# Meeting Details

Title: Solutions with Vortices of a Semi-Stiff Boundary Value Problem for the Ginzburg-Landau Equation CCMA PDEs and Numerical Methods Seminar Series Leonid Berlyand, PSU We study solutions of 2D Ginzburg-Landau (GL) equation for a complex valued order parameter $u$ $$-\Delta u+\frac{1}{\varepsilon^2}u(|u|^2-1)=0, \, \, x \in A \subset R^2$$ which is lies in the core of Ginzburg-Landau theory of superconductivity and superfluidity. For a 2D domain $A$ with holes we consider the so-called "semi-stiff" boundary conditions: the Dirichlet condition for the modulus, $|u|=1$, and the homogeneous Neumann condition for the phase. The principal result of this work shows that there are stable solutions of this problem with vortices. These are vortices of a novel type: they approach the boundary and have bounded energy in the limit of small $\varepsilon$. In the well studied case of the Dirichlet problem for GL PDE stable solutions contain vortices distant from boundary with energy that blows up as $\varepsilon \to 0$. % For the %Dirichlet boundary condition ("stiff" problem) the existence of stable solutions with vortices whose energy blows up as $\varepsilon \to 0$ is well known. These vortices are distant from the boundary. By contrast, in the case of homogeneous Neumann ("soft") problem stable solutions with vortices are not established. In this work we develop variational method which allows to construct local minimizers of the corresponding Ginzburg-Landau energy functional. We introduce approximate bulk degree which is the key ingredient of this method, and it is preserved in the weak $H^1$-limit unlike the standard degree over a curve. This is a joint work with V. Rybalko.