Abstract

We study an old variational problem formulated by Euler as Proposition 53 of his Scientia Navalis by means of the direct method of the calculus of variations. Precisely, through relaxation arguments, we prove the existence of minimizers. We fully investigate the analytical structure of the minimizers in dependence of the geometric parameters and we identify the ranges of uniqueness and non-uniqueness.

中文翻译：

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欧拉最优轮廓问题

摘要

我们通过变分的直接方法研究了由Euler提出的，作为其Scientia Navalis命题53的旧变分问题。确切地说，通过松弛论证，我们证明了极小值的存在。我们根据几何参数充分研究了最小化器的分析结构，并确定了唯一性和非唯一性的范围。