The Problem of Minimal Resistance for Functions and Domains

Abstract

Here we solve the problem posed by Comte and Lachand-Robert in [SIAM J. Math. Anal., 34 (2002), pp. 101--120]. Take a bounded domain $\Omega \subset \mathbb{R}^2$ and a piecewise smooth nonpositive function $u : \bar\Omega \to \mathbb{R}$ vanishing on $\partial\Omega$. 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 $R(u;\Omega) = \frac{1}{|\Omega|} \int_\Omega (1 + |\nabla u(x)|^2)^{-1} dx$. We need to find $\inf_{\Omega,u} R(u;\Omega)$. One can easily see that $|\nabla u(x)| < 1$ for all regular $x \in \Omega$, and therefore one always has $R(u;\Omega) > 1/2$. We prove that the infimum of $R$ is exactly 1/2. This result is somewhat paradoxical, and the proof is inspired by, and partly similar to, the paradoxical solution given by Besicovitch to the Kakeya problem [Amer. Math. Monthly, 70 (1963), pp. 697--706].