Euler's optimal profile problem (Q2300553)

The authors study, with a modern point of view, an old problem raised by L. Euler about the optimal profile (of a boat for example). In our modern language, it is the following problem of calculus of variations: \[ \min J(\gamma):=\int_0^1 \frac{(\gamma_2'(t))^3_+}{{\gamma'_1(t)}^2+{\gamma'_2(t)}^2} dt \] with the area constraint \(\int_0^1 \gamma_1(t) \gamma'_2(t) dt=ah-L\), in the class of simple planar curves \(\gamma=(\gamma_1,\gamma_2)\) where we assume \(\gamma:[0,1]\to [0,a]\times [0,h]\) absolutely continuous, \(\gamma(0)=(0,0), \gamma(1)=(a,h)\) where \(a,h,L\) are three positive constants (\(L<ah\)). In this general setting, it might happen that the problem has no solution as the authors show with an example with strong changing-sign oscillations of \(\gamma'_1\). Therefore, they impose moreover that \(\gamma'_1\geq 0\) that is rather natural. They are able to prove (through the passage to a relaxed formulation) existence of an optimal curve. They also prove \begin{itemize} \item uniqueness of the optimal curve when \(2L\notin [a^2,2ah-a^2]\) and they give the explicit solution, \item there exist an infinite number of optimal curves when \(2L\in [a^2,2ah-a^2]\) and they describe a family of solutions. \end{itemize}