The problem of minimal resistance for functions and domains (Q2927867)

The author studies the following problem concerning a flow of particles falling vertically down and reflected elastically by the boundary of a bounded domain \(\Omega\subset \mathbb{R}^2\). It is assumed that the upper part of the body \(\Omega\) is given by the graph of the function \(u:\Omega \to \mathbb{R}\), vanishing on \(\partial \Omega\). The resistance of the graph to the flow is expressed as \(R(u,\Omega)={{1}\over {|\Omega|}} \int_\Omega(1+|\nabla u(x)|^2)^{-1}\). The author solves a problem posed by \textit{M. Comte} and \textit{T. Lachand-Robert} [SIAM J. Math. Anal. 34, No. 1, 101--120 (2002; Zbl 1034.49013)]. The main results are: Theorem 1.2. \(\inf_{u,\Omega}R(u,\Omega)=1/2\), Theorem 1.3. For any domain \(\Omega\) one has \(\inf_u R(u,\Omega) =1/2\). These results are closely related to the solution to the Kakeya problem given by \textit{A. S. Besicovitch} [Am. Math. Mon. 70, 697--706 (1963; Zbl 0117.39402)].