© Hindawi Publishing Corp. 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 mo-tion 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 in-terest 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 problem