Analog of the Löwner-Kufarev equation for the semigroup of conformal mappings of the disk into itself with fixed points and invariant diameter (Q1688340)

The author proposes analogues of the Loewner-Kufarev representations for conformal self-maps \(f:\mathbb D\to\mathbb D\) of the unit disk \(\mathbb D=\{z\in\mathbb C:|z|<1\}\) in the case when \(f(0)=0\), \(f\) have two boundary fixed points \(z=\pm1\), \(f'(\pm1)<\infty\), and \(f\) preserves the real diameter. Let \(\mathcal D[-1;0;1]\) denote a semigroup of conformal self-maps \(f:\mathbb D\to\mathbb D\) such that \(f(0)=0\), \(\text{Im}\,f(x)=0\) for \(x\in(-1,1)\), \(\lim_{z\to\pm1}=\pm1\), \(f'(\pm1)<\infty\). A two-parametric family \(\{w_{t,s}:0\leq s\leq t\leq T\}\) of functions \(w_{t,s}\in\mathcal D[-1;0;1]\) is an evolution family in \(\mathcal D[-1;0;1]\) on \([0,T]\) if the following conditions hold: \(w_{t,t}(z)\equiv z\) for \(0\leq t\leq T\), \(w_{t,s}=w_{t,\tau}\circ w_{\tau,s}\) for \(0\leq s\leq\tau\leq t\leq T\) and \(w_{t,s}\to z\) locally uniformly in \(\mathbb D\) for \((t-s)\to0\). The author proves two theorems. Theorem 1: Let \(\{w_{t,s}:0\leq s\leq t\leq T\}\) be an evolution family in \(\mathcal D[-1;0;1]\) and let \(\alpha(t)=w'_{t,0}(-1)\) and \(\beta(t)=w'_{t,0}(1)\) be increasing and absolutely continuous on \([0,T]\). Then, for all \(z\in\mathbb D\) and \(s\in[0,T]\), the function \(t\mapsto w_{t,s}(z)\) is absolutely continuous on \([s,T]\) and solves the differential equation \[ \frac{dw}{dt}=\frac{-w}{\lambda_1(t)((1-w)/(1+w))+\lambda_2(t)((1+w)/(1-w))+p(w,t)} \] with the initial data \(w|_{t=s}=z\), where the functions \(\lambda_1(t)\) and \(\lambda_2(t)\) are measurable in \(t\) and positive, \(1/\lambda_1(t),1/\lambda_2(t)\in L([0,T])\), the function \(p:\mathbb D\times[0,T]\to\mathbb C\) is holomorphic in \(w\in\mathbb D\) and measurable in \(t\in[0,T]\), \(\text{Re}\,p(\cdot,t)\geq0\) and \(p^{(n)}(0,t)\in\mathbb R\), \(n=0,1,2,\dots,\) for almost every \(t\in[0,T]\). Theorem 2: Let \(p:\mathbb D\times[0,T]\to\mathbb C\), \(T>0\), and \(\lambda_1(t)\) and \(\lambda_2(t)\) be defined as in Theorem 1. Then, for all \(z\in\mathbb D\) and \(s\in[0,T]\), there is a unique absolutely continuous solution \(w=w(t,z,s;p)\), \(s\leq t\leq T\), to the differential equation of Theorem 1 with \(w|_{t=s}=z\). Besides, for every \(t\in[s,T]\), the mapping \(w_{t,s}^p:z\mapsto w(t,z,s;p)\) is univalent and belongs to \(\mathcal D[-1;0;1]\), and \(\{w_{t,s}^p:0\leq s\leq t\leq T\}\) is an evolution family in \(\mathcal D[-1;0;1]\).