# Problems of optimal aerodynamic resistance, billiards and optimal mass transportation

by Alexander Plakhov (Universidade de Aveiro, Portugal)

A body moves through a medium composed of point particles at rest. The medium is very rare, so that the mutual interaction of the particles is neglected. Interaction of the particles with the body is absolutely elastic. We consider the following problem: find the body, from a given class of bodies, such that the force of resistance of the medium to its motion is minimal or maximal.

The minimization problem was firstly considered by Newton (1686) in classes of convex axially symmetric bodies. Since 1993, it has been studied by Buttazzo, Kawohl, Lachand-Robert et al in classes of convex (not necessarily symmetric) bodies.

We consider this problem in wider classes of (generally nonconvex and non-symmetric) bodies. We also study various kinds of the body's motion: translational motion, translation with rotation, etc. The problem amounts to studying billiard scattering on a compact obstacle. In several cases, the problem can be reduced to the Monge-Kantorovich optimal mass transport problem and then explicitly solved.

The following results will be presented:

- construction of bodies of arbitrarily small resistance (case of translational motion);
- "rough circles" of maximal and minimal resistance (case of translation with slow rotation).

Possible applications may concern

- artificial satellites of Earth moving on low altitudes (100--200 km) and experiencing the drag force from the rest of the atmosphere (minimizing the drag force);
- solar sails (maximizing the force of pressure of solar photons).