : R N R N+1, with v 1, the energy integrand F p (u, v) := f p (ċ u, v) dt, f p (ċ u, v) := ċ u 1 p ċ u p + v p so that F p (u, τ u ) = E 0 p (u), and to give the following: Definition 3.1. We denote by F p : R N+1 x R N+1 y integrand f p in the sense of [9]. [0, + ] the parametric convex l.s.c. extension of the More precisely, denoting by ξ 0 R the first component of a vector ξ R N+1 x R N+1 y, we set f p (ξ) := ξ (x) 1 p ξ (x) p + ξ (y) p if ξ 0 = 1 we extend by homogeneity f p (ξ) := ξ 0 f p ξ ξ 0 and we define f p : R N+1 x R N+1 y [0, + ] by if ξ 0 > 0 f p (ξ) := f p (ξ) if ξ 0 > 0 + if ξ 0 0. Then by definition F p is the greatest convex and lower semicontinuous function that is lower than or equal to f p. Proposition 3.2. For every p > 1 one has: ξ (x) + ξ (x) 1 p ξ (y) p if ξ (x) > 0 and ξ 0 0 F p (ξ) = + otherwise. (3.2) Proof. We denote by ξ 0 the complementary components to ξ 0 of any vector ξ R N+1 x R N+1 y, so that ξ = (ξ 0, ξ 0 ). By convexity arguments one has F p (ξ) = sup{a + b ξ 0 a R, b R 2N+1, a + b G f p (1, G) G R 2N+1 }. (3.3) By convexity of f p one has F p (ξ) = f p (ξ) if ξ 0 > 0, and F p (ξ) = ξ (x) + ξ (x) 1 p ξ (y) p if ξ 0 0 provided that ξ (x) > 0. By homogeneity, for each λ > 0 and G R 2N+1 this yields f p (λ, λg) = Π x (λ, G) + Π y(λ, G) p Π x (λ, G) p 1. Choosing G so that Π x (0, G) = 0 and Π y (0, G) 0, we get f p (λ, λg) + as λ 0 +. By using (3.3), this implies that F p (ξ) = + if ξ (x) = 0. Finally, one similarly obtains that F p (ξ) = + if ξ 0 < 0.

8 14 E. Acerbi, D. Mucci / Nonlinear Analysis 153 (2017) 7 34 Remark 3.3. f p = 1, instead, one clearly has ξ (x) + ξ (y) if ξ 0 0 F 1 (ξ) = + otherwise. Definition 3.4. The p-curvature energy functional is given on currents Σ = [[M, θ, ξ ] by Ep 0 (Σ) := F p (ξ) θ dh 1. Therefore, in (3.1) we have just shown that in the smooth case E 0 p (GG u ) = E p (u). M Regularity. Let Σ = [[M, θ, ξ ]. For our purposes we shall tacitly assume w.l.g. that ξ 0 0. We now see that the functional Ep 0 is regular, i.e., every current Σ = [[M, θ, ξ ] with finite energy has finite mass, too. For p = 1, since F 1 (ξ) = ξ (x) + ξ (y) ξ we trivially have M(Σ) = ξ θ dh 1 F 1 (ξ) θ dh 1 = E1 0 (Σ), M yielding the regularity property. f p > 1, we observe that 1 + t p 2 1 (1 + t) for any t 0. We now show the regularity property. Proposition 3.5. f Σ = [[M, θ, ξ ], then M(Σ) 2 E 0 p (Σ) for all exponents p > 1. M Proof. For any ξ R N+1 x R N+1 y we get F p (ξ) 2 1 F 1 (ξ). n fact, the above inequality is trivial if ξ (x) = 0, as F p (ξ) = +. Assuming instead ξ (x) > 0, we estimate F p (ξ) = ξ (x) 1 + ξ(y) p ξ (x) ξ(y) = 2 1 ( ξ (x) + ξ (y) ) = 2 1 F ξ (x) p ξ (x) 1 (ξ). Since F 1 (ξ) ξ, we deduce that 2 F p (ξ) ξ, whence the mass estimate readily follows. Remark 3.6. For p = 1 we also have F 1 (ξ) 2 1/2 ξ, whence 2 1/2 E 0 1 (Σ) M(Σ) E 0 1 (Σ). Therefore, for currents Σ = [[M, θ, ξ ] such that ξ 0 0 H 1 -a.e., it turns out that Σ has finite mass if and only if it has finite energy E 0 1 (Σ). However, as the following example shows, it may happen that a sequence {Σ h } has equibounded masses but sup h E 0 p (Σ h ) = for every p > 1. Example 3.7. Let = [ 1, 1] and N = 1. For 0 < R < 1, let γ R : [ 1, 1 + π/2] U S 1 given by (θ, R, 1, 0) if 1 θ 0 γ R (θ) := (R sin θ, R cos θ, cos θ, sin θ) if 0 θ π/2 (R, θ π/2, 0, 1) if π/2 θ 1 + π/2 and define Σ R := γ R# [[ 1, 1 + π/2] R 1 (U S 1 ), so that Σ R = [[M R, 1, ξ R ] where M R = γ R ([ 1, 1 + π/2]) and ξ R (z) = γ R (θ)/ γ R (θ) if z = γ R (θ). Since (1, 0, 0, 0) if 1 θ < 0 γ R (θ) = (R cos θ, R sin θ, sin θ, cos θ) if 0 < θ < π/2 (0, 1, 0, 0) if π/2 < θ 1 + π/2

