We consider a class of integral functionals defined in a Sobolev space of functions vanishing at the boundary of a nonempty bounded connected open n-dimensional set. We prove that, if the functional admits a minimizer depending only on the distance from the boundary, then that set must be a ball.
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Graziano Crasta, A symmetry problem in the calculus of variations. J. Eur. Math. Soc. 8 (2006), no. 1, pp. 139–154DOI 10.4171/JEMS/41