The interplay between variational functionals and the Brunn–Minkowski Theory is a well-established phenomenon widely investigated in the last thirty years. In this work, we prove the existence of solutions to the even logarithmic Minkowski problems arising from variational functionals, such as the first eigenvalue of the Laplacian and the torsional rigidity. In particular, we lay down a blueprint showing that the same result holds for more generic functionals by adapting the volume case from Böröczky, Lutwak, Yang, and Zhang. We show how these results imply the existence of self-similar solutions to variational flow problems à la Firey’s worn stone problem.

中文翻译：

---

（自相似，变分）滚石乐队

变分泛函与 Brunn-Minkowski 理论之间的相互作用是过去三十年来广泛研究的一种公认现象。在这项工作中，我们证明了由变分泛函引起的偶对数 Minkowski 问题解的存在性，例如拉普拉斯算子的第一特征值和扭转刚度。特别是，我们制定了一个蓝图，表明通过调整 Böröczky、Lutwak、Yang 和Zhang 的体积案例，相同的结果适用于更通用的泛函。我们展示了这些结果如何暗示变分流问题（如 Firey 磨损的石头问题）存在自相似解。