9 E. Acerbi, D. Mucci / Nonlinear Analysis 153 (2017) we compute 1 if 1 θ < 0 γ R (θ) = 1 + R 2 if 0 < θ < π/2 1 if π/2 < θ 1 + π/2 and hence ξ (x) R > 0 on M R, as (1, 0, 0, 0) if 1 θ < 0 ξ R (γ R (θ)) = (1 + R 2 ) 1/2 (R cos θ, R sin θ, sin θ, cos θ) if 0 < θ < π/2 (0, 1, 0, 0) if π/2 < θ 1 + π/2. Now, the mass of Σ R is equal to the length of the simple curve γ R, and by the area formula M(Σ R ) = γ R (θ) dθ = 2 + π 1 + R2. 2 As before, we let γ (x) R γ (x) R (θ) = [ 1,1+π/2] := Π x γ R and γ (y) R := Π x γ R, so that γ R = (γ (x) R, γ(y) R ). Using that 1 if 1 θ < 0 0 if 1 θ < 0 R if 0 < θ < π/2 γ (y) R (θ) = 1 if 0 < θ < π/2 1 if π/2 < θ 1 + π/2 0 if π/2 < θ 1 + π/2 for every p > 1, by Proposition 3.2 and by the area formula we compute (x) γ Ep 0 R (Σ R ) = γ R (θ) (θ) + γ(x) R (θ) 1 p γ(x) R (θ) p [ 1,1+π/2] γ R (θ) γ R (θ) 1 p γ R (θ) p dθ (x) = γ R (θ) + γ(x) R (θ) 1 p γ (y) R (θ) p dθ where and whence Similarly, one has whence we recover the estimate n particular, we have [ 1,1+π/2] [ 1,1+π/2] [ 1,1+π/2] γ (x) R (θ) dθ = 2 + π 2 R γ (x) R (θ) 1 p γ (y) R (θ) p dθ = π 2 R1 p, E 0 p (Σ R ) = 2 + π 2 R + π 2 R1 p. E1 0 (Σ R ) = 2 + π (R + 1), 2 2 1/2 E 0 1 (Σ R ) M(Σ R ) E 0 1 (Σ R ), 2 E 0 p (Σ R ) E 0 1 (Σ R ) p > 1. lim M(Σ R) = 2 + π R 0 + 2, lim E 0 R (Σ R ) = 2 + π 2, lim R 0 + E 0 p (Σ R ) = + p > 1. The cited example is given by Σ h = Σ Rh for a sequence of radii R h 0.

10 16 E. Acerbi, D. Mucci / Nonlinear Analysis 153 (2017) 7 34 Closure property. By definition it turns out that the functional Σ E 0 p (Σ) is lower semicontinuous w.r.t. the weak convergence in D 1 (U S N ). 3 We now point out a sequential weak closure property. Fix p > 1 and choose an i.m. rectifiable current Σ 0 = [[M, θ, ξ ] R 1 (U S N ) with finite energy, E 0 p (Σ 0 ) <. Setting Γ := Σ 0, we introduce the non-empty class F Γ := {Σ R 1 (U S N ) Σ = Γ, E 0 p (Σ) < }. Proposition 3.8. The class F Γ is closed along weakly converging sequences with equibounded energies. Therefore, the minimum of the problem inf{ep 0 (Σ) Σ F Γ } is attained. Proof. f {Σ h } h F Γ satisfies sup h Ep 0 (Σ h ) < and Σ h Σ, by the regularity property we have sup h M(Σ h ) <. We thus may apply Federer Fleming s closure theorem [7] to deduce that Σ R 1 (U S N ). The weak convergence conserving the boundary condition, we also have Σ = Γ. Finally, by lowersemicontinuity Ep 0 (Σ) lim inf h Ep 0 (Σ h ) <, whence Σ F Γ. Choosing an energy minimizing sequence {Σ h } h F Γ, possibly passing to a subsequence we have that Σ h Σ for some Σ D 1 (U S N ). The second assertion follows as Σ F Γ. 4. Gauss graphs with finite energy n this section we report from [1] the structure properties of the class of currents that naturally arise in the relaxation process. The class Gcart. For C 2 -functions u we have E p (u) 2 1 E 1 (u). As a consequence, see (2.4), if u E p (, R N ) then u E 1 (, R N ), and we may use some results proved in [1] for the case p = 1. We thus recall from [1] the class Gcart(U S N ), defined by Gcart(U S N ) := {Σ D 1 (U S N ) {u h } C 2 (, R N ) such that GG uh Σ in D 1 (U S N ), sup M(GG uh ) < }. (4.1) h Federer Fleming s closure theorem [7] yields that Σ is an i.m. rectifiable current in R 1 (U S N ) satisfying the null-boundary condition Σ = 0 on C c (U S N ). The weak convergence with equibounded masses implies the strong convergence u k u in L 1 to some function u BV(, R N ), that will be sometimes denoted by u Σ. Moreover, by Remark 3.6 we infer that actually u E 1 (, R N ). Therefore, each current Σ in Gcart(U S N ) decomposes as and we correspondingly have Σ = GG a u + GG C u + Σ, u = u Σ (4.2) M(Σ) = M(GG a u) + M(GG C u ) + M( Σ) <. The absolute continuous and Cantor components only depend on u = u Σ, and their definition makes sense because in [1], see also Section 8, we proved that the mapping Φ u (t) := (c u (t), τ u (t)) satisfies Φ u BV(, U S N ) u E 1 (, R N ) 3 The weak convergence Σh Σ of currents in D 1 (U S N ) is defined by duality as Σ h, ω Σ, ω ω D 1 (U S N ).

