Symmetry of minimizers with a level surface parallel to the boundary

  • Giulio Ciraolo

    Università di Palermo, Italy
  • Rolando Magnanini

    Università di Firenze, Italy
  • Shigeru Sakaguchi

    Tohoku University, Sendai, Japan


We consider the functional

IΩ(v)=Ω[f(Dv)v]dx,\mathcal I_{\Omega} (v) = \int_{\Omega} [f(|Dv|) - v] dx,

where Ω\Omega is a bounded domain and ff is a convex function. Under general assumptions on ff, Crasta [Cr1] has shown that if IΩ\mathcal I_{\Omega} admits a minimizer in W01,1(Ω)W_0^{1,1}(\Omega) depending only on the distance from the boundary of Ω\Omega, then Ω\Omega must be a ball. With some restrictions on ff, we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance).

We then discuss how these results extend to more general settings, in particular to functionals that are not differentiable and to solutions of fully nonlinear elliptic and parabolic equations.

Cite this article

Giulio Ciraolo, Rolando Magnanini, Shigeru Sakaguchi, Symmetry of minimizers with a level surface parallel to the boundary. J. Eur. Math. Soc. 17 (2015), no. 11, pp. 2789–2804

DOI 10.4171/JEMS/571