No abstract

We consider an inverse problem for the Poisson equation with point sources in dimension 2. This problem has been studied extensively for the data in the form of the values of both the solution and its normal derivative given on the boundary. Our focus is on another type of data, namely the absolute value of the gradient and its derivatives. We present a method for converting the boundary data of the latter type to the boundary data of the former type. For the resulting system of ordinary differential equations, we establish the necessary and sufficient conditions for the existence of periodic solutions. To guarantee the uniqueness of the solution we introduce an additional constraint. Next, we present some numerical examples to illustrate the method and comment on the rate of convergence of the algorithm. Finally, we show that the method in its present form cannot be extended to dimension 3.

Building on the results of Chapman et al. (Z Angew Math Phys 47:410-431, 1996) on the behavior of minimizers in the Ginzburg-Landau thin-film model, we show that the vortices in the three-dimensional superconducting thin films are located in the cylinders whose cross sections coincide with the disks that contain the vortices in the two-dimensional model. To arrive at this conclusion, we prove that the three-dimensional minimizers converge to the two-dimensional counterparts in H 1 and in C α . We also give examples of regimes in which the vortex structure of the two-dimensional minimizers is well understood. Our results, in particular, provide insight into the behavior of the three-dimensional vortices in these regimes. (2000). Primary 35Q56. Mathematics Subject Classification

We study the numerical solutions of a system of Ginzburg-Landau type equations arising in the thin film model of superconductivity. These solutions are obtained by the Mountain Pass algorithm that was originally developed for semilinear elliptic equations. We prove a key hypothesis of the Mountain Pass theorem and investigate the physical features of the solutions such as the presence, the number, and the location of vortices and the numerical properties such as stability.