11 E. Acerbi, D. Mucci / Nonlinear Analysis 153 (2017) where U = R N. More precisely, if u E 1 (, R N ), not only u BV(, R N ), but also the Gauss map τ u := ċ u / ċ u belongs to BV(, S N ), provided that it is defined in terms of the approximate gradient u. Remark 4.1. Since the first component τ 0 u of the tantrix of a smooth Cartesian curve c u is positive, by weak BV-convergence we infer that the first component of τ u is a.e. non-negative for each u E 1 (, R N ), whence τ u takes values into the half-sphere S N + := {y = (y 0, y 1,..., y N ) R N+1 y : y = 1, y 0 0}. At any point t one has τ 0 u(t ± ) = 0 if and only if u(t ± ) = +. We thus correspondingly define S N 1 0 := {y S N + y 0 = 0} (4.3) so that τ 0 u(t ± ) = 0 τ u (t ± ) S N 1 0. n a similar way we deduce that the support sptσ U S N + for every Σ Gcart(U S N ). A geometric property. n [1] we proved that currents in Gcart(U S N ) preserve the geometry of Gauss graphs: indeed, for a regular Gauss graph (c u, τ u ) the second component τ u is the normalization of the derivative of the first component c u. More precisely, we proved that when the first component of the tangent vector to Σ at a point z = (x, y) U S N + is non zero, then it has to be parallel (and with the same verse) to the second component y of the point z, see (4.4). n fact, for every Σ Gcart(U S N ) we showed the existence of a Lipschitz function Ψ Lip( L, U S N ), where L := [0, L], such that the image current Ψ # [[ L ] agrees with Σ. Moreover, for a.e. s L such that Π x ( Ψ(s)) 0, we have Π x ( Ψ(s)) Π x ( Ψ(s)) = Π y(ψ(s)) S N +. (4.4) The a.c. component. According to Example 2.1, the current GG a u D 1 (U S N ) is given by GG a u, ω := ω(φ u (t)), Φ u (t) dt ω D 1 (U S N ) (4.5) so that GG a u := Φ u# [[ ] as in the smooth case, but this time the pull-back is defined a.e. in terms of the approximate gradient of the BV-function Φ u, whence the formulas (2.6) hold with GG u = GG a u. The Cantor component. The current GG C u D 1 (U S N ) is defined by linear extension of its action on basic forms. For any g Cc (U S N ) we set: (i) GG C u, g(x, y) dx 0 := 0 (ii) GG C u, g(x, y) dx j := g(φ u+) dd C u j, j = 1,..., N (iii) GG C u, g(x, y) dy j := g(φ u+) dd C τu, j j = 0, 1,..., N. We shall see, Theorem 8.1, that the Gauss map τ u of a function u with finite relaxed energy has no Cantor part, D C τ u = 0, and hence it is a special function with bounded variation. This will allow us to prove that the energy contribution of the Cantor component GG C u reduces to the total variation D C u (), see Remark 8.2. Boundary. We have: (GG a u + GG C u ) = δφu(t +) δ Φu(t ) on Cc (U S N ) t J Φu

12 18 E. Acerbi, D. Mucci / Nonlinear Analysis 153 (2017) 7 34 where δ P is the unit Dirac mass at the point P. n fact, for each f Cc (U S N ) we compute GG a u, f := GG a u, df = f(φ u+ ) dd C Φ u f(φu (t + )) f(φ u (t )) GG C u, f := GG C u, df = f(φ u+ ) dd C Φ u. Since the composition function f Φ u belongs to BV (), by definition of distributional derivative we deduce that D(f Φ u) = 0, whereas by choosing Φ u+ (t) = Φ u (t + ) as a precise representative, by the chain-rule formula we get t J Φu D(f Φ u ) = f(φ u ) Φ u dt + f(φ u+ ) D C Φ u + f(φ u+ ) f(φ u ) H 0 J Φu and hence (GG a u + GG C u ), f = f(φ u+ ) f(φ u ) H 0 J Φu. Jump-corner and corner components. As a consequence, the third component Σ in (4.2) satisfies the verticality condition Σ, g(x, y) dx 0 = 0 g C c (U S N ) and the boundary condition Σ = δφu(t +) δ Φu(t ) on C c (U S N ). t J Φu Also, the following decomposition in mass holds: Σ = Σ + t J Φu Γ t,σ, M( Σ) = M( Σ) + t J Φu M(Γ t,σ ), (4.6) where the current Σ R 1 (U S N ) satisfies the null-boundary condition Σ = 0, and Γ t,σ is for each t J Φu an a-cyclic i.m. rectifiable current in R 1 (U S N ), supported in {t} R N S N +, and with boundary Recalling that J Φu = J u (J u \ J u ), we denote by Σ Jc := Γ t,σ = δ Φu(t +) δ Φu(t ). (4.7) t J u Γ t,σ, Σ c := the jump-corner and corner components of a current Σ correspondingly call J u \ J u the set of corner points of u. t J u\j u Γ t,σ (4.8) in Gcart(U S N ), respectively. We shall Relation with Cartesian currents. For each Σ Gcart(U S N ), the projection T = T (Σ) := Π x# Σ is a Cartesian current in cart( R N ) in the sense of Giaquinta Modica Souček [9]. More precisely, according to (4.2), (4.6), and (4.8) we can write where one respectively has T = T a u + T C u + T J + T s, u = u Σ Π x# GG a u = T a u, Π x# GG C u = T C u, Π x# Σ Jc = T J, Π x# Σ c = 0, Π x# Σ = T s.

