Rybalko V. Existence and asymptotic behavior of solutions to problems of mathematical physics

The first section of the thesis is devoted to the study of the variational problem for the Ginzburg-Landau functional in a class of complex valued functions with unit absolute value on the boundary and given degrees on its connected components. A new theory of such problems is developed. In particular, we study the issues of existence/nonexistence of global minimizers with given degrees; we develop a variational method for constructing local minimizers with zeros (vortices); we study singular behavior of minimizers and establish limiting locations of vortices near the boundary; we study the structure of Palais-Smale sequences and prove existence of mountain pass type critical points. In the second section we develop methods which allow one to study spectral problems for singularly perturbed nonsymmetric elliptic operators. We find an effective problem describing asymptotic behavior of ground states for Dirichlet problem with strongly oscillating locally periodic coefficients, establish improved asymptotic formulas for first eigenvalues and propose a selection algorithm for the limiting eigenfunctions. Analogous questions are considered for the Neumann problem with smoothly varying coefficients. Also, we study a spectral problem in a thin cylinder with Fourier conditions on its bases. In the third section we study two homogenization problems for which we discover new collective effect due to the inhomogeneities. In the final forth section we study bifurcation of traveling wave solutions in a free boundary problem modeling motility of living cells on a substratum.