Linear and nonlinear perturbations of the operator

The perturbation theory for the operator div is of particular interest in the study of boundary-value problems for the general nonlinear equation F(\dot y,y,x)=0. Taking as linearization the first order operator Lu=C_{ij}u_{x_j}^i+C_iu^i, one can, under certain conditions, regard the operator L as a compact perturbation of the operator div. This book presents results on boundary-value problems for L and the theory of nonlinear perturbations of L. Specifically, necessary and sufficient solvability conditions in explicit form are found for various boundary-value problems for the operator L. An analog of the Weyl decomposition is proved. The book also contains a local description of the set of all solutions (located in a small neighborhood of a known solution) to the boundary-value problems for the nonlinear equation F(\dot y, y, x) = 0 for which L is a linearization. A classification of sets of all solutions to various boundary-value problems for the nonlinear equation F(\dot y, y, x) = 0 is given. The results are illustrated by various applications in geometry, the calculus of variations, physics, and continuum mechanics.