Eigenvalues of strings and cylinders with variable mass density
نویسندگان
چکیده
This paper proves sharp inequalities for the frequencies of vibration of strings and cylindrical membranes, under either fixed or fixed-free boundary conditions. We establish such inequalities also on the "generalized" cylinder, which is the cross-product of an interval with a homogeneous Riemannian manifold. The goal is to better understand the effect that inhomogeneities in the mass distribution have on the frequencies of vibration. Theorem 2, for example, says roughly that for a vibrating string fixed at one end and free at the other, moving mass towards the free end will tend to decrease the frequencies. To make this more precise, suppose w is a positive function on the interval (0, L) and \j(w) is the j-th eigenvalue of the weighted Laplacian w~d/dx on that interval, under Neumann boundary conditions at x = 0 and Dirichlet boundary conditions at x = L; this means that for some eigenfunction ^j,
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