Abstract
We consider a relaxed notion of energy of non-parametric codimension one surfaces that takes into account area, mean curvature, and Gauss curvature. It is given by the best value obtained by approximation with inscribed polyhedral surfaces. The BV and measure properties of functions with finite relaxed energy are studied. Concerning the total mean and Gauss curvature, the classical counterexample by Schwarz-Peano to the definition of area is also analyzed.
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References
E. Acerbi and D. Mucci, Curvature-dependent energies: a geometric and analytical approach. Proc. Roy. Soc. Edinburgh147A (2017), 449–503
Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. Oxford Math. Monographs, Oxford (2000)
G. Anzellotti, Functionals depending on curvatures. Rend. Sem. Mat. Univ. Pol. Torino. Fascicolo speciale 1989: P.D.E. and Geometry (1988), 47–62
G. Anzellotti and M. Giaquinta, Funzioni BV e tracce. Rend. Sem. Mat. Univ. Padova60 (1978), 1–21
G. Anzellotti, R. Serapioni and I. Tamanini, Curvatures, Functionals, Currents. Indiana Univ. Math. J.39 (1990), 617–669
A. Cordoba, Bounded variation and differentiability of functions. Collectanea Mathematica26 (1975), 227–238
M. Giaquinta, G. Modica and J. Souček, Cartesian currents in the calculus of variations, Vol. I. Ergebnisse Math. Grenzgebiete (III Ser) 37, Springer, Berlin, 1998
K. Hildebrandt, K. Polthier and M. Wardetzky, On the convergence of metric and geometric properties of polyhedral surfaces. Geom. Dedicata123, 2006, 89–112
J.W. Milnor, On the total curvature of knots. Ann. of Math.52 (1950), 248–257
D. Mucci, On the curvature energy of Cartesian surfaces. Preprint (2017). Available at the web page http://cvgmt.sns.it/paper/3665/
J.M. Sullivan, Curves of finite total curvature. In: Discrete Differential Geometry (Bobenko, Schröder, Sullivan, and Ziegler, eds.), Oberwolfach Seminars, vol. 38, Birkäuser, 2008
J.M. Sullivan, Curvature of smooth and discrete surfaces. In: Discrete Differential Geometry(Bobenko, Schröder, Sullivan, and Ziegler, eds.), Oberwolfach Seminars, vol. 38, Birkäuser, 2008
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The research of D.M. was partially supported by PRIN 2010-2011 “Calcolo delle Variazioni” and by the GNAMPA of INDAM.
The research of A.S. was partially supported by PRIN 2010-2011 “Varietà reali e complesse: geometria, topologia e analisi armonica” and by the GNSAGA of INDAM
We wish to thank the referee for his or her helpful remarks which helped to increase the readability of the paper.
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Mucci, D., Saracco, A. Bounded Variation and Relaxed Curvature of Surfaces. Milan J. Math. 88, 191–223 (2020). https://doi.org/10.1007/s00032-020-00311-w
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DOI: https://doi.org/10.1007/s00032-020-00311-w