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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