We minimize a linear combination of the length and the \(L^2\)-norm of the curvature among networks in \(\mathbb {R}^d\) belonging to a given class determined by the number of curves, the order of the junctions, and the angles between curves at the junctions. Since this class lacks compactness, we characterize the set of limits of sequences of networks bounded in energy, providing an explicit representation of the relaxed problem. This is expressed in terms of the new notion of degenerate elastic networks that, rather surprisingly, involves only the properties of the given class, without reference to the curvature. In the case of \(d=2\) we also give an equivalent description of degenerate elastic networks by means of a combinatorial definition easy to validate by a finite algorithm. Moreover we provide examples, counterexamples, and additional results that motivate our study and show the sharpness of our characterization.

我们最小化长度和\(L^2\) -网络之间曲率的线性组合\(\mathbb {R}^d\)属于给定类别的网络，该类别由曲线数量决定交点的角度，以及交点处曲线之间的角度。由于此类缺乏紧凑性，我们描述了以能量为界的网络序列的限制集，提供了松弛问题的明确表示。这是用退化弹性网络的新概念表达的，令人惊讶的是，它只涉及给定类的属性，而不涉及曲率。在\(d=2\)的情况下我们还通过易于通过有限算法验证的组合定义给出了退化弹性网络的等效描述。此外，我们提供了示例、反例和其他结果，以激发我们的研究并展示我们表征的敏锐性。