Overdetermined problems and the $p$-Laplacian

Kawohl, B.

Overdetermined problems and the $p$ -Laplacian.

Kawohl, Bernd (2007)

Acta Mathematica Universitatis Comenianae. New Series

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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 $p \to 1$ . The associated eigenfunction $u_{p}$ converges to the characteristic function of the Cheeger set, i.e. a subset of $Ω$ which minimizes the ratio $| \partial D | / | D |$ among all simply connected $D \subset \subset Ω$ . As a byproduct we prove that for convex $Ω$ the Cheeger set $ω$ is also convex.

The second eigenvalue of the $p$ -Laplacian as $p$ goes to 1.

Parini, Enea (2010)

International Journal of Differential Equations

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