Abstract
We deduce variational formulae for domains close to a disk which generalize mappings from the unit disk and from the exterior of the unit disk onto a circular digon. These results lead to the asymptotic conformal welding for such domains. Besides we deduce the asymptotic conformal welding for circular quadrangles with inner angles απ. The by-product of the main consideration is a contribution to the Kufarev type examples in the Lowner theory. The mapping from the unit disk onto a circular digon symmetric with respect to the real axis satisfies the Lowner type equation though it is not a slit mapping.
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