Abstract:
The driving function $\lambda$ of the Loewner equation that generates a non-tangential slit is Hölder continuous with exponent $1/2$. For a tangential slit emanating from a corner, the behavior of the driving function $\lambda(\tau)$ in a neighborhood of $\tau=0$ depends on the tan-gency order of the slit and on the angle of the corner. In this paper we investigate a family of mappings $f = f (z, \tau)$, $\tau\in[0,T]$. For a fixed $\tau$, the mapping $f$ takes the half-plane onto a digon with a slit (the length of the slit depends on $\tau$) along a circular arc emanating tangentially from a vertex of the digon with an angle an. We obtain the form of the expansion of the driving function $\lambda$ at the point $\tau=0$, which generates the slit in the digon. We construct a mapping of the half-plane onto a triangle with tangential slit emanating from a corner of the triangle, assuming that the driving function has the same form as for the digon. We propose the following conjecture: if $\lambda$ generates in a simply connected domain $D$ a slit along a circular arc emanating tangentially from a corner with an interior angle $\alpha\pi$, then the function $\lambda$ expands into the series $\lambda(\tau)=\sum\limits_{k=0}^\infty\lambda_k\tau^{\frac{1+k\alpha}{2+\alpha}}$, $\lambda_0\ne0$.
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