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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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