94-306 Evans M. Harrell II

ON THE SECOND EIGENVALUE OF THE LAPLACE OPERATOR PENALIZED BY CURVATURE (13K, amstex) Oct 6, 94

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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.

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