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

Abstract

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 ∞. Let p tend to ∞. There is a critical radius R C of the ball such that the principal eigenvalue goes to ∞ for 0 < R R C and to 0 for R > R C. The critical radius is R C = 1 for any N ∈ N for the p-Laplacian and R C = √ 2N in the case of the p-biharmonic operator. When p approaches 1, the principal eigenvalue of the Dirichlet p-Laplacian is N R −1 × (1 − (p − 1) log R(p − 1)) + o(p − 1) while the asymptotics for the principal eigenvalue of the Navier p-biharmonic operator reads 2N/R 2 + O(−(p − 1) log(p − 1)).

Hardy inequalities with double singular weights

“…In Benedikt and Drábek [18], Theorem 2, and in Benedikt and Drábek [19], Theorem 2, by Picone's identity the following estimate for p > 1 was proved ,2) p,n (B R ), for p ≥ 2, see Proposition 6.4, the estimate…”

“…There are estimates for λ p,n (Ω) by means of the Cheeger's constant, Cheeger [27], Lefton and Wei [68], Kawohl and Fridman [60], by the Picone's identity Benedikt and Drábek [18,19], with the Sobolev inequality Maz'ja [77], Ludwig et al [76], with estimates in parallelepiped Lindqvist [75] and others. However, Hardy inequality with double singular weights allows one to get better analytical estimates for λ p,n (Ω).…”

Pointwise bounds for positive supersolutions of nonlinear elliptic problems involving the $p$-Laplacian and application

“…As it mentioned in [15], while upper bounds for λ 1 (Ω) can be obtained by choosing particular test function v in (1.3), but lower bounds are more challenging. For more details on estimates and asymptotics of the principal eigenvalue and eigenfunction of the p-Laplacian operator, we refer the reader to [3,4,5,15]. For example when Ω = B we shall prove the following lower bound, which is better than those given in [3,4,15], for some range of p and N (see end of Section 3).…”

“…For more details on estimates and asymptotics of the principal eigenvalue and eigenfunction of the p-Laplacian operator, we refer the reader to [3,4,5,15]. For example when Ω = B we shall prove the following lower bound, which is better than those given in [3,4,15], for some range of p and N (see end of Section 3).…”

“…Benedikt and Derábek in [4] also presented upper and lower bounds for λ 1 (Ω) on a bounded domain Ω ⊂ R N . In particular, when Ω = B they proved that…”