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Displaying similar documents to “Overdetermined problems and the p -Laplacian”

Isoperimetric estimates for the first eigenvalue of the p -Laplace operator and the Cheeger constant

Bernhard Kawohl, V. Fridman (2003)

Commentationes Mathematicae Universitatis Carolinae

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First we recall a Faber-Krahn type inequality and an estimate for λ p ( Ω ) in terms of the so-called Cheeger constant. Then we prove that the eigenvalue λ p ( Ω ) converges to the Cheeger constant h ( Ω ) as p 1 . The associated eigenfunction u p converges to the characteristic function of the Cheeger set, i.e. a subset of Ω which minimizes the ratio | D | / | D | among all simply connected D Ω . As a byproduct we prove that for convex Ω the Cheeger set ω is also convex.