5 Dec 2007; Mat-Nat Faculty (UiTø), room A228. Wednesday 10:15-12:00

Volodymyr Rybalko (Inst. Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, Kharkov)

"Solutions with Vortices of a Semi-Stiff Boundary Value Problem for the Ginzburg-Landau Equation"

We study solutions of the 2D Ginzburg-Landau equation ε² Δu=u (|u|²-1) subject to "semi-stiff" boundary conditions: the Dirichlet condition for the modulus, |u|=1, and the homogeneous Neumann condition for the phase. Our principal result shows there are stable solutions of this problem with zeros (vortices), which are located near the boundary and have bounded energy in the limit of small ε. For the Dirichlet boundary condition ("stiff" problem), the existence of stable solutions with vortices, whose energy blows up as ε→0, is well known. By contrast, stable solutions with vortices are not established in the case of the homogeneous Neumann ("soft") boundary condition. (nonexistence is proved for simply connected domains).

We develop a variational method which allows one to construct local minimizers of the corresponding Ginzburg-Landau energy functional. We introduce an approximate bulk degree as the key ingredient of this method, and, unlike the standard degree over the curve, it is preserved in the weak H¹-limit.

NB: On this seminar we will have other guest researchers from Narvik, working in the same direction: Yury Holovatyy, Iryna Pankratova, Andrey Piatnitski.