13 E. Acerbi, D. Mucci / Nonlinear Analysis 153 (2017) The absolute continuous component Tu a =: G u is given by Tu a = c u# [[ ], so that Tu a, φ(x) dx 0 = Φ(c u (t)) dt, Tu a, φ(x) dx j = Φ(c u (t)) u j (t) dt, for every φ C c ( R N ). The Cantor component T C u satisfies j = 1,..., N T C u, φ(x) dx 0 = 0, T C u, φ(x) dx j = D C u j, φ c u+, j = 1,..., N. The Jump component Tu J is given by Tu J, φ(x) dx 0 = 0, Tu J, φ(x) dx j = φ(x) dx j, t J γ u t(t ) j = 1,..., N where γ t (T ) is for every t J u an oriented, simple, and rectifiable arc in {t} R N with end points given by the one sided limits c u (t ± ), so that [[γ t (T )] = δ cu(t +) δ cu(t ). We remark that property Π x# Σ c = 0 follows from the fact that Γ t,σ is for each t J u \ J u an a-cyclic i.m. rectifiable current supported in {c u (t)} S N +. Arguing as before, one obtains the null-boundary condition (T a u + T C u + T J ) = 0. As a consequence, the singular component T s is a (vertical) i.m. rectifiable current such that T s = 0. Finally, a decomposition in mass holds, i.e. M(T ) = M(T a u ) + M(T C u ) + M(T J ) + M(T s ) where one has M(Tu a ) = ċ u dt, M(Tu C ) = D C u (), M(T J ) = M([[γ t (T )]). t J u 5. Energy lower bound n this section we introduce a class of minimal currents associated to our relaxation problem. n fact, we prove an energy lower bound: the relaxed energy is greater than the energy of the corresponding minimal current. The case p = 1. According to (4.2), for every u BV(, R N ) we define Gcart u := {Σ Gcart(U S N ) u Σ = u}, u BV(, R N ). (5.1) n the case p = 1, in [1] we in fact proved the following. Proposition 5.1. Let u L 1 (, R N ), where N 1. Then u E 1 (, R N ) Gcart u. n this case, moreover, we have E 1 (u) = min{e1 0 (Σ) Σ Gcart u }. Minimal currents. n order to obtain a similar formula in the case p > 1, for every u E p (, R N ) we now introduce an optimal current Σ p u of the type: Σ p u := GG a u + GG C u + S p u, (5.2)

14 20 E. Acerbi, D. Mucci / Nonlinear Analysis 153 (2017) 7 34 where the third component S p u R 1 (U S N ) is given by S p u := t J Φu Γ p t for suitable minimal currents Γ p t that we now describe, see Definition 5.4. For any point t J Φu we first introduce a suitable class F(u, t) of a-cyclic i.m. rectifiable currents Γ in R 1 (U S N ), supported in {t} R N S N +, with boundary Γ = δ Φu(t +) δ Φu(t ) (5.3) and such that Ep 0 (Γ ) <. By Federer s structure theorem [6, ], for any such Γ we find a Lipschitz function γ = γ Γ : [ 1/2, 1/2] U S N with constant velocity γ(s) = M(Γ ) for a.e. s, and γ # [[ 1/2, 1/2] = Γ. Therefore, we deduce that γ (x) (s) > 0 a.e., and as in Example 3.7 we obtain that Ep 0 (Γ ) = γ (x) (s) + γ (x) (s) 1 p γ (y) (s) p ds. (5.4) [ 1/2,1/2] According to the geometric property (4.4), we say that Γ F(u, t) if in addition one has γ (x) (s) γ (x) (s) = γ(y) (s) for a.e. s [ 1/2, 1/2], γ = γ Γ. (5.5) Remark 5.2. Let Γ F(u, t) for some t J Φu. By property (5.5) we deduce that the second component γ (y) := Π y (γ Γ ) is the tantrix of the first component γ (x) := Π x (γ Γ ), and that the curve γ (x) has positive length, as γ (x) (s) > 0 a.e. Therefore, formula (5.4) yields that the p-energy of Γ is equal to the p-curvature functional of the curve Π x (γ Γ ), i.e. Ep 0 (Γ ) = (1 + kc p ) dh 1, c := Π x (γ Γ ). (5.6) c Finally, notice that the vertical cycle Σ defined in Example 6.2 satisfies the above property (5.5), whence Σ F(u, t), with t = 0 J u. Existence. We have: Proposition 5.3. For every u E p (, R N ) and t J Φu, the minimum of the problem is attained. inf{e 0 p (Γ ) Γ F(u, t)} Proof. n fact, by (4.7) and the structure property (4.4), we deduce that the class F(u, t) is non-empty for each t J Φu. We then choose a minimizing sequence {Γ h } h in the class F(u, t). On account of Proposition 3.8, and using that sup h M(Γ h ) 2 sup h Ep 0 (Γ h ), by Federer Fleming s theorem we deduce that (up to a subsequence) {Γ h } h weakly converges to a current Γ in R 1 (U S N ), supported in {t} R N S N +, and satisfying the boundary-condition (5.3). By lower-semicontinuity one has Ep 0 (Γ ) = inf{ep 0 (Γ ) Γ F(u, t)}. Therefore, denoting γ = γ Γ : [ 1/2, 1/2] U S N the parameterization of Γ as above, we only have to show that the geometric property (5.5) holds for γ = γ. Since Ep 0 (Γ ) <, we deduce as above that γ (x) (s) > 0 for a.e. s. Moreover, by assumption we know that (5.5) holds for γ h := γ Γh for each h, and we thus have to prove that such a geometric property is preserved by weak convergence. This property has been proved in [1, Thm. 6.3], and for completeness we report here the same argument.

