 94306 Evans M. Harrell II
 ON THE SECOND EIGENVALUE OF THE LAPLACE OPERATOR PENALIZED BY CURVATURE
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Oct 6, 94

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Abstract. Consider the operator $\Q^2  q(\k)$, where $\Q^2$ is the (positive)
LaplaceBeltrami operator on a closed manifold of the topological type of
the
twosphere $S^2$ and $q$ is a symmetric nonnegative quadratic form in the
principal
curvatures. Generalizing a wellknown theorem of J.\ Hersch for the
LaplaceBeltrami 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 twophase
systems obeying
the AllenCahn equation.
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