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Some results on surface measures in calculus of variations. (English) Zbl 0890.49020
Starting from the identity, for every smooth bounded open subset $$E$$ of $$\mathbb{R}^n$$, of the notions of perimeter, $$(n-1)$$-dimensional Hausdorff measure, Minkowski content and the approximability in the sense of $$\Gamma$$-convergence of the perimeter by a sequence of elliptic functionals with a double equal well potential term, in the paper a similar study is carried out when $$\mathbb{R}^n$$ is endowed with a convex Finsler metric depending continuously on the position.
Given such a metric $$\phi$$ and introduced the integrated distance $$\delta_\phi$$ associated to $$\phi$$ and the $$(n-1)$$-dimensional Hausdorff measure $$H^{n- 1}_\phi$$ with respect to $$\delta_\phi$$, a representation formula for $$H^{n-1}_\phi$$ as an integral in the measure $$H^{n-1}$$ is first proved. Then the perimeter $$P_\phi(E)$$ of a set $$E$$ with respect to $$\phi$$ is introduced and some inequalities between $$P_\phi$$ and $$H^{n-1}_\phi$$ are proved. It is also shown that $$H^{n- 1}_\phi$$ is a perimeter with respect to a suitable metric $$\psi$$ constructed starting from $$\phi$$. Then, for every set $$E$$, $$P_\phi(E)$$ is compared with the Minkowski content $$M^-_\phi(\partial E)$$ with respect to $$\phi$$ and it is proved that $$P_\phi(E)\leq M^-_\phi(\partial E)$$ and that equality holds under suitable regularity assumptions.
Finally, the approximability of $$P_\phi(E)$$ in the sense of $$\Gamma$$-convergence by a sequence of elliptic functionals with a double equal well potential term is discussed and the notions of normal and of mean curvature of a smooth hypersurface with respect to $$\phi$$ are introduced.

##### MSC:
 49Q20 Variational problems in a geometric measure-theoretic setting 49J45 Methods involving semicontinuity and convergence; relaxation 28A78 Hausdorff and packing measures
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##### References:
 [1] Abraham, R.; Marsden, J. E.; Ratiu, T., Manifolds, Tensor Analysis, and Applications (1983), Reading, Mass.: Addison-Wesley, Reading, Mass. · Zbl 0875.58002 [2] Amar, M.; Bellettini, G., A notion of total variation depending on a metric with discontinuous coefficients, Ann. Inst. H. Poinearé Anal. Non Linéaire, 11, 91-133 (1993) · Zbl 0842.49016 [3] Amar, M.; Bellettini, G., Approximation by Γ-convergence of a total variation with discontinuous coefficients, Asymptotic Anal., 10, 225-243 (1995) · Zbl 0844.49013 [4] Asanov, G. S., Finsler Geometry. Relativity and Gauge Theories (1985), Dordrecht: D. Reidel Publishing Company, Dordrecht · Zbl 0576.53001 [5] Barroso, A. C.; Fonseca, I., Anisotropic singular perturbations.—The vectorial case, Proc. Royal Soc. Edinburgh, 124A, 527-571 (1994) · Zbl 0804.49013 [6] Benjancu, A., Finsler Geometry and Applications (1990), Chichester: Ellis Horwood Limited, Chichester [7] G.Bellettini - M.Paolini,Anisotropic motion by mean curvature in the context of Finsler geometry, preprint Univ. of Bologna (1994), to appear on Hokkaido Mathematical Journal. · Zbl 0873.53011 [8] Bellettini, G.; Paolini, M., Quasi-optimal error estimates for the mean curvature flow with a forcing term, Differential Integral Equations, 8, No. 4, 735-752 (1995) · Zbl 0820.49019 [9] Bouchitté, G., Singular perturbations of variational problems arising from a two-phase transition model, Appl. Math. Opt., 21, 289-315 (1990) · Zbl 0695.49003 [10] Burago, Y. D.; Zalgaller, V. A., Geometric Inequalities (Grundleheren Math. Wiss., Bd. 285) (1988), Berlin, Heidelberg, New York: Springer, Berlin, Heidelberg, New York [11] Busemann, H., Intrinsic area, Ann. Math., 48, 234-267 (1947) · Zbl 0029.35301 [12] Busemann, H., A