Bloch principle for elliptic differential operators with periodic coefficients

Abstract

Differential operators corresponding to elliptic equations of divergent type with 1-periodic coefficients are considered. The equations are put in Sobolev spaces with an arbitrary 1-periodic Borel measure on the entire space R (d) . In the study of the spectrum of operators of this kind, the Bloch principle is of fundamental importance. According to this principle, all points of the desired spectrum are obtained when studying the equation on the unit cube with quasiperiodic boundary conditions. The proof of the Bloch principle for problems in the above formulation is proved, in several versions of the principle. Examples of the application of the principle to finding the spectrum of specific operators, for example, for the Laplacian in a weighted space or on a singular structure of lattice type.