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On the curvature energy of Cartesian surfaces. (English) Zbl 1479.49093

The author studies (in codimension one) the lower semicontinuous envelope of Cartesian surfaces under the functional curvature functional defined by G. Anzellotti et al. [Indiana Univ. Math. J. 39, No. 3, 617–669 (1990; Zbl 0718.49030)] which for a smooth surface \(M\subseteq\mathbb{R}^3\) can be defined as follows using the elementary symmetric curvatures \(k_1\) and \(k_2\): \[ \|M\|:=\mathcal{H}^2(M)+\int_M \sqrt{k_1^2+k_2^2}\,d \mathcal{H}^2 +\int_M |k_1k_2|\,d\mathcal{H}^2. \] To this aim, following the approach of [loc. cit.], he studies the class of currents that naturally arise as weak limits of Gauss graphs of smooth functions. The curvature measures are then studied in the non-parametric case. Concerning homogeneous functions, some model examples are studied in detail. Finally, a new gap phenomenon is observed.

MSC:

49Q15 Geometric measure and integration theory, integral and normal currents in optimization
53A05 Surfaces in Euclidean and related spaces
49J45 Methods involving semicontinuity and convergence; relaxation

Citations:

Zbl 0718.49030
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[1] Acerbi, E.; Dal Maso, G., New lower semicontinuity results for polyconvex integrals, Calc. Var., 2, 329-372 (1994) · Zbl 0810.49014 · doi:10.1007/BF01235534
[2] Acerbi, E.; Mucci, D., Curvature-dependent energies: a geometric and analytical approach, Proc. R. Soc. Edinb., 147A, 449-503 (2017) · Zbl 1381.49010 · doi:10.1017/S0308210516000202
[3] Acerbi, E.; Mucci, D., Curvature-dependent energies: the elastic case, Nonlinear Anal., 153, 7-34 (2017) · Zbl 1361.53007 · doi:10.1016/j.na.2016.05.012
[4] Allard, WK, First variation of a varifold, Ann. Math., 95, 417-491 (1972) · Zbl 0252.49028 · doi:10.2307/1970868
[5] Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs, Oxford (2000) · Zbl 0957.49001
[6] Anzellotti, G.: Functionals depending on curvatures. Rend. Sem. Mat. Univ. Pol. Torino. Fascicolo speciale 1989: P.D.E. and Geometry, pp. 47-62 (1988) · Zbl 0738.49021
[7] Anzellotti, G.; Serapioni, R.; Tamanini, I., Curvatures, Functionals, Currents, Indiana Univ. Math. J., 39, 617-669 (1990) · Zbl 0718.49030 · doi:10.1512/iumj.1990.39.39033
[8] Ball, JM, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., 63, 337-403 (1977) · Zbl 0368.73040 · doi:10.1007/BF00279992
[9] Buttazzo, G.; Mizel, VJ, Interpretation of the Lavrentiev phenomenon by relaxation, J. Funct. Anal., 110, 434-460 (1992) · Zbl 0784.49006 · doi:10.1016/0022-1236(92)90038-K
[10] Dal Maso, G., Integral representation on \(\text{BV}({\varOmega })\) of \(\text{ GG } \)-limits of variational integrals, Manuscr. Math., 30, 387-416 (1980) · Zbl 0435.49016 · doi:10.1007/BF01301259
[11] Dal Maso, G.; Fonseca, I.; Leoni, G.; Morini, M., A higher order model for image restoration: the one dimensional case, Siam J. Math. Appl., 40, 6, 2351-2391 (2009) · Zbl 1193.49012 · doi:10.1137/070697823
[12] De Giorgi, E.; Letta, G., Une notion générale de convergence faible pour des fonctions croissantes d’ensemble, Ann. S.N.S. Pisa Cl. Sci., 4, 61-99 (1977) · Zbl 0405.28008
[13] Delladio, S., Special generalized Gauss graphs and their application to minimization of functionals involving curvatures, J. Reine Angew. Math., 486, 17-43 (1997) · Zbl 0871.49034
[14] Federer, H.: Geometric Measure Theory. Grundlehren der mathematischen Wissenschaften, vol. 153. Springer, New York (1969) · Zbl 0176.00801
[15] Federer, H.; Fleming, W., Normal and integral currents, Ann. Math., 72, 458-520 (1960) · Zbl 0187.31301 · doi:10.2307/1970227
[16] Giaquinta, M., Modica, G., Souček, J.: Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 106, 97-159 (1989). Erratum and addendum. Arch. Ration. Mech. Anal. 109, 385-392 (1990) · Zbl 0712.73009
[17] Giaquinta, M., Modica, G., Souček, J.: Cartesian Currents in the Calculus of Variations, vol. I. Ergebnisse Math. Grenzgebiete (III Ser) 37. Springer, Berlin (1998) · Zbl 0914.49001
[18] Giusti, E., Minimal Surfaces and Functions of Bounded Variations (1984), Boston: Birkauser, Boston · Zbl 0545.49018 · doi:10.1007/978-1-4684-9486-0
[19] Hutchinson, JE, Second fundamental form for varifolds and existence of surfaces minimizing curvature, Indiana Univ. Math. J., 35, 45-71 (1986) · Zbl 0561.53008 · doi:10.1512/iumj.1986.35.35003
[20] Krantz, SG; Parks, HR, Geometric Integration Theory. Cornerstones (2008), Boston: Birkhäuser, Boston · Zbl 1149.28001 · doi:10.1007/978-0-8176-4679-0
[21] Mantegazza, C., Curvature varifolds with boundary, J. Differ. Geom., 43, 807-843 (1996) · Zbl 0865.49030 · doi:10.4310/jdg/1214458533
[22] Müller, S., Det = det. A remark on the distributional determinant, C. R. Acad. Sci. Paris Sér. I Math., 311, 13-17 (1990) · Zbl 0717.46033
[23] Simon, L.: Lectures on Geometric Measure Theory. Proc. C.M.A. 3, Australian National University, Canberra (1983) · Zbl 0546.49019
[24] Sullivan, JM; Bobenko, AI; Schröder, P.; Sullivan, JM; Ziegler, GM, Curvature of smooth and discrete surfaces, Discrete Differential Geometry (2008), Boston: Birkäuser, Boston · doi:10.1007/978-3-7643-8621-4_9
[25] Sullivan, JM; Bobenko, AI; Schröder, P.; Sullivan, JM; Ziegler, GM, Curves of finite total curvature, Discrete Differential Geometry (2008), Boston: Birkäuser, Boston · Zbl 1151.53305 · doi:10.1007/978-3-7643-8621-4_9
[26] Truesdell, C., The influence of elasticity on analysis: the classical heritage, Bull. Am. Math. Soc., 9, 293-310 (1983) · Zbl 0555.73030 · doi:10.1090/S0273-0979-1983-15187-X
[27] Willmore, T.: Note on embedded surfaces. An. Stiint. Univ. “Al. I. Cusa” Iasi Sect. I A Math. II, 443-446 (1965)
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