Symmetry of minimizers with a level surface parallel to the boundary

2012/03/23 by Ciraolo, Giulio, Magnanini, Rolando, Sakaguchi, Shigeru · 1 citation
#35B06 #35J70 #35K55 #49K20 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1203.5295

Abstract

We consider the functional I_Ω(v) = ∫_Ω[f(|Dv|) - v] dx, where Ω is a bounded domain and f is a convex function. Under general assumptions on f, G. Crasta [Cr1] has shown that if I_Ω admits a minimizer in W₀^1,1(Ω) depending only on the distance from the boundary of Ω, then Ω must be a ball. With some restrictions on f, 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.

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• Hessian operators, overdetermined problems, and higher order mean curvatures: symmetry and stability results

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