15 E. Acerbi, D. Mucci / Nonlinear Analysis 153 (2017) Setting a(s) := γ (x) (s) and b(s) := γ (y) (s) we prove that the two vectors a(s) and b(s) are parallel and pointing the same way, for a.e. s [ 1/2, 1/2]. Now, given two vectors a, b R N+1, with a 0 and b = 1, they are parallel and pointing the same way if and only if a/ a = b, that is equivalent to a b = a, since (a/ a ) b 2 = a b a. Since M(Γ h ) 2 Ep 0 (Γ h ), and γ h (s) = M(Γ h ) for a.e. s [ 1/2, 1/2], by Ascoli s theorem we may assume that {γ h } h uniformly converges to the Lipschitz function γ. Therefore, letting a h (s) := γ (x) h (s) and b h (s) := γ (y) h (s), we infer that b h(s) b(s) S N uniformly. Since moreover γ h γ weakly-* in L, we deduce that a h (s) a(s) weakly-* in L, whence lim h [ 1/2,1/2] a h (s) b h (s) ds = [ 1/2,1/2] a(s) b(s) ds. Since moreover a h (s) a(s) weakly in L 1, the lower semicontinuity a(s) ds lim inf a h (s) ds h holds, and hence [ 1/2,1/2] [ 1/2,1/2] [ 1/2,1/2] a(s) a(s) b(s) ds lim inf ah (s) a h (s) b h (s) ds. h [ 1/2,1/2] Since (5.5) holds for γ = γ h, we have seen that b h (s) = 1, a h (s) 0, a h (s)/ a h (s) = b h (s) for a.e. s and for all h. Therefore, we obtain a(s) a(s) b(s) ds 0. [ 1/2,1/2] The integrand being non-negative by the Schwartz inequality a b a b = a, we deduce that a(s) a(s) b(s) = 0 for a.e. s [ 1/2, 1/2], whence the two vectors a(s) and b(s) are parallel and pointing the same way, as required. Definition 5.4. For each t J Φu we denote by Γ p t a minimum point for Ep 0 in the class F(u, t). n particular, Γ p t is an a-cyclic current in R 1 (U S N ). From our definition (5.2) we thus obtain: Ep 0 (Σu) p := Ep 0 (GG a u) + Ep 0 (GG C u ) + Ep 0 (Su), p Ep 0 (Su) p = Ep 0 (Γ p t ). (5.7) t J Φu Remark 5.5. Examples 6.2 and 6.5 will show that in high codimension N 2 in general ċ u (t ) ċ u (t + ) at some point t J Φu, even if we always have ċ u (t ± ) =. As a consequence, differently to what happens in the case p = 1, it turns out that if t J Φu the optimal current Γ p t is not supported on a straight line-segment, in general. Euler equation. For the sake of completeness, we report here the result of the computation of the corresponding Euler Lagrange equation.

16 22 E. Acerbi, D. Mucci / Nonlinear Analysis 153 (2017) 7 34 Proposition 5.6. For any non-negative smooth function f of the curvature k of curves γ, the Euler equation of the functional F(γ) := γ f(k) dh1 is f(k) k ċ +... f (k) k 2 ċ f(k) k (ċ c) ċ 3 + k k f(k) f(k) ċ = 0. (5.8) n particular, for f(k) = 1 + k 2, so that F(γ) = L(γ) + γ k2 dh 1, the above equation takes the simpler form: 2 k ċ 2 k (ċ c) ċ 3 + k (k 2 1) ċ = 0. Therefore, even for p = 2 it seems a difficult task to write explicitly the above minima Γ p t. Notice that by choosing the arc-length parameterization, one has ċ = 1 and (ċ c) = 0 a.e., whence Eq. (5.8) reduces to the classical one f(k) k +... f (k) k 2 + k k f(k) f(k) = 0 compare [8, Ch. 1, Sec. 5]. For f(k) = 1 + k p, where p > 1, so that F(γ) = L(γ) + γ k p dh 1, the above equation takes the simpler form: p k p 2 k + p(p 2) k p 4 k k 2 + k k p k (p 1) = 0. Now, searching for smooth closed planar curves with constant curvature, i.e. for minimal circles of radius R, since k R 1 we deduce that the above equation is solved when R = (p 1) 1/p. More generally, even in the presence of first order boundary conditions, minimizing curves in general depend on the choice of the exponent p > 1. Energy lower bound. As a consequence, we readily obtain the following energy lower bound: Corollary 5.7. Let p > 1. For every u E p (, R N ) we have E p (u) E 0 p (GG a u) + E 0 p (GG C u ) + E 0 p (S p u) where E 0 p (S p u) = t J Φu E 0 p (Γ p t ). Proof. By lower semicontinuity we get E p (u) inf{ep 0 (Σ) Σ Gcart(U S N ), u Σ = u} see (4.2), where Ep 0 (Σ) = Ep 0 (GG a u) + Ep 0 (GG C u ) + Ep 0 ( Σ) and according to (4.6) E 0 p ( Σ) = E 0 p ( Σ) + t J Φu E 0 p (Γ t,σ ). The energy lower bound follows as by minimality E 0 p (Γ t,σ ) E 0 p (Γ p t ) for each t J Φu.

