Micol Amar

Virginia De Cicco

Comptes Rendus Mathématique

1/ftionutile poursult isionsariousw-kes intoconti-imizersfudedC. R. Acad. Sci. Paris, Ser. I 340 (2005) 735–738http://france.elsevier.com/direct/CRASSFunctional AnalysisA new approximation result for BV-functionsMicol Amar, Virginia De CiccoUniversità di Roma “La Sapienza”, Dipartimento di Metodi e Modelli Matematici, Via A. Scarpa 16, 00161 Roma, ItalyReceived 5 November 2004; accepted after revision 22 March 2005Presented by Philippe G. CiarletAbstractThis Note is devoted to obtaining an approximation result for BV-functions by means of aquasi-polyhedralsequence oBV-functions. This approximation could have interesting applications in some problems of the Calculus of Variations.To citethis article: M. Amar, V. De Cicco, C. R. Acad. Sci. Paris, Ser. I 340 (2005). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméUn nouveau résultat d’approximation pour fonctions BV. On démontre un théorème de approximation pour une foncqui appartient à l’espace BV avec une suite quasi-polyédriques de fonctions BV. Cette approximation peut être trèsquelques problèmes du Calcul des Variations.Pour citer cet article : M. Amar, V. De Cicco, C. R. Acad. Sci. Paris, Ser. I 340(2005). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.1. IntroductionIn this Note, we prove an approximation result for BV-functions. In this direction, the first classical redue to Anzellotti–Giaquinta, who proved that BV-functions can be approximated by means of smooth functuεwhich are essentially obtained by mollification, so that their main interest lies on theC∞-regularity.In the framework of the Calculus of Variations, this type of approximation has been usefully applied in vproblems concerning relaxation andΓ -convergence, where{uε} plays the role of the ‘recovery sequence’. Hoever, in some recent problems it seems more useful to have an approximation of BV-functions which taaccount not so much theC∞-regularity of the approximating sequence, as the geometric properties of its disnuity set. This idea has been firstly developed by Dibos and Séré, in the context of the approximation of minfor the Mumford–Shah functional (see [5]). They proved an approximation result forSBV-functions by means ofunctions, still belonging toSBV, with their jump set contained in a finite union of smooth hypersurfaces, inclin hyperplanes.E-mail addresses:amar@dmmm.uniroma1.it (M. Amar), decicco@dmmm.uniroma1.it (V. De Cicco).1631-073X/$ – see front matter 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2005.03.027736 M. Amar, V. De Cicco / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 735–738nctions,etions,moreet” ad byfrit toofmations inFollowing the outlines of Dibos and Séré, Cortesani and Toader in [4] (see also [3]) proved that those fuwhich have a polyhedral jump set and are of classC∞ outside, are dense inSBVp ∩ L∞, in an appropriate sensconnected with the Mumford–Shah functional.In view of similar possible applications, we propose a new approximation result for general BV-funcwhich therefore