Weyl's spectral asymptotics formula for nonelliptic operators of the Dirichlet problem

One considers the asymptotic behavior of the spectrum of the Dirichlet problem for a class of operators with constant coefficients, including in it the hypoelliptic operators. For this class one obtains the classical Weyl formula of spectral asymptotics. The residual of the distribution function of the spectrum is estimated in terms of the measure of the interior boundary layer of the level surface of the operator symbol. These results can be carried over also to the vectorial case. One considers separately the class of operators whose quadratic form corresponds to the differential norm of the Sobolev type. For the indicated class one describes all possible elementary generalizations of the Weyl formula.