Bounded variation and relaxed curvature of surfaces
In the non-parametric case, the surface \(\Sigma \) is assumed to be the graph \(G_{u}=\left\{ \left( x,u\left( x\right) \right) \left\vert x\in Q\right. \right\} \ \)of a continuous and real-valued function \(u\) defined on a closed and bounded domain \(Q\subset \mathbb{R}^{2}\), e.g., \(Q=\left[ 0,1\right] ^{2}\), the unit square. In his celebrated paper of [Atti Accad. Naz. Lincei, Rend., VI. Ser. 3, 357--362 (1926; JFM 52.0251.01)], \textit{L. Tonelli} showed that the graph surface \(\Sigma \) has finite relaxed area in Lebesgue's sense if and only if \(u\) is a function of bounded variation. The aim of this paper is to extend (at least partially) Tonelli's result concerning the area to a similar notion of total mean and Gauss curvature. In correspondence to a relaxed formula that takes into account both area and curvatures, one expects that if \(u\) has finite relaxed energy, then both \(u\) and the outward unit normal \(v_{u}\) are function of bounded variation. Moreover, the non-smooth counterpart of the density of the total mean and Gauss curvature energy of smooth functions \(u\), suggests that suitable distributions (depending on the approximate derivative of \(u\), and of the unit normal \(v_{u}\)) are expected to be measures with finite total variation, too.
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