We report our recent results from [1, 2] on the total curvature of graphs of curves in high codimension Euclidean space. We introduce the corresponding relaxed energy functional and provide an explicit representation formula. In the case of continuous Cartesian curves, i.e., of graphs \({c_{u}}\) of continuous functions u on an interval, the relaxed energy is finite if and only if the curve \({c_{u}}\) has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We also deal with the "elastic" case, corresponding to a superlinear dependence on the pointwise curvature. Different phenomena w.r.t. the "plastic" case are observed. A p-curvature functional is well-defined on continuous curves with finite relaxed energy, and the relaxed energy is given by the length plus the p-curvature. We treat the wider class of graphs of one-dimensional BV-functions.

中文翻译：

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曲率依赖性能量

我们从[1，2]报告了我们在高维次欧空间中曲线图的总曲率方面的最新结果。我们介绍了相应的松弛能量泛函​​，并提供了一个明确的表示公式。在连续的笛卡尔曲线（即连续函数u的图\（{c_ {u}} \）的间隔上）的情况下，当且仅当曲线\（{c_ {u}} \ ）具有有限的变化和有限的总曲率。此外，在这种情况下，总曲率不取决于u的导数的Cantor部分。我们还处理“弹性”情况，这对应于对点曲率的超线性依赖性。观察到“可塑”情况的不同现象。一个p曲率函数在具有有限松弛能量的连续曲线上定义良好，并且松弛能量由长度加上p曲率给出。我们处理一维BV函数的更广泛的图类。