Homogenization of harmonic maps with large number of vortices and applications in superconductivity and superfluidity. (Q5954473)

The authors deal with a homogenization problem for harmonic maps with vortices which models physical problems. The main gloal of the authors is to describes asymptotic behaviour of the fluxes NEWLINE\[NEWLINEv_\varepsilon= \text{Im}(u^*_\varepsilon\nabla u_\varepsilon)\tag{1}NEWLINE\]NEWLINE as \(\varepsilon\to 0\), when the number of holes becomes large and their diameters become small. Here Im stands for the imaginary part of complex number and (*) denotes the complex conjugate. The homogenized problem is described in terms of effective vorticity and the effective anisotropy tensor. The converges of the fluxes is rigorously proved.