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Estimates of the principal eigenvalue of the $p$ -Laplacian and the $p$ -biharmonic operator

Jiří Benedikt — 2015

Mathematica Bohemica

We survey recent results concerning estimates of the principal eigenvalue of the Dirichlet $p$ -Laplacian and the Navier $p$ -biharmonic operator on a ball of radius $R$ in $ℝ^{N}$ and its asymptotics for $p$ approaching $1$ and $\infty$ . Let $p$ tend to $\infty$ . There is a critical radius $R_{C}$ of the ball such that the principal eigenvalue goes to $\infty$ for $0 < R \leq R_{C}$ and to $0$ for $R > R_{C}$ . The critical radius is $R_{C} = 1$ for any $N \in ℕ$ for the $p$ -Laplacian and $R_{C} = \sqrt{2 N}$ in the case of the $p$ -biharmonic operator. When $p$ approaches $1$ , the principal eigenvalue of the Dirichlet...

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Estimates of the principal eigenvalue of the $p$ -Laplacian and the $p$ -biharmonic operator

Uniqueness theorem for $p$ -biharmonic equations.

On the discreteness of the spectra of the Dirichlet and Neumann $p$ -biharmonic problems.

Prüfer transformation for the $p$ -Laplacian.

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Formula preview

Currently displaying 1 – 4 of 4

Estimates of the principal eigenvalue of the p -Laplacian and the p -biharmonic operator

Uniqueness theorem for p -biharmonic equations.

On the discreteness of the spectra of the Dirichlet and Neumann p -biharmonic problems.

Prüfer transformation for the p -Laplacian.

Estimates of the principal eigenvalue of the $p$ -Laplacian and the $p$ -biharmonic operator

Uniqueness theorem for $p$ -biharmonic equations.

On the discreteness of the spectra of the Dirichlet and Neumann $p$ -biharmonic problems.

Prüfer transformation for the $p$ -Laplacian.