A symmetry problem in the calculus of variations

Abstract

Let Ω be a ball in ℝ^N, centered at zero, and letu be a minimizer of the nonconvex functional$R(v) = \int_\Omega {\tfrac{1}{{1 + |\nabla v(x)|^2 }}dx}$ over one of the classesC _M := {w ∈W ^1,∞ _loc (∖) ∣ 0 ≤w(x) ≤M inΩ,w concave} orE _M := {w ∈W ^1,2 _loc (Ω) ∣ 0 ≤w(x) ∖M in≤,Δw ∖ 0 inL′(∖)}of admissible functions. Thenu is not radial and not unique. Therefore one can further reduce the resistance of Newton's rotational “body of minimal resistance“ through symmetry breaking.