On the weak formulation of Prandtl's minimum drag problem

Abstract

We study Prandtl's classical problem on minimising the induced drag for a finite wing with fixed span. The induced drag is given by a singular quadratic functional of the circulation, with admissible functions satisfying the prescribed lift and second-moment conditions. We formulate the problem in the fractional Sobolev space \(H^{1/2}\), which is the natural energy space for the functional, prove existence and uniqueness of minimisers by variational methods, and derive the corresponding Euler--Lagrange equation. % Passing to a periodic formulation on the one-dimensional torus, we identify the drag functional with the quadratic form of the half-Laplacian and solve the resulting singular integral equation explicitly and recover Prandtl's bell-shaped circulation profile.