The problem of minimal resistance for functions and domains

Abstract

Here we solve the problem posed by Comte and Lachand-Robert in (2001). Take a bounded domain \Omega in R^2 and a piecewise smooth non-positive function u : \bar{\Omega} \to R vanishing on the boundary of \Omega. Consider a flow of point particles falling vertically down and reflected elastically from the graph of u. It is assumed that each particle is reflected no more than once (no multiple reflections are allowed); then the resistance of the graph to the flow is expressed as F(u;\Omega) = \frac{1}{|\Omega|} \int_\Omega (1 + |\nabla u(x)|^2)^{-1} dx. It is required to find \inf_{\Omega,u} F(u;\Omega). One can easily see that |\nabla u(x)| < 1 for all regular x \in \Omega, and therefore one always has F(u;\Omega) > 1/2. We prove that the infimum of F is exactly 1/2. This result is somewhat paradoxical, and the proof is inspired by, and is partly similar to, the paradoxical solution given by Besicovitch to the Kakeya problem.