A symmetry theorem revisited (Q2758981)

Let \(D\) be a bounded domain in \(\mathbb{R}^n\) such that \(0\in D\) and \(D\) is regular for the Dirichlet problem, let \(\mu\) denote harmonic measure on \(\partial D\), and let \(H^{n-1}\) denote \((n-1)\)-dimensional Hausdorff measure. The open ball of centre \(x\) and radius \(r\) in \(\mathbb{R}^n\) is denoted by \(B(x,r)\). It is proved that if there exist numbers \(r_0>0\) and \(L\geq 1\) such that \(\mu(B(x,r)\cap \partial D)\leq Lr^{n-1}\) for all \(x\in \partial D\) and all \(r\in (0,r_0]\) and if \(\mu= aH^{n-1}\) on \(\partial D\) for some positive constant \(a\), then \(D\) is a ball of centre 0. The authors [Nonlinear diffusion equations and their equilibrium states, Proc. 3rd Conf. 1989, Prog. Nonlinear Differ. Equ. 7, 347-374 (1992; Zbl 0792.35009)] earlier obtained the same conclusion under more restrictive hypotheses.