Non-slit and singular solutions to the Löwner equation

We consider the Löwner differential equation in ordinary derivatives generating univalent self-maps of the unit disk or of the upper half-plane. If the solution to this equation represents a one-slit map, then the driving term is a continuous function. The reverse statement is not true in general, as a famous Kufarevʼs example shows. Lind, Marshall and Rohde found a sufficient condition for the driving term in the Löwner equation which guarantees a slit solution. The 1/2 Lipschitz norm of this term must be less than 4. We construct a family of non-slit solutions to the Löwner equation whose driving term is of 1/2 Lipschitz norm which admits the whole spectrum of values [4,∞). Then we turn to the properties of singular slit solutions in the half-plane. In particular, we prove that an analytic orthogonal slit is 1/2 Lipschitz with the vanishing norm.