17 E. Acerbi, D. Mucci / Nonlinear Analysis 153 (2017) Functions with finite relaxed energy n this section we outline some features concerning functions with finite relaxed energy. n particular, differently to what happens for N = 1, we see that corner points may occur in high codimension. However, we prove that the set of corner points of a function with finite relaxed energy is always finite, Theorem 6.3. The Gauss map. We first point out a property of the Gauss map at the Jump set J Φu. Notice that the following proposition is false in the case p = 1, compare [1] and Remark 6.4. Proposition 6.1. Let N 1 and u E p (, R N ) for some p > 1. Then τ 0 u(t ± ) = 0 for every t J Φu. Proof. Let {u h } C 2 (, R N ) be such that sup h Ep 0 (u h ) < and u h u strongly in L 1. Using that sup h E1 0 (u h ) <, possibly passing to a subsequence we find the existence of a current Σ Gcart(U S N ) such that GG uh Σ weakly in D 1 (U S N ), see (4.1). By lower semicontinuity we have Ep 0 (Σ) <. Since Ep 0 (Σ) = Ep 0 (GG a u)+ep 0 (GG C u )+Ep 0 ( Σ), according to the decomposition (4.6) we have Ep 0 (Γ t,σ ) < for every t J Φu. Assume that τu(t 0 ± ) > 0 for some point t J Φu. Since Γ t,σ is an a-cyclic i.m. rectifiable current in R 1 (U S N ), supported in {t} R N S N +, the boundary condition (4.7) joined with the structure property (4.4) implies that the unit tangent vector ξ has zero component ξ (x) at a subset of points in set(γ t,σ ) with positive H 1 -measure. By the structure (3.2) of the energy integrand, this yields that Ep 0 (Γ t,σ ) = +, a contradiction. Corner points. Condition τ 0 u(t ± ) = 0 is equivalent to u(t ± ) = +. Therefore, in case of codimension N = 1, Proposition 6.1 implies that u(t + ) = u(t ) {± } for each t J Φu, otherwise one would obtain as above that E 0 p (Γ t,σ ) = +, a contradiction. This implies that J u = and hence, compare (4.8), that the corner component Σ c = 0 if N = 1, as already shown in [5]. Example 3.7 may suggest the absence of corner points for Cartesian curves c u of continuous functions u E p (, R N ). We now see that this is not the case in high codimension N 2. Example 6.2. For = [ 1, 1] and N = 2, consider the continuous W 1,1 -function u : R 2 given by (0, t u(t) := 2 2t) if t 0 ( t 2 + 2t, 0) if t 0 so that the graph curve c u is the union of two quarters of unit circles meeting at the point 0 R 3, centered at the points ( 1, 0, 0) and (1, 0, 0) and lying in the hyperplanes x 1 = 0 and x 2 = 0, respectively. Since ċ u (0 ) = (1, 0, + ) and ċ u (0 + ) = (1, +, 0), we have τ u (0 ) = (0, 0, 1) and τ u (0 + ) = (0, 1, 0), thus a corner point with turning angle equal to π/2 appears at the point 0 R 3, whence J u \ J u = {0}. t is not difficult to check that u Ep 0 (, R 2 ) for each p > 1. n fact, one may approximate u by a smooth sequence {u h } such that the Gauss graphs GG uh weakly converge to a current Σ in Gcart(U S 2 ) given by Σ = GG a u + Σ, where Σ is the 1-current integration of a smooth curve in {0} R 2 S 1 0, see (4.3), with end points Φ u (0 ± ) = (0 R 3, τ u (0 ± )) and satisfying the geometric property (4.4). Since sup h Ep 0 (u h ) <, we deduce that u Ep 0 (, R 2 ). When p = 1, one simply has to smoothen the angle by means of an arc with small curvature radius. For p > 1, instead, good approximations (i.e., with small energy contribution) are performed by means of vertical arcs which have curvature radius greater than a positive constant, depending on p, see Fig. 1 on the left. Differently to the case p = 1 analyzed in [1], the vertical current Σ is not supported in {c u (0)} S 1 0, compare (4.3). An example is Σ := γ # [[π/2, π ], where γ : [π/2, π] R 3 x R 3 y is the regular curve defined in

18 24 E. Acerbi, D. Mucci / Nonlinear Analysis 153 (2017) 7 34 Fig. 1. To the left, the curve c u and (dashed) the current Σ which lies in the vertical plane {t = 0}; to the right, one term of the smooth approximating sequence c uh. Fig. 2. The small notches on the t and x 2 axes are 1/h away from the origin; the angled lines thus have slopes 1/h and h. components γ (x) := Π x γ and γ (y) := Π y γ by γ (x) (θ) := (0, γ x (θ)) and γ (y) (θ) := (0, γ y (θ)), where γ x (θ) := sin θ cos 2 θ, cos θ sin 2 θ, γ y (θ) := γ(x) (θ) γ (x) (θ), so that the geometric property (4.4) holds true. More explicitly, one has cos θ(3 sin 2 θ 1), sin θ(1 3 cos 2 θ) γ (y) (θ) = (3 sin 4 θ 3 sin 2 θ + 1) 1/2 θ [π/2, π] whence γ(π/2) = (0 R 3, τ u (0 )) = Φ u (0 ) and γ(π) = (0 R 3, τ u (0 + )) = Φ u (0 + ), and according to (4.7) Σ = γ # [[π/2, π ] = δ γ(π) δ γ(π/2) = δ Φu(0 +) δ Φu(0 ). An approximating sequence u h : [ 1, 1] R 2 satisfying sup h E 0 p (u h ) < is defined by widening the base of the vertical arc so that the loop is made on the interval 1/h t 1/h; the resulting arc does no longer begin and end vertically, but with slope h, so we cannot simply glue the two parts of the graph of u to it (conveniently separated); the problem is easily solved by cutting away a suitable (and small) terminal portion of each arc and translating it so that the three pieces fit, see Fig. 2. The resulting curve is defined on a very slightly larger interval than [ 1, 1] but a reparameterization does the trick, see Fig. 1 on the right. Since {u h } h W 1,1 (, R N ), by applying Step 2 of Theorem 7.1 to each u h, a diagonal argument yields the existence of a smooth sequence {u h } h C 1 (, R N ) with sup h E 0 p (u h ) < and u h u strongly in L 1. We omit any further detail.

