Exact solutions of the multiple Loewner equation
The author studies the generalized Loewner equation \[ \frac{dg(z,t)}{dt}=\sum_{k=1}^n\, \frac{2\mu_k}{g(z,t)-\lambda_k(t)},\;\;\;g(z,0)=z, \] where \(\mu_k>0,\;\sum_{k=1}^n\mu_k=1\), and \(\lambda_k\) are real, continous functions on \([0,+\infty)\). In several special cases, he provides implicit, integral solutions of the equation and describes geometric properties of these solutions. He considers are the following cases: (a) The functions \(\lambda_k\) are constants, \(\lambda_k(t)=\lambda_k\) and \(\lambda_1<\lambda_2<\dots<\lambda_n\). (b) \(n=2\), \(\mu_1=\mu_2=\frac{1}{2}\), \(\lambda_{1,2}=\pm A\;\sqrt{t}\). (c) \(n=2\), \(\mu_1=\mu_2=\frac{1}{2}\), \(\lambda_{1,2}=\pm B\;e^{t}\). Moreover, the author considers an inverse problem: certain conformal maps on the upper half-plane minus symmetric circular curves are represented as solutions to a certain Loewner equation.
scientific article; zbMATH DE number 7291216
scientific article