In this thesis; we invastigate the inequality etwen eigenvalues of Dirichlet-Laplacian and Stokes operators on the one hand, and between Dirichlet-Laplacian and Neumann-Laplacian on the other hand. We show that the k-th eigenvalue of the Dirichlet-Laplacian is strictly less than the k-th eigenvalue of the We use Filonov’s original idea, used first in proving the inequality between Dirichlet- and Neumann- Laplacian operators, to proves both inequalities. This paper is organized as follows. We describe the necessary function spaces, trace operators, and related lemmas in Section 2. In Section 4, we define the Filonov’s strict inequality is a strengthening of the partial inequality µ K+ 1 ? ? K proved by L. Friedlander in [1991] using very different techniques. A fairly direct variational argument shows that ? K ? ? K . A simple proof of the inequality µ K+1 ? K is given. Here the ? K (respectively, µ K ) are the eigenvalues of the Dirichlet (respectively, Neumann) problem for the Laplace operator in an arbitrary domain of finite measure in R , d 1. For d = 2 and for a domain bounded by an analytic curve, Polya and Szego proved that µ 2 ? ? ? 1 , where is an absolute constant less than one (expressed in terms of zeros of Bessel functions). Since their proofs involve conformal mappings, they do not work in higher dimensions. Developing an idea used by Payne Levine and Weinberger established for an arbitrary dimension, a series of inequalities of the form µ k+r ? k , r = 1,…, d, under some conditions on the principal curvatures of the C 2+? - smooth boundary ?? of a bounded domain ?. In particular, µ k+1 ? k if the mean curvature is nonnegative, and µ k+d ? ? k for all convex domains.