The problem of minimal resistance, old and new
Giuseppe Buttazzo
TL;DR
The paper surveys Newton's classical minimal resistance problem and its modern reformulations, presenting both the Cartesian graph formulation with $F(u)=\int_\Omega\frac{1}{1+|\nabla u|^2}\,dx$ and the intrinsic boundary-integral formulation $F(E)=\int_{\partial E} f(x,\nu)\,d\mathcal{H}^d$, and discusses existence results under height, volume, and surface constraints. It highlights key phenomena such as the nonexistence of globally smooth minimizers, symmetry breaking (radial vs nonradial optimal shapes), and the role of convexity in guaranteeing compactness and lower semicontinuity. The article then surveys generalizations to lift constraints and to a positive-temperature fluid model, where the resistance is determined by a velocity distribution $\rho(v)$ and reduces to Newton's model when $\rho = \delta_{-Vn}$. Finally, it reviews related problems including one-impact relaxations, multiple reflections, tangential friction, rotation, and visibility considerations, outlining directions for future research.
Abstract
Since its original formulation by Isaac Newton in 1685, the problem of determining bodies of minimal resistance moving through a fluid has been one of the classical problems in the calculus of variations. Initially posed for cylindrically symmetric bodies, the problem was later extended to general convex shapes, as explored in \cite{BK93}, \cite{BFK95}. Since then, this broader formulation has inspired a number of articles dedicated to the study of the geometric and analytical properties of optimal shapes, with particular attention to their structure, regularity, and behavior under various constraints. In this article, we provide a comprehensive overview of the principal results that have been established, highlighting the main theoretical advancements. Furthermore, we introduce some new directions of research, some of which were described in \cite{P12}, that offer promising perspectives for future investigation.