On the curvature energy of Cartesian surfaces (Q2050633)

The author studies (in codimension one) the lower semicontinuous envelope of Cartesian surfaces under the functional curvature functional defined by \textit{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.