could also be of Cantor type. This implies that the approximating functions are slightlygeneral than those proposed by Cortesani and Toader. More precisely, givenu ∈ BV∩L∞ we construct a sequencof BV-functions, strictly converging tou, such that their set of approximate discontinuity points is “almospolyhedron, in the sense that theHN−1-measure of the non-polyhedral part is small. This result is obtainerefining a classical theorem, due to Federer, of approximation of a countableHN−1-rectifiable set by means osmooth compact manifolds which are arbitrarily close to a polyhedron, apart from a set of smallHN−1-measure.We think that this type of approximation could be useful to study problems of relaxation andΓ -convergence fofunctionals defined on the wholeBV. On the other hand, this approach, being very intrinsic, should also permsimplify the study of similar problems onSBV.2. PreliminariesIn the following, we assume thatN  2 be a fixed integer andΩ be an open bounded set inRN .Let u ∈ L1loc(Ω); we say thatu has anapproximate limitat x ∈ Ω , if there exists a unique valuẽu(x) ∈ R suchthatlimr→0+−∫Br (x)∣∣u(y) − ũ(x)∣∣dx = 0, (1)where−∫Br (x)stands for 1LN(Br (x))∫Br (x)andBr(x) denotes the ball inRN centered inx with radiusr . Let Su bethe set of points where the previous property does not hold, the so-calledapproximate discontinuity set. Note thatSu is a Borel set and̃u :Ω \ Su → R is a Borel function. Clearly, ifx is a Lebesgue point ofu, then (1) holds withũ(x) replaced byu(x). Moreover, we recall thatLN -almost everyx ∈ Ω is a Lebesgue point ofu.Let us recall a useful property for the composition of BV-functions with Lipschitz functions.Theorem 2.1(see Theorem 3.16 in [2]).LetΩ,Ω ′ be open subsets ofRN andφ :Ω ′ → Ω be a bijective Lipschitzfunction, whose inverse is Lipschitz, too. Letu ∈ BV(Ω) andv = u ◦ φ. Thenv ∈ BV(Ω ′) and1[Lip(φ)]N−1 |Du|(φ(B)) |Dv|(B)  [Lip(φ−1)]N−1|Du|(φ(B))for every Borel subsetB of Ω ′.3. Main resultsIn the first theorem, we improve a fine property of countableHN−1-rectifiable sets stated in Theorem 4.2.19[6] (see also Theorem 3.2 in [1]). This result is the crucial tool in order to obtain the quasi-polyhedral approxiof BV-functions, which will be stated in Theorem 3.3.Theorem 3.1.LetBR be a ball inRN andS ⊂ BR be a countableHN−1-rectifiable set. Then for everyε > 0 thereexists a diffeomorphismφε :RN → RN satisfying the following properties:(i) φε :BR → BR and, outsideBR , φε = Id (whereId denotes the identity functionId(x) = x);(ii) Lip (φε), Lip(φ−1ε )  1+ ε;(iii) there exists a compact setKε ⊂ BR , such thatKε is a polyhedron composed by a finite number of cube(N − 1) dimensions andHN−1(S  φ (K )) < ε.