Asymptotics of the spectrum of problems with constraints

By means of the method of an approximate spectral projection operator the classical asymptotic formula for the distribution function of eigenvalues with an estimate of the remainder is proved both for problems with an unsolvable constraint such as an incompressibility condition (the Navier–Stokes and Maxwell systems) and those with a solvable constraint (an example is the spectral problem of the theory of electroelasticity). Problems in a bounded Lipschitz domain are considered. We note that an estimate of the remainder for the linearized Navier–Stokes system was obtained earlier only for the case of a domain with boundary of class $C^\infty$, while for problems with solvable constraints only the leading term of the asymptotics was known; the asymptotics of the spectrum in the problem of the theory of electroelasticity has not been studied previously.
Bibliography: 13 titles.