94-306 Evans M. Harrell II
ON THE SECOND EIGENVALUE OF THE LAPLACE OPERATOR PENALIZED BY CURVATURE (13K, amstex) Oct 6, 94
Abstract , Paper (src), View paper (auto. generated ps), Index of related papers

Abstract. Consider the operator \$-\Q^2 - q(\k)\$, where \$-\Q^2\$ is the (positive) Laplace-Beltrami operator on a closed manifold of the topological type of the two-sphere \$S^2\$ and \$q\$ is a symmetric non-negative quadratic form in the principal curvatures. Generalizing a well-known theorem of J.\ Hersch for the Laplace-Beltrami operator alone, it is shown in this note that the second eigenvalue \$\la_1\$ is uniquely maximized, among manifolds of fixed area, by the true sphere. This problem arises in stability analysis of two-phase systems obeying the Allen-Cahn equation.

Files: 94-306.tex