Minimal harmonic measure on complementary regions
Let \(\omega (z,E,D)\) denote the harmonic measure of the set \(E\subset \partial D\) with respect to the domain \(D\), evaluated at the point \(z\). The authors study the following extremal problem: Let \(\Delta\) be an open disk, \(a_1,a_2\in \Delta\) and \(l\) be a simple arc in \(\Delta\) dividing it into two domains \(D_1, D_2\) with \(a_1\in D_1\) and \(a_2\in D_2\); find \(\inf \frac{1}{2}(\omega (a_1, l,D_1)+\omega (a_1, l,D_1))\), where the infimum is taken over all possible arcs \(l\). By the conformal invariance of harmonic measure, this problem is equivalent to that of finding \[ \Omega(r):=\inf \tfrac{1}{2}(\omega (r, l,D_1)+\omega (-r, l,D_1)), \] where \(0<r<1\) and the infimum is taken over all arcs \(l\) dividing the unit disk \(D\) into two domains \(D_1, D_2\) with \(r\in D_1\) and \(-r\in D_2\). The authors provide a complete solution of the above problem. The extremal configuration may be described as follows: There exists a certain value \(r_o\) of \(r\) (explicitly evaluated in the paper) such that (a) if \(0<r<r_o\), then the extremal configuration is essentially unique and it is degenerate, i.e. \(l=[r,1], D_2=D\setminus l, D_1=\emptyset\); (b) if \(r_o<r<1\), the extremal configuration is unique and symmetric, i.e. \(l=[-i,i]\); (c) if \(r=r_o\), then there are exactly two extremal configurations: the degenerate and the symmetric. An important step towards the above result is the consideration and solution of another extremal problem which is to find \(\Omega(r)\) under the additional constraint that \(\omega(-r,l,D_2)\) is fixed. This problem is solved by the method of extremal length. Another tool used in the paper is the explicit computation of harmonic measures and extremal lengths in terms of complete elliptic integrals and the application of calculus techniques on functions involving complete elliptic integrals and their derivatives.
scientific article; zbMATH DE number 1995503
scientific article