19 E. Acerbi, D. Mucci / Nonlinear Analysis 153 (2017) Finiteness of corner points. We now see that for every p > 1, the set J u \J u of corner points of a function u E p (, R N ) with finite relaxed energy is always finite. On account of Definition 5.4 and Corollary 5.7, it clearly suffices to prove the following. Theorem 6.3. Let u E p (, R N ) for some p > 1. For every t J u \ J u we have E 0 p (Γ p t ) π/2. Proof. Let Γ F(u, t) and consider the curve c := Π x (γ Γ ) where γ Γ : [ 1/2, 1/2] U S N is the Lipschitz function with constant velocity such that γ Γ # [[ 1/2, 1/2] = Γ. The boundary condition (5.3), where Φ u (t ± ) = (c u (t), τ u (t ± )) by the continuity of u at the point t, yields that c is a closed curve with initial and final velocities parallel to the unit vectors τ u (t ± ) S N 1 0, and τ u (t ) τ u (t + ). By (5.6) we estimate Ep 0 (Γ ) = + kc c(1 p ) dh k c ) dh 2 c(1 1 1 k c dh 1. 2 c Therefore, it suffices to observe that the total curvature c k c dh 1 of any closed arc with positive length is at least π, compare [10] and also [3, Lemma 3.2]. Remark 6.4. Theorem 6.3 fails to hold in the case p = 1. t suffices to consider a piecewise affine and continuous function u : [0, 1] R with a countable set of corner points J u = {t j := 1 2 j j N + } such that (setting t 0 = 0) the slope of u at the interval ]t j 1, t j [ is equal to 2 j for every j. n fact, the length of the Cartesian curve c u is lower than 2, and its total curvature is equal to π/4, whence u E 0 1 (, R). Notice that by Proposition 6.1 we get E p (u) = + for every p > 1. Discontinuity points. n general a function with finite relaxed energy may have a non-trivial jump set J u. This was already observed in [5] when N = 1, and we give here an example for N = 2. Example 6.5. Similarly to Example 6.2, consider the BV-function u : [ 1, 1] R 2 given by (0, t u(t) := 2 2t) if t 0 ( t 2 + 2t, 3) if t > 0 so that u has a jump point at the origin, as u(0 ) = (0, 0) and u(0 + ) = (0, 3). This time the graph of u is the union of two quarters of unit circles centered at the points ( 1, 0, 0) and (1, 1, 3) and lying in the hyperplanes x 1 = 0 and x 2 = 3, respectively. We again have ċ u (0 ) = (1, 0, + ) and ċ u (0 + ) = (1, +, 0), so that τ u (0 ) = (0, 0, 1) and τ u (0 + ) = (0, 1, 0). Again, it is not difficult to check that u Ep 0 (, R 2 ) for each p > 1. n fact, one may approximate u by a smooth sequence {u h } such that the Gauss graphs GG uh weakly converge to a current Σ in Gcart(U S 2 ) given by Σ = GG a u + Σ, where Σ is the 1-current integration of a smooth curve in {0} R 2 S 1 0 with end points Φ u (0 ) = ((0, 0, 0), (0, 0, 1)) and Φ u (0 + ) = ((0, 0, 3), (0, 1, 0)) and satisfying the geometric property (4.4). Since sup h Ep 0 (u h ) <, we deduce again that u E p (, R 2 ). An explicit formula for Σ can be obtained by slightly modifying the definition of γ from Example 6.2, this time defining a regular curve γ : [π/2, π] R 3 x R 3 y such that γ y (θ) = γ(x) (θ) γ (x) (θ) θ [π/2, π] whereas γ(π/2) = ((0, 0, 0), τ u (0 )) = Φ u (0 ) and γ(π) = ((0, 0, 3), τ u (0 + )) = Φ u (0 + ). Whence the geometric property (4.4) is satisfied and according to (4.7) we again have Σ = δ Φu(0 +) δ Φu(0 ). Countable discontinuity set. The lower bound in Theorem 6.3 is false if t J u, i.e. on the Jump component Σ J u. As a consequence, in general the Jump set J u of a function u E p (, R N ) is countable, for any p > 1. We sketch here an example in codimension N = 2.

