Inserted: 18 nov 2017
Last Updated: 7 feb 2021
Journal: The Journal of Geometric Analysis
Published online: 05 February 2021.
We analyze the lower semicontinuous envelope of the curvature functional of Cartesian surfaces in codimension one. To this aim, following the approach by Anzellotti-Serapioni-Tamanini, we study 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.
Keywords: Curvature of surfaces; Cartesian currents; Gauss graphs; Gap phenomenon