Harmonic measures of sides of a slit perpendicular to the domain boundary
Let the functions \(f(z,t)\), \(t\geq 0\), normalized near infinity by \(f(z, t)= z+ 2t/z+ O(1/z^2)\), map subdomains \(\mathbb{D}_t\) of the upper half-plane \(\mathbb{H}\) into \(\mathbb{R}\), and satisfy the chordal Löwner equation NEWLINE\[NEWLINE\frac{df(z, t)}{dt} =\frac{2}{f(z,t) \lambda(t)},NEWLINE\]NEWLINE where \(\lambda(t)\) is a continuous real-valued driving term.NEWLINENEWLINE In the case when the subdomains \(\mathbb{D}_t\) are a decreasing family consisting of \(\mathbb{H}\) minus an increasing vertical slit from the origin on \(\partial\mathbb{H}\), the authors show that the harmonic measures of the two sides of the slit are asymptotically equal. They also look at singular solutions of the chordal Löwner equation when the driving term has a higher Lipschitz order.
scientific article; zbMATH DE number 6062860