Dmitri Prokhorov

Infinite lifetime for the starlike dynamics in Hele-Shaw cells

One of the folklore questions in the theory of free boundary problems is the lifetime of the starlike dynamics in a Hele-Shaw cell. We prove precisely that, starting with a starlike analytic phase domain Omega(0), the Hele-Shaw chain of subordinating domains Omega( t), Omega(0) = Omega(0), exists for an infinite time under injection at the point of starlikeness.

Singular and Tangent Slit Solutions to the Löwner Equation

We consider the Lowner differential equation generating univalent maps of the unit disk (or of the upper half-plane) onto itself minus a single slit. We prove that the circular slits, tangent to the real axis are generated by Holder continuous driving terms with exponent 1/3 in the Lowner equation. Singular solutions are described, and the critical value of the norm of driving terms generating quasisymmetric slits in the disk is obtained.

On the local extremum property of the Koebe function

We discuss and compare several necessary criteria for the local extremality of the Koebe mapping in extremal problems for univalent functions. These criteria are applied to study Robertson type inequalities and also to investigate a conjecture of Bombieri.

Convex dynamics in Hele-Shaw cells

We study geometric properties of a contracting bubble driven by a homogeneous source at infinity and surface tension. The properties that are preserved during the time evolution are under consideration. In particular, we study convex dynamics of the bubble and prove that the rate of the area change is controlled by variation of the bubble logarithmic capacity. Next we consider injection through a single finite source and study some isoperimetric inequalities that correspond to the convex and α-convex dynamics. 2000 Mathematics Subject Classification: 30C45, 76S05, 76D99, 35Q35, 30C35. 1. Hele-Shaw problem. We are concerned with the one-phase Hele-Shaw problem in two space dimensions. Hele-Shaw [13] was the first who described in 1898 the motion of a fluid in a narrow gap between two parallel plates. A significant contribution after his work was made in 1945 by Polubarinova-Kochina [26, 27] and Galin [11], and then, by Saffman and Taylor [31] who discovered viscous fingering in 1958. New interest to this problem is reflected, for example, in a more than 600 item bibliography made by Gillow and Howison in the workshop of Hele-Shaw free boundary problems (http://www.maths.ox.ac.uk/∼howison/Hele-Shaw).

Coefficients of Functions Close to the Identity Function

We are looking for algorithms to solve extremal coefficient problems within the class of bounded univalent functions close to the identity. We use the Loewner differential equations and optimization methods to show that in some cases the necessary extremum conditions involving the Pontryagin maximum principle appear to be sufficient. There are some applications to the known and new coefficient estimates.

Conformal mapping asymptotics at a cusp

We describe the asymptotic behavior of the mapping function at an analytic cusp compared with Kaiser’s results for cusps with small perturbation of angles and the known explicit formulae for cusps with circular boundary curves. Saying “analytic cusp” here we mean that the boundary curves are real analytic away probably from the cusp. We propose a boundary curve parametrization by generalized power series which allows us to give explicit representations for locally univalent mapping functions with given asymptotic properties and for cusp boundary curves having an arbitrary order of tangency.

The compositions of hyperbolic triangle mappings

The behaviour of the compositions of linear fractional transformations of the unit disk and their limits has been studied on several occasions (e.g. [1,2,7]). In this note we observe the interesting property of the compositions of mappings of the unit disk onto a special hyperbolic triangle.

Coefficient products for bounded univalent functions

It is shown that the Pick functions are extremal for the functional |a 2a3| in the class of bounded univalent functions f(z)=z+ a 2z2+ …, |f(z)|< M,|z|<1Mis sufficiently large. This is a special case of a conjecture of Z. Jakubowski for the functional |a kan | where either kor nis even. Moreover it is shown that the Pick functions do not maximize | k an| if kand nare odd.

Directional convexity of level lines for functions convex in a given direction

Let K(φ) be the class of functions f(z) = z + a 2 z 2 +... which are holomorphic and convex in direction e iφ in the unit disk D, i.e. the domain f(D) is such that the intersection of f(D) and any straight line {w: w = w 0 + te iφ , t ∈ R} is a connected or empty set. In this note we determine the radius rψ ,φ of the biggest disk |z| < r ψ,φ with the property that each function f ∈ K(ψ) maps this disk onto the convex domain in the direction e iφ .

Harmonic Measures of Slit Sides, Conformal Welding and Extremum Problems

We present a list of 3 conjectures and 12 questions in geometric function theory. All of them have a connection with the Loewner theory for subordinate chains of conformal maps.