Local minima in Newton's aerodynamical problem and inequalities between norms of partial derivatives

TL;DR

This paper investigates the conditions under which a flat surface is a local minimum for Newton's aerodynamic problem, linking it to inequalities between norms of derivatives of concave functions and deriving a complete geometric criterion.

Contribution

It introduces a new criterion based on inequalities between partial derivatives norms to characterize local minima in Newton's problem, extending understanding beyond surfaces of revolution.

Findings

Complete criterion for local minimality depending on domain geometry

Reduction of the problem to inequalities between $L_2$-norms of derivatives

Analysis of how domain shape influences the ratio of derivative norms

Abstract

The problem considered first by I. Newton (1687) consists in finding a surface of the minimal frontal resistance in a parallel flow of non-interacting point particles. The standard formulation assumes that the surface is convex with a given convex base $\Omega$ and a bounded altitude. Newton found the solution for surfaces of revolution. Without this assumption the problem is still unsolved, although many important results have been obtained in the last decades. We consider the problem to characterize the domains $\Omega$ for which the flat surface gives a local minimum. We show that this problem can be reduced to an inequality between $L_2$-norms of partial derivatives for bivariate concave functions on a convex domain that vanish on the boundary. Can the ratio between those norms be arbitrarily large? The answer depends on the geometry of the domain. A complete criterion is derived, which also solves the local minimality problem.