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A notion of total variation depending on a metric with discontinuous coefficients. (English) Zbl 0842.49016
The authors consider the problem of defining a perimeter of a set when $$\mathbb{R}^n$$ is endowed with a general distance. More precisely, if $$\Omega$$ is a Lipschitz bounded open subset of $$\mathbb{R}^n$$ and $$\phi: \Omega\times \mathbb{R}^n\to [0, +\infty[$$ is a Borel function with $$\phi(x,\cdot)$$ convex and one-homogeneous, and such that $$\phi(x, z)\leq C|z|$$, for every $$u\in \text{BV}(\Omega)$$ the total variation of $$u$$ with respect to $$\phi$$ is defined by $\int_\Omega |Du|_\phi= \sup\Biggl\{ \int_\Omega u\text{ div } \sigma dx\Biggr\},$ where the supremum is taken over all vector fields $$\sigma\in L^\infty(\Omega; \mathbb{R}^n)$$ with compact support in $$\Omega$$ such that $$\text{div } \sigma\in L^n(\Omega)$$ and $$\phi^0(x, \sigma(x))\leq 1$$ on $$\Omega$$, $$\phi^0$$ being the dual function of $$\phi$$.
It is proven that $$\int_\Omega |Du|_\phi$$ can be represented in an integral form with respect to the measure $$|Du|$$. Moreover, if $${\mathcal F}[\phi]: \text{BV}(\Omega)\to [0, +\infty]$$ is the functional defined by ${\mathcal F}[\phi](u)= \begin{cases} \int_\Omega \phi(x, \nabla u(x))dx\quad & \text{if } u\in W^{1,1}(\Omega),\\ + \infty\quad & \text{otherwise},\end{cases}$ and if $$\overline{{\mathcal F}[\phi]}: \text{BV}(\Omega)\to [0, +\infty]$$ denotes the relaxed functional of $${\mathcal F}[\phi]$$ in the topology $$L^1(\Omega)$$, we have $\int_\Omega |Du|_\phi= \overline{{\mathcal F}[\phi]}(u)\qquad \forall u\in \text{BV}(\Omega).$ Finally, setting for all $$u\in \text{BV}(\Omega)$$ $$\nu^u(x)= {dDu\over d|Du|} (x)$$ and $${\mathcal I}[\phi](u)= \int_\Omega \phi(x, \nu^u(x)) d|Du|$$ it is shown that the relaxed functional $$\overline{{\mathcal I}[\phi]}$$ of $${\mathcal I}[\phi]$$ on $$\text{BV}(\Omega)$$ coincides with $$\overline{{\mathcal F}[\phi]}$$ provided $$\phi$$ is upper semicontinuous. On the contrary, when $$\phi$$ is only a Borel function, denoting by $${\mathcal N}(\Omega)$$ the class of Lebesgue negligible subsets of $$\Omega$$ and setting for every $$N\in {\mathcal N}(\Omega),$$ $\phi_N(x, z)= \begin{cases} \phi(x, z)\quad & \text{if } x\in \Omega\backslash N,\\ C|z|\quad & \text{if } x\in N,\end{cases}$ then for all $$u\in \text{BV}(\Omega)$$, $\int_\Omega |Du|_\phi= \sup_{N\in {\mathcal N}(\Omega)} \overline{{\mathcal I}[\phi_N]} (u).$
Reviewer: G.Buttazzo (Pisa)

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation 49Q20 Variational problems in a geometric measure-theoretic setting
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##### References:
 [1] Alberti, G., A Lusin type theorem for gradients, J. Fund. Anal., Vol. 100, 1, 110-118, (1991) · Zbl 0752.46025 [2] G. Anzellotti, Traces of Bounded Vectorfields and the Divergence Theorm, preprint Univ. Trento, 1983. [3] Anzellotti, G., Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura e Appl., Vol. 135, 4, 293-318, (1983) · Zbl 0572.46023 [4] Barroso, A. C.; Fonseca, I., Anisotropic singular perturbations. the vectorial case, (1992), Carnegie Mellon University, Research Report n.92-NA-015 [5] Bouchitté, G.; Valadier, M., Integral representation of convex functionals on a space of measures, J. Funct. Anal., Vol. 80, 398-420, (1988) · Zbl 0662.46009 [6] Bouchitté, G., Singular perturbations of variational problems arising from a two-phase transition model, Appl. Math. and Opt., Vol. 21, 289-315, (1990) · Zbl 0695.49003 [7] G. Bouchitté and G. Dal Maso, Integral Representation and Relaxation of Convex Local Functionals on BV(Ω), preprint SISSA, April 1991, to appear on Ann. Scuola Norm. Sup., Pisa. [8] H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Masson, Paris, 1983. [9] Busemann, H.; Mayer, W., On the foundations of calculus of variations, Trans. Amer. Math., Soc., Vol. 49, 173-198, (1941) · JFM 67.1036.03 [10] Busemann, H., Metric methods in Finsler spaces and in the foundations of geometry, Ann. of Math. Studies, Vol. 8, (1942), Princeton Univ. Press Princeton · Zbl 0063.00672 [11] G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Longman, Harlow, 1989. · Zbl 0669.49005 [12] Buttazzo, G.; Dal Maso, G., Integral representation and relaxation of local functionals, Nonlinear Analysis, Theory, Methods & Applications, Vol. 9, 515-532, (1985) · Zbl 0527.49008 [13] Castaing, C.; Valadier, M., Convex analysis and measurable multifunctions, Lect. Not. in Math, Vol. 580, (1977), Springer-Verlag Berlin · Zbl 0346.46038 [14] Dacorogna, B., Direct methods in the calculus of variations, (1989), Springer-Verlag Berlin · Zbl 0703.49001 [15] Dal Maso, G., Integral representation on BV(ω) of г-limits of variational integrals, Manuscripta Math., Vol. 30, 387-416, (1980) · Zbl 0435.49016 [16] G. Dal Maso, An Introduction to Г-Convergence, preprint SISSA, 1992, Trieste, to appear on Birkhäuser. [17] G. Dal Maso and L. Modica, A General Theory of Variational Functionals, in: Topics in Functional Analysis (1980-1981), Quaderni Scuola Norm. Sup. Pisa, Pisa, 1980, pp. 149-221. [18] G. De Cecco, Geometria sulle Varietà di Lipschitz, preprint Univ. di Roma “La Sapienza”, 1992, pp. 1-38. [19] De Cecco, G.; Palmieri, G., Distanza intriseca su una varietà riemanniana di Lipschitz, Rend. Sem. Mat. Univ. Pol. Torino, Vol. 46, 157-170, (1988) [20] De Cecco, G.; Palmieri, G., Lenght of curves on lip manifolds, Rend. Mat. Acc. Lincei, Vol. 1, 9, 215-221, (1990) · Zbl 0719.53046 [21] De Cecco, G.; Palmieri, G., Integral distance on a Lipschitz Riemannian manifold, Math. Zeitschrift, Vol. 207, 223-243, (1991) · Zbl 0722.58006 [22] E. De Giorgi, Nuovi Teoremi Relativi alle Misure (r-1)-dimensionali in uno Spazio a r Dimensioni, Ricerche Mat. IV, 1955, pp. 95-113. · Zbl 0066.29903 [23] De Giorgi, E., Conversazioni di matematica, Quaderni Dip. Mat. Univ. Lecce, Vol. 2, (1990), 1988-1990 [24] De Giorgi E. Su Alcuni Problemi Comuni all’Analisi e alla Geometria. Note di Matematica, IX-Suppl., 1989, pp. 59-71. [25] E. De Giorgi, Alcuni Problemi Variazionali della Geometria, Conference in onore di G. Aquaro, Bari, Italy, 9 Nov. 1990, 1991, pp. 112-125, Conferenze del Sem. Mat. Bari. [26] E. De Giorgi, F. Colombini and L. C. Piccinini, Frontiere Orientate di Misura Minima e Questioni Collegate, Quaderni della Classe di Scienze della Scuola Normale Superiore di Pisa, Pisa, 1972. [27] Ekeland, I.; Теmam, R., Convex analysis and variational problems, (1976), North-Holland New York [28] Federer, H., Geometric measure theory, (1968), Springer-Verlag Berlin [29] Fusco, N.; Moscariello, G., L^2-lower semicontinuity of functionals of quadratic type, Ann. di Mat. Pura e Appl., Vol. 129, 305-326, (1981) · Zbl 0483.49008 [30] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, 1984. · Zbl 0545.49018 [31] Goffman, C.; Serrin, J., Sublinear functions of measures and variational integrals, Duke Math. J., Vol. 31, 159-178, (1964) · Zbl 0123.09804 [32] Luckhaus, S.; Modica, L., The Gibbs-thomson relation within the gradient theory of phase transitions, Arch. Rat. Mech. An, Vol. 1, 71-83, (1989) · Zbl 0681.49012 [33] Massari, U.; Miranda, M., Minimal surfaces of codimension one, (1984), North-Holland New York · Zbl 0565.49030 [34] Μaz’ya, V. G., Sobolev spaces, (1985), Springer-Verlag Berlin [35] Miranda, M., Superfici cartesiane generalizzate ed insiemi di perimetro localmente finito sui prodotti cartesiani, Ann. Scuola Norm. Sup., Vol. 3, 515-542, (1964), Pisa · Zbl 0152.24402 [36] C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin, 1966. · Zbl 0142.38701 [37] J. Neveu, Bases Mathématiques du Calcul des Probabilités, Masson, Paris, 1980. [38] Owen, N., Non convex variational problems with general singular perturbations, Trans. Am. Math. Soc., Vol. 310, 393-404, (1988) · Zbl 0718.34075 [39] Owen, N.; Sternberg, P., Non convex variational problems with anisotropic perturbations, Nonlin. Anal., Vol. 16, 705-719, (1991) · Zbl 0748.49034 [40] C. Pauc, La Méthode Métrique en Calcul des Variations, Hermann, Paris, 1941. [41] Reshetnyak, Yu. G., Weak convergence of completely additive vector functions on a set, Siberian Math. J., Vol. 9, 1039-1045, (1968) · Zbl 0176.44402 [42] Rinow, W., Die innere geometrie der metrischen Räume, (1961), Springer-Verlag Berlin · Zbl 0096.16302 [43] Rockafellar, R. T., Convex analysis, (1972), Princeton University Press Princeton, New Jersey · Zbl 0224.49003 [44] Teleman, N., The index of signature operators on Lipschitz manifolds, Inst. Hautes Études Sci. Publ. Math., Vol. 58, 39-78, (1983) [45] S. Venturini, Derivations of Distance Functions on ℝ^n, preprint, 1993. [46] Vol’pert, A. I., The space BV and quasilinear equation, Math. USSR Sbornik, Vol. 2, 225-267, (1967) · Zbl 0168.07402
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