Analysis on fingering problems in fluids and phase transitions

The existence problem of symmetric 3-dimensional finger solutions of Mullins-Sekerka equation is studied. The finger solutions are traveling wave solutions whose finger-shaped interfaces are moving along a certain direction at a constant speed within a cylindrical domain. The existence of at least one symmetric 3-dimensional finger solutions is obtained through a fixed point argument of the Hilbert Transformation.;The numerical calculation of a Mullins-Sekerka problem in 2-dimension is carried out.