A uniqueness theorem for the symmetrization of arbitrary condensers
Let D be a domain in the complex plane such that complement of D relative to the compactified complex plane is the union of disjoint closed sets \(E_ 0\) and \(E_ 1\), where \(E_ 0\) is unbounded and \(E_ 1\) is bounded. The triple \((D,E_ 0,E_ 1)\) is called a condenser. Let H be the generalized solution of the Dirichlet problem in D for the function h defined on \(\partial D=\partial E_ 0\cup\partial E_ 1\) to be equal to j on \(\partial E_ j (j=1,2)\). The integral \(| (D)| =\int_{D}\{(\partial H/\partial y)^ 2+(\partial H/\partial x)^ 2\}\quad dx\quad dy\) is called the capacity of the condenser \((D,E_ 0,E_ 1)\). Let \((D^*,E^*_ 0,E^*_ 1)\) be the condenser obtained from \((D,E_ 0,E_ 1)\) by circular symmetrization relative to some ray originating at 0. \textit{G. Pólya} and \textit{G. Szegö} [Isoperimetric inequalities in mathematical physics (1951; Zbl 0044.383)] showed that \(| (D^*)|\leq | (D)|\). In the present paper it is proved that equality holds here if and only if \((D^*,E^*_ 0,E^*_ 1)\) can be obtained from \((D,E_ 0,E_ 1)\) by some rotation about 0. This was established by \textit{J. A. Jenkins} [Ann. Math., II. Ser. 75, 223-230 (1962; Zbl 0106.049)] in the case of a regular condenser.
scientific article; zbMATH DE number 3873685