On Shape Optimizing the Ratio of the First Two Eigenvalues of the Laplacian

Cover On Shape Optimizing the Ratio of the First Two Eigenvalues of the Laplacian
On Shape Optimizing the Ratio of the First Two Eigenvalues of the Laplacian
Jean Pierre a Haeberly
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86] if we show that the functions 'iv{A{x), B{x)) are regular and locally Lips- chitz of some rank / for all V € (S* . They certainly are regular since they are smooth functions of x. Moreover, the differentiability of ^viA{x), B{x)) implies that it is Lipschitz of rank ly in a neighborhood of x. More precisely, by the Mean Value Theorem, I ^v{Aix'), B{x'))-^v{A{x"\ Bix")) I

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