Native minima in Newton’s aerodynamical drawback and inequalities between norms of partial derivatives
Authors: Alexander Plakhov, Vladimir Protasov
Summary: The issue thought of first by I. Newton (1687) consists to find a floor of the minimal frontal resistance in a parallel circulation of non-interacting level particles. The usual formulation assumes that the floor is convex with a given convex base Ω and a bounded altitude. Newton discovered the answer for surfaces of revolution. With out this assumption the issue continues to be unsolved, though many necessary outcomes have been obtained within the final a long time. We take into account the issue to characterize the domains Ω for which the flat floor provides an area minimal. We present that this drawback may be decreased to an inequality between L2-norms of partial derivatives for bivariate concave features on a convex area that vanish on the boundary. Can the ratio between these norms be arbitrarily giant? The reply relies on the geometry of the area. An entire criterion is derived, which additionally solves the native minimality drawback.
