A symmetry problem in the calculus of variations

Abstract

We consider the integral functional J(u) = ∫Ω [f(|Du|) - u] dx, u ∈ W01,1(Ω), where Ω ⊂ ℝ, n ≥ 2, is a nonempty bounded connected open subset of ℝn with smooth boundary, and ℝ ∋ s → f(|s|) convex, differentiable function. We prove that if J admits a minimizer in W 01,1 (Ω) depending only on the distance from the boundary of Ω, then Ω must be a ball. © European Mathematical Society 2006.