# Euler's optimal profile problem

## TL;DR

This paper investigates Euler's classical variational problem, proving the existence of minimizers using relaxation methods and analyzing their structure, uniqueness, and dependence on geometric parameters.

## Contribution

It provides a rigorous existence proof for minimizers and characterizes their analytical structure and uniqueness conditions based on geometric parameters.

## Key findings

- Existence of minimizers established via relaxation methods.
- Detailed analysis of minimizers' structure depending on parameters.
- Identification of parameter ranges for uniqueness and non-uniqueness.

## 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.