Symmetrization and harmonic measure
This paper is concerned with symmetrization inequalities for \(\alpha \)-harmonic measures \((0<\alpha <2)\) associated with domains in \(\mathbb{R}^{n}\). The main results establish when equality occurs. They are formulated in terms of \(1\)-dimensional Steiner symmetrization, and the cases of equality essentially involve symmetry about a hyperplane. The proofs are carried through by reducing symmetrization results to polarization results, and then using probabilistic methods to establish the latter. In the classical case where \(\alpha =2\) equality statements had been previously been established when \(n=2\) by \textit{M. Essén} and \textit{D. F. Shea} [Ann. Acad. Sci. Fenn., Ser. A I 4, 311--340 (1979; Zbl 0392.31001)], and in all dimensions by \textit{A. Yu. Solynin} [Algebra Anal. 8, No. 6, 148--185 (1996; Zbl 0890.31004)], under additional regularity assumptions with which the author can now dispense.
scientific article; zbMATH DE number 5615676