20 26 E. Acerbi, D. Mucci / Nonlinear Analysis 153 (2017) 7 34 Consider u : [0, 1] R 2 whose components u j are increasing and bounded functions that are discontinuous on the countable set J u = {t i = 1 2 i i N + } and smooth outside J u, in such a way that the integral ċ u p 1 ( ċ u p + τ u p ) dt is finite, the series i u(t i+) u(t i ) is convergent, and u is a BV-function with no Cantor-part, D C u () = 0. We define u with unbounded right and left derivatives at the discontinuity set, i.e. u(t i ±) = + for each i, and the Gauss map τ u : S 1 is a BV-function with no Cantor part, D C τ u = 0, and discontinuity set J τu = J u, i.e. u has no corner points. Then τ u (t i ±) S 1 0 for each i, see Remark 4.1. Denoting by d i the geodesic distance between τ u (t i ) and τ u (t i +) in S 1 0, we can define u in such a way that the series i d i is convergent. Finally, denoting by Γ p t i a minimum point for Ep 0 in the class F(u, t i ), see Definition 5.4, we can define u in such a way that also E 0 p (Γ p t i ) c p ( u(t i +) u(t i ) + d i ) i for some constant c p > 0 not depending on i. By our relaxation result, Proposition 9.2, we have E p (u) = ċ u 1 p ċ u p + τ u p dt + Ep 0 (Γ p t i ) < i=1 and hence u E 0 p (, R 2 ), but H 0 (J u ) = +. We omit any further detail. 7. Energy upper bound Let p > 1 and N 1. Recalling that the optimal current Σ p u is defined by (5.2), in this section we prove the following density result. Theorem 7.1 (Energy Upper Bound). For every u E p (, R N ), there exists a sequence of smooth functions {u h } C 2 (, R N ) such that u h u strongly in L 1, GG uh Σ p u weakly in D 1 (U S N ) and E p (u h ) E 0 p (Σ p u) as h. Convolution argument. As a preliminary step, we report here a density argument from Step 1 in [1, Thm. 8.1], readapted to the case p > 1. Proposition 7.2. Let u E p (, R N ) be a Sobolev function in W 1,1 (, R N ), with u L (, R N ). Then τ u W 1,p (, S N ). Moreover, there exists a smooth sequence {u h } C 2 (, R N ) such that u h u strongly in W 1,1 and lim ċ uh 1 p ċ uh p + τ uh p dt = ċ u 1 p ċ u p + τ u p dt. (7.1) h Proof. Since u E 1 (, R N ), we know that Φ u = (c u, τ u ) is a BV-function, where c u (t) = (t, u(t)) and ċ u = 1 + u 2, τu 0 := 1 ċ u, τ u j := uj, j = 1,..., N. ċ u Moreover, if {u h } C 2 (, R N ) is such that u h u in L 1 and sup h E p (u h ) <, since sup h E 1 (u h ) <, in [1] we showed that (possibly passing to a subsequence) u h u a.e. in. f u W 1, (, R N ), we may and do assume that sup h ( u h + u h ) <. Therefore, using that sup h ċ uh < and sup h ċ u h 1 p τ uh p dt <, we deduce that sup h τ u h p dt <. Therefore, possibly passing to a subsequence we conclude that τ uh τ u weakly in W 1,p and hence τ u W 1,p (, S N ). For each j 1 we choose a sequence {v j h } C () that converges strongly in W 1,p to τu. j Since moreover u <, we may assume that v j h τu j < 1 for each h and j. Denoting v h = (vh 1,..., vn h ), and

21 E. Acerbi, D. Mucci / Nonlinear Analysis 153 (2017) vh 0 := 1 v h 2, we compute v h 0 = v h v h 1 vh. 2 Therefore, by dominated convergence we deduce that the sequence {vh 0} C () converges strongly in W 1,p to τu. 0 Setting then for j = 1,..., N w j v j h h (t) := (t) 1 vh (t), 2 uj h (t) := uj (a) + we now check the following convergences as h : t a w j h (s) ds, t (i) w j h uj strongly in L 1, for each j; (ii) u j h (t) uj (a) + t a uj (s) ds = u j (t) a.e. and strongly in L 1 (); (iii) 1 + uh 2 dt 1 + u 2 dt, hence u h u in W 1,1 (, R N ); (iv) vh 2 + ( v h 0)2 p/2 dt τ u p dt. Differently to the cited case p = 1, we point out that property (iv) holds true as the sequence (vh 0, v h) : S N converges to τ u strongly in W 1,p. n Step 1 from [1, Thm. 8.1], we also computed τ uh = ċ uh k uh = v h 2 + (v h v h ) 2 1/2 = 1 v h 2 v h 2 + ( v h 0)2. By property (iv) this implies that Using that sup h ċ uh 1 p and the claim follows. lim τ uh p dt = τ u p dt. h <, by dominated convergence we get ċ uh 1 p τ uh p dt = ċ u 1 p τ u p dt lim h Remark 7.3. f u E p (, R N ) W 1,, then Σu p = GG a u, whence Ep 0 (Σu) p = Ep 0 (GG a u). Moreover, arguing as in the smooth case, compare (3.1), we deduce that Ep 0 (GG a u) = ċ u 1 p ċ u p + τ u p dt u E p (, R N ). (7.2) Therefore, Proposition 7.2 yields the validity of Theorem 7.1 for the subclass of Lipschitz functions u in W 1, (, R N ). Proof of Theorem 7.1. According to the previous remark, by a diagonal argument it suffices to find an approximating sequence {u h } in W 1, (, R N ). Let u E p (, R N ) and write Σ p u = [[M, θ, ξ ], so that the multiplicity θ 1. By Corollary 5.7 we have E 0 p (Σ p u) <. On account of (4.4), Definition 5.4, and (3.2), we deduce that for H 1 -a.e. z M the following geometric property holds: ξ (x) ξ (x) (z) = y, z = (x, y) U SN.