ε εM. Amar, V. De Cicco / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 735–738 737tseesor to1,sowMoreover for any family{φε} satisfying(i) and (ii) we have:(iv) asε → 0+, φε → Id, uniformly onRN ;(v) asε → 0+, |J (φ−1ε )| and|J (φε)| tend to1 uniformly onΩ , whereJ (φ−1ε ) andJ (φε) denote the determinanof the Jacobian matrices ofφ−1ε andφε, respectively.Proof. The existence of the diffeomorphismφε with the properties (i)–(iii) is proved in Theorem 4.2.19 of [6] (salso Theorem 3.2 in [1]). Assume now that{φε} is a family satisfying (i) and (ii); we will show that it satisfies al(iv) and (v). By (ii), it follows (1 + ε)−1|x − y|  |φε(x) − φε(y)|  (1 + ε)|x − y| and |φε(x)|  2R + 3|y0|,wherex ∈ BR andy0 ∈ RN \ BR ; the same holds forφ−1ε . Hence, the sequences{φε}, {φ−1ε } are equiboundedand equicontinuous, then there existφ0, ψ0 :RN → RN such that, up to a subsequence,φε → φ0 andφ−1ε → ψ0uniformly onBR (and so on the whole ofRN , sinceφε = φ−1ε = Id outsideBR). It is easy to see thatφ0(ψ0(x)) =ψ0(φ0(x)) = x, so thatψ0 = φ−10 . Moreover,|φ0(x) − φ0(y)| = |x − y|, which implies thatφ0 (and henceφ−10 ,too) is a linear isometric map. Indeed, assume firstly thatφ0(0) = 0; then we have|φ0(x)| = |x|, which impliesalso that〈x, y〉 = 〈φ0(x),φ0(y)〉, where〈·, ·〉 denotes the usual scalar product inRN ; hence, if{ei}i=1,...,N is anorthonormal basis inRN , then{φ0(ei)}i=1,...,N is an orthonormal basis inRN , too. Givenx = ∑αiei , we canwrite φ0(x) = ∑βiφ0(ei), whereαi = 〈x, ei〉 = 〈φ0(x),φ0(ei)〉 = βi . This proves thatφ0 is a linear isometricmap, which coincides with the identity outsideBR , henceφ0 ≡ Id. If φ0(0) = x0 = 0, it is enough to replaceφ0with φ̃0(·) = φ0(·) − x0. Hence, the whole sequence{φε}, and not only a subsequence, tends to Id. In ordeprove (v), we note that, by (ii) and recalling that for every invertible matrixA, the det(A−1) = 1/det(A), it follows1(1+ε)N  |J (φε)|  (1+ ε)N and 1(1+ε)N  |J (φ−1ε )|  (1+ ε)N , which concludes the proof.In order to state our main result (Theorem 3.3), we need the following definition.Definition 3.2.Givenu ∈ BV(Ω)∩L∞(Ω), we say that a sequence{uε} ⊆ BV(Ω)∩L∞(Ω) is aquasi-polyhedralapproximation ofu, if the following properties hold:• the sequence{uε} is equibounded inL∞(Ω);• uε → u strongly inL1(Ω), asε → 0;• |Duε|(Ω) → |Du|(Ω), asε → 0;• there exists a sequence of polyhedra{Kε} ⊆ Ω such thatHN−1(Suε  Kε) → 0, asε → 0;• HN−1(Suε ) → HN−1(Su), asε → 0.Theorem 3.3.Let Ω = BR . For everyu ∈ BV(Ω) ∩ L∞(Ω), there exists a quasi-polyhedral approximation{uε}of u. More precisely, for everyε > 0, settinguε := u ◦ φε (whereφε is given in Theorem3.1), it follows thatuε ∈ BV(Ω) ∩ L∞(Ω) and(i) ‖uε‖∞ = ‖u‖∞ and‖uε‖BV  (1+ ε)N‖u‖BV;(ii) uε → u strongly inL1(Ω), asε → 0;(iii) |Duε|(Ω) → |Du|(Ω), asε → 0;(iv) there exists a polyhedronKε ⊂ Ω and a constantc > 0 such thatHN−1(Suε  Kε)  cε;(v) (1+ ε)1−NHN−1(Su)  HN−1(Suε )  (1+ ε)N−1HN−1(Su).Proof. For everyε > 0, defineuε := u ◦ φε, whereφε :RN → RN is the diffeomorphism given in Theorem 3.with S := Su, which is a countableHN−1-rectifiable subset ofΩ . Clearly, by definition,‖uε‖∞ = ‖u‖∞. ByTheorem 2.1 the functionuε belongs to BV(Ω) ∩ L∞(Ω) and, for every 0< ε  1, an easy calculations yield‖uε‖BV  c, wherec is a positive constant independent ofε. Hence, there existsu0 ∈ BV(Ω) such that, up to asubsequence,u → u strongly inL1(Ω) andDu ⇀ Du weakly∗ in the sense of measures. Firstly, we will shε 0 ε 0738 M. Amar, V. De Cicco / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 735–738ve alsod (ii)of thew York,–49.9–162.thatu0 = u, which implies also that the whole sequence, not only a subsequence, converges tou. Let N ⊂ Ω , withLN(N) = 0, be such that for everyx ∈ Ω \ N , up to another subsequence,uε(x) → u0(x) andu0(x) = limr→01ωNrN∫Br(x)u0(y)dy and u(x) = limr→01ωNrN∫Br (x)u(y)dy, (2)whereωN denotes the Lebesgue measure of the unit ball inRN . Recalling Theorem 3.1, we have that|Jφ−1ε | and|Jφε| tend to 1, asε → 0+, uniformly onΩ . Moreover, for everyr > 0 and everyx ∈ Ω \ N , we have∣∣uε(x) − u(x)∣∣ ∣∣∣∣uε(x) − 1ωNrN∫Br(x)uε(y)dy∣∣∣∣ +∣∣∣∣ 1ωNrN∫φε(Br (x))u(y)[∣∣Jφ−1ε (y)∣∣ − 1]dy∣∣∣∣+∣∣∣∣ 1ωNrN[ ∫φε(Br (x))u(y)dy −∫Br(x)u(y)dy]∣∣∣∣ +∣∣∣∣ 1ωNrN∫Br (x)u(y)dy − u(x)∣∣∣∣.Passing to the lim sup asε → 0+ and taking into account that{uε} tends tou0 strongly inL1(Ω) and pointwise inΩ \ N , we obtainlim supε→0+∣∣uε(x) − u(x)∣∣ ∣∣∣∣u0(x) − −∫Br(x)u0(y)dy∣∣∣∣ + ‖u‖∞ωNrN lim supε→0+ LN(φε(Br(x)))[supΩ∣∣∣∣Jφ−1ε (y)∣∣ − 1∣∣]+ ‖u‖∞ωNrNlim supε→0+LN(φε(Br(x))  Br(x)) +∣∣∣∣ 1ωNrN∫Br(x)u(y)dy − u(x)∣∣∣∣=∣∣∣∣u0(x) − 1ωNrN∫Br(x)u0(y)dy∣∣∣∣ +∣∣∣∣ 1ωNrN∫Br(x)u(y)dy − u(x)∣∣∣∣,for everyr > 0. Now, lettingr → 0+ and taking into account (2), it follows|uε(x) − u(x)| → 0, whenε → 0.This implies thatuε(x) → u(x) for a.e.x ∈ Ω and since we have also thatuε(x) → u0(x) for everyx ∈ Ω \ N ,it follows thatu = u0 almost everywhere and that the whole sequence{uε} tends tou in L1(Ω). Now, taking intoaccount the lower semicontinuity of the total variation, (i) and (ii) of Theorem 3.1, and Theorem 2.1, we hathat |Duε|(Ω) → |Du|(Ω), whenε → 0. Hence, (iii) is proven.Finally, by (iii) of Theorem 3.1, there exists a polyhedronKε ⊂ Ω such thatHN−1(Su  φε(Kε)) < ε . Hence,taking into account thatSuε = φ−1ε (Su), the properties of Hausdorff measures (see Proposition 2.49 in [2]), anof Theorem 3.1, it followsHN−1(Suε  Kε) = HN−1(φ−1ε (Su  φε(Kε)))  (1 + ε)N−1HN−1(Su  φε(Kε)) (1+ ε)N−1ε. In a similar way we obtain (v), which concludes the proof.AcknowledgementsWe are grateful to N. Fusco for having indicated us Theorem 3.2 in his paper [1].References[1] L. Ambrosio, N. Fusco, J.E. Hutchinson, Higher integrability of the gradient and dimension of the singular set for minimizersMumford–Shah functional, Calc. Var. Partial Differential Equations (2) 16 (2003) 187–215.[2] L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Ne2000.[3] G. Cortesani, Strong approximation ofGSBVfunctions by piecewise smooth functions, Ann. Univ. Ferrara Sez. VII (N.S.) 43 (1998) 27[4] G. Cortesani, R. Toader, A density result inSBVwith respect to non-isotropic energies, Nonlinear Anal. 38 (1999) 585–604.[5] F. Dibos, E. Séré, An approximation result for the minimizers of the Mumford–Shah functional, Boll. Un. Mat. Ital. 11-A (1997) 14[6] H. Federer, Geometric Measure Theory, Springer-Verlag, Berlin, 1969.

A new approximation result for BV-functions

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A new approximation result for BV-functions

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