theorem on convex bodies of the Brunn-Minkowsky type, Proc. Nat. Acad. Sci. U.S.A., 35, 27-31 (1949) · Zbl 0032.19001 [13] Busemann, H.; Mayer, W., On the foundations of calculus of variations, Trans. Amer. Math. Soc, 49, 173-198 (1941) · JFM 67.1036.03 [14] Buttazzo, G., Semicontinuity, Relaxation and Integral Representation in the Calculus of Variation (1989), Harlow: Longman Scientific & Technical, Harlow · Zbl 0669.49005 [15] Chen, Y. G.; Giga, Y.; Goto, S., Uniqueness and existence of viscosity solutions of generalized mean curvature flow equation, J. Differential Geom., 33, 749-786 (1991) · Zbl 0696.35087 [16] Maso, G. Dal, An Introduction to Γ-Convergence (1993), Boston: Birkhäuser, Boston · Zbl 0816.49001 [17] De Giorgi, E.; Franzoni, T., Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 58, 842-850 (1975) · Zbl 0339.49005 [18] De Giorgi, E.; Franzoni, T., Su un tipo di convergenza variazionale, Rend. Sem. Mat. Brescia, 3, 63-101 (1979) [19] L. C.Evans - R. F.Gariepy,Measure Theory and Fine Properties of Functions, CRC Press (1992). · Zbl 0804.28001 [20] Federer, H., Geometric Measure Theory (1968), Berlin: Springer-Verlag, Berlin [21] Fonseca, I., The Wulff theorem revisited, Proc. Roy. Soc. London Ser. A, 432, 125-145 (1991) · Zbl 0725.49017 [22] Ponseca, I.; Müller, S., A uniqueness proof for the Wulff theorem, Proc. Roy. Soc. London Ser. A, 119, 125-136 (1991) · Zbl 0752.49019 [23] Giusti, E., Minimal Surfaces and Functions of Bounded Variation (1984), Boston: Birkhäuser, Boston · Zbl 0545.49018 [24] Gurtin, M. E., Towards a nonequilibrium thermodynamics of two-phase materials, Arch. Rational Mech. Anal., 100, 275-312 (1988) · Zbl 0673.73007 [25] Luckhaus, S.; Modica, L., The Gibbs-Thomson relation within the gradient theory of phase transitions, Arch. Rational Mech. Anal., 107, no. 1, 71-83 (1989) · Zbl 0681.49012 [26] Lutwak, E.; Gruber, P. M.; Wills, J. N., Selected Affine-Isoperimetric Inequalities, Handbook of Convex Geometry, 151-176 (1993), Amsterdam: North-Holland, Amsterdam · Zbl 0847.52006 [27] Matsumoto, M., Foundations of Finsler Geometry and Special Finsler Spaces (1986), Otsu: Kaiseicha Press, Otsu · Zbl 0594.53001 [28] Modica, L., Gradient theory of phase transitions with boundary contact energy, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4, 487-512 (1987) · Zbl 0642.49009 [29] Modica, L., The gradient theory of phase transitions and minimal interfaces criterion, Arch. Rational Mech. Anal., 98, 123-142 (1987) · Zbl 0616.76004 [30] Modica, L.; Mortola, S., Un esempio di г-convergenza, Boll. Un. Mat. Ital., B (5), 14, 285-299 (1977) · Zbl 0356.49008 [31] Owen, N., Non convex variational problems with general singular perturbations, Trans. Amer. Math. Soc, 310, 393-404 (1988) · Zbl 0718.34075 [32] Owen, N.; Sternberg, P., Non convex variational problems with anisotropic perturbations, Nonlinear Anal., 16, 705-719 (1991) · Zbl 0748.49034 [33] Reshetnyak, Yu. G., Weak convergence of completely additive functions on a set, Siberian Math. J., 9, 1039-1045 (1968) · Zbl 0176.44402 [34] Rockafellar, R. T., Convex Analysis (1972), Princeton, New Jersey: Princeton University Press, Princeton, New Jersey · Zbl 0224.49003 [35] R.Schneider,Convex Bodies: the Brunn-Minkowsky Theory, Encyclopedia of Mathematics and its Applications, Cambridge Univ. Press (1993). [36] Z.Shen,Length and Finsler manifolds with curvature bounded from below, preprint. [37] Taylor, J., Mean curvature and weighted mean curvature II, Acta Metall., 40, 1475-1485 (1992) [38] S.Venturini,Derivations of distance functions inR^n, preprint. [39] Vol’Pert, A. I., The space BV and quasilinear equations, Math. USSR Sbornik, 2, 225-267 (1967) · Zbl 0168.07402
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