This construction can be viewed as a generalization of geometric prequantization to a degenerate situation. Abstract in TeX Laugesen, Richard Snyder. Assume that on each concentric subdisk the inhomogeneous membrane has at least as much mass as the homogeneous membrane. Furthermore, the zeta function of the eigenvalues is minimal for the homogeneous disk. Corollaries follow for simply connected surfaces with nonnegative curvature.
Stronger results hold in one dimension, for vibrating strings.
Abstract in TeX Goodey, Paul. These are used to solve the intermediate cases of a problem of Shephard and to give other applications to projection functions of convex bodies of revolution. Abstract in TeX Hayashi, Nakao. We prove the existence of modified scattering states and the sharp time decay estimate in the uniform norm of solutions to the Cauchy problem with small initial data. In order to derive the desired estimates we introduce a certain phase function since the previous methods, based solely on a priori estimates of the operator acting on the solution without specifying any phase function, do not work for the critical case under consideration.