The Problem of Minimal Resistance for Functions and Domains

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 Ω ⊂ R² and a piecewise smooth nonpositive function u : Ω → R vanishing on ∂Ω. 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;Ω) = 1/|Ω| ∫_Ω (1 + |∇ u(x)|²)⁻¹ dx. We need to find infΩ,u R(u;Ω). One can easily see that |∇ u(x)| < 1 for all regular x ∈ Ω, and therefore one always has R(u;Ω) > 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].