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A chain rule formula in \(BV\) and application to lower semicontinuity. (English) Zbl 1136.49011

Introduction: It is well known that if \(u\) is a \(BV\) function from a bounded open set \(\Omega\subset\mathbb R^N\) and \(B:\mathbb R\to\mathbb R\) is Lipschitz, the composition \(v=B\circ u\) is also in \(BV(\Omega)\) and the following chain rule formula holds
\[ Dv= B'(\widetilde{u})\nabla u{\mathcal L}^N+ B'(\widetilde{u}) D^cu+ (B(u^+)- B(u^-)) \nu_u{\mathcal H}^{N-1}\llcorner J_u, \]
where \(\nabla u\) is the absolutely continuous part of \(Du\), \(D^cu\) is the Cantor part of \(Du\) and \(J_u\) is the jump set of \(u\) (for the definition of these and other relevant quantities, see Sect. 2). A delicate issue about this formula concerns the meaning of the first two terms on the right hand side. In fact, in order to understand why they are well defined, one has to take into account that \(B'(t)\) exists for \({\mathcal L}^1\)-a.e. \(t\) and that, if \(E\) is an \({\mathcal L}^1\)-null set in \(\mathbb R\), not only \(\nabla u\) vanishes \({\mathcal L}^N\)-a.e. on \(\widetilde{u}^{-1}(E)\), but also \(|D^cu| (\widetilde{u}^{-1}(E))=0\) (see [L. Ambrosio, N. Fusco and D. Pallara “Functions of bounded variation and free discontinuity problems”, Oxford: Clarendon Press (2000; Zbl 0957.49001)], Theorem 3.92). The difficulty of giving a correct meaning to the various parts in which the derivative of a \(BV\) function can be split is even greater when \(u\) is a vector field, a case where a chain rule formula has been proved by L. Ambrosio and G. Dal Maso in [Proc. Am. Math. Soc. 108, No. 3, 691–702 (1990; Zbl 0685.49027)]. In particular, their result applies to the composition of a scalar \(BV\) function with a Lipschitz function \(B\) depending also on \(x\), namely to the function \(B(x, u(x))\), where \(B:\Omega\times\mathbb R\to\mathbb R\) is Lipschitz. In many applications, however, \(B\) has the special form
\[ B(x,t)= \int_0^t b(x,s)\,ds, \tag{1} \]
but, on the other hand, one would like to assume only a weak differentiability of \(B\) with respect to \(x\) (or even less). In this spirit, V. De Cicco and G. Leoni [Calc. Var. Partial Differ. Equ. 19, No. 1, 23–51 (2004; Zbl 1056.49019)] have obtained a chain rule formula in the case \(B(x,t)\) is a vector field such that \(\operatorname{div}_xB(\cdot,t)\) belongs to \(L^1(\Omega)\), uniformly with respect to \(t\), and \(u\) is in \(W^{1,1}(\Omega)\). In the same paper they prove an \(L^1\)-lower semicontinuity result in \(W^{1,1}\) by applying their formula to a vector field \(B\) of the type (1).
In this paper we extend these results to the case where \(u\) is a \(BV\) function and replace the assumption that \(\operatorname{div}_xB\) is in \(L^1\) with a \(BV\) dependence with respect to \(x\).

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
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References:

[1] Ambrosio L, Dal Maso G, (1990) A general chain rule for distributional derivatives. Proc. Amer. Math. Soc. 108, 691–702 · Zbl 0685.49027 · doi:10.1090/S0002-9939-1990-0969514-3
[2] Ambrosio L, Fusco N., Pallara D.(2000) Functions of bounded variation and free discontinuity problems. Oxford University Press, Oxford · Zbl 0957.49001
[3] Chen G.-Q., Frid H. (1999) Divergence-measure fields and hyperbolic conservation laws. Arch. Rational Mech. Anal. 147, 89–118 · Zbl 0942.35111 · doi:10.1007/s002050050146
[4] Dal Maso G. (1980) Integral representation on BV({\(\Omega\)}) of {\(\Gamma\)}–limits of variational integrals. Man. Math. 30, 387–416 · Zbl 0435.49016 · doi:10.1007/BF01301259
[5] De Cicco V. (1991) Lower semicontinuity for certain integral functionals on BV({\(\Omega\)}). Boll. U.M.I. 5-B: 291–313 · Zbl 0738.46012
[6] De Cicco V., Leoni G. (2004) A chain rule in L 1(div;{\(\Omega\)}) and its applications to lower semicontinuity. Calc. Var. Partial Differential Equations 19(1): 23–51 · Zbl 1056.49019 · doi:10.1007/s00526-003-0192-2
[7] De Cicco V., Fusco N., Verde A. (2005) On L 1-lower semicontinuity in BV. J. Convex Anal. 12, 173–185 · Zbl 1115.49011
[8] De Giorgi E. (1954) Su una teoria generale della misura (r 1)-dimensionale in uno spazio a r dimensioni. Ann. Mat. Pura Appl. 36(4): 191–213 · Zbl 0055.28504 · doi:10.1007/BF02412838
[9] De Giorgi E.: Teoremi di semicontinuità nel calcolo delle variazioni. Istituto Nazionale di Alta Matematica (1968–1969)
[10] Federer H. (1969) Geometric measure theory. Springer, Berlin Heidelberg New York · Zbl 0176.00801
[11] Fonseca I., Leoni G. (2001) On lower semicontinuity and relaxation. Proc. R. Soc. Edim., Sect. A, Math. 131, 519–565 · Zbl 1003.49015 · doi:10.1017/S0308210500000998
[12] Fusco N., Giannetti F., Verde A. (2003) A remark on the L 1 lower semicontinuity for integral functionals in BV, Manusc. Math. 112, 313–323 · Zbl 1030.49014 · doi:10.1007/s00229-003-0400-6
[13] Fusco N., Gori M., Maggi F.: A remark on Serrin’s Theorem. NoDEA (to appear) · Zbl 1215.49024
[14] Gori M., Marcellini P. (2002) An extension of the Serrin’s lower semicontinuity theorem. J. Convex Anal. 9, 475–502 · Zbl 1019.49021
[15] Gori M., Maggi F., Marcellini P. (2003) On some sharp conditions for lower semicontinuity in L 1. Diff. Int. Eq. 16, 51–76 · Zbl 1028.49012
[16] Serrin J. (1961) On the definition and properties of certain variational integrals. Trans. Amer. Math. Soc. 161, 139–167 · Zbl 0102.04601 · doi:10.1090/S0002-9947-1961-0138018-9
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