Abstract of the paper ``Bounds on decay constants for diffusion through inhomogeneous media''

The decay constants for diffusion through inhomogeneous media are known to be proportional to the eigenvalues of the corresponding elliptic operator. A new method of obtaining a hierarchy of upper bounds on sums and products of these eigenvalues as well as the eigenvalues themselves is presented. The first member of this hierarchy is just the usual Rayleigh-Ritz quotient. The other members of the hierarchy are generalized Rayleigh-Ritz quotients which can be derived simply using properties of integrals of the solutions of the diffusion equation. Explicit bounds are presented for the first three eigenvalues, but general methods of obtaining bounds for higher order eigenvalues are also outlined. For fixed time t, many of the bounds reduce to results given by the classical method of moments. The hierarchy of rigorous variational bounds on the eigenvalues studied may be generated using simple recursion relations based on properties of the characteristic orthogonal polynomials. The conditions on the trial functions used to obtain bounds on eigenvalues higher than the first are much simpler than those required by the traditional Rayleigh-Ritz procedure.

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