Convergence of the eigenvalues of the $$p$$ -Laplace operator as $$p$$ goes to 1

Littig, Samuel; Schuricht, Friedemann

doi:10.1007/s00526-013-0597-5

Cited by 28 publications

(23 citation statements)

References 22 publications

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“…The next lemma, where Id stands for the identity function, is extracted of the proof of [18,Lemma 3.2]. It helps us to overcome the fact that C ∞ c (Ω) is not dense in C 0,s 0 (Ω).…”

Section: Lemmamentioning

confidence: 99%

Asymptotic behavior of extremals for fractional Sobolev inequalities associated with singular problems

Ercole

Pereira

Sanchis

2019

Annali di Matematica

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Let Ω be a smooth, bounded domain of R N , ω be a positive, L 1 -normalized function, and 0 < s < 1 < p. We study the asymptotic behavior, as p → ∞, of the pair p Λ p , u p , where Λ p is the best constant C in the Sobolev type inequalityand u p is the positive, suitably normalized extremal function corresponding to Λ p . We show that the limit pairs are closely related to the problem of minimizing the quotient |u| s / exp Ω (log |u|)ωdx , where |u| s denotes the s-Hölder seminorm of a function u ∈ C 0,s 0 (Ω).

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“…The next lemma, where Id stands for the identity function, is extracted of the proof of [18,Lemma 3.2]. It helps us to overcome the fact that C ∞ c (Ω) is not dense in C 0,s 0 (Ω).…”

Section: Lemmamentioning

confidence: 99%

Asymptotic behavior of extremals for fractional Sobolev inequalities associated with singular problems

Ercole

Pereira

Sanchis

2019

Annali di Matematica

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“…The results of this paper can be applied also to the singular case p = 1, which must be treated separately. In [20] the authors defined a sequence of variational eigenvalues and proved that they can be approximated by the corresponding eigenvalues of the p-Laplacian as p → 1. The second variational eigenvalue of the 1-Laplacian can be characterized geometrically, as a consequence of [20,Theorem 2.4] and [21,Theorem 5.5] (see also [7]).…”

Section: Final Remarks and Open Questionsmentioning

confidence: 99%

“…In [20] the authors defined a sequence of variational eigenvalues and proved that they can be approximated by the corresponding eigenvalues of the p-Laplacian as p → 1. The second variational eigenvalue of the 1-Laplacian can be characterized geometrically, as a consequence of [20,Theorem 2.4] and [21,Theorem 5.5] (see also [7]). In particular, if Ω = B 2 is a disc, it holds λ 2 (1; B 2 ) = λ (1; B 2 ), and therefore λ 2 (1; B 2 ) = λ 3 (1; B 2 ) = λ (1; B 2 ) by reasoning as in Proposition 3.1.…”

Section: Final Remarks and Open Questionsmentioning

confidence: 99%

On multiplicity of eigenvalues and symmetry of eigenfunctions of the $p$-Laplacian

Audoux¹,

Bobkov²,

Parini³

2018

TMNA

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Abstract. We investigate multiplicity and symmetry properties of higher eigenvalues and eigenfunctions of the p-Laplacian under homogeneous Dirichlet boundary conditions on certain symmetric domains Ω ⊂ R N . By means of topological arguments, we show how symmetries of Ω help to construct subsets of W 1,p 0 (Ω) with suitably high Krasnosel'skiȋ genus. In particular, if Ω is a ball B ⊂ R N , we obtain the following chain of inequalities:Here λi(p; B) are variational eigenvalues of the p-Laplacian on B, and λ (p; B) is the eigenvalue which has an associated eigenfunction whose nodal set is an equatorial section of B. If λ2(p; B) = λ (p; B), as it holds true for p = 2, the result implies that the multiplicity of the second eigenvalue is at least N . In the case N = 2, we can deduce that any third eigenfunction of the p-Laplacian on a disc is nonradial. The case of other symmetric domains and the limit cases p = 1, p = ∞ are also considered.

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“…This does not allow to replace λ by λ p where the latter one is the first eigenvalue of the p-Laplacian. For other types of approximations of eigenvalues for 1-Laplacian we refer to [8,13,15,18,20]. Also, besides the different choices of λ may lead to a lack of compactness, we observe that the model functional in (3.1) can be taken into account also in the case of thin domains of the type Ω ε , i.e.…”

Section: Proofmentioning

confidence: 99%

Some remarks about dimension reduction for −Δ1

Amendola

Gargiulo

Zappale

2015

Asymptotic Analysis

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The paper is devoted to study 3D-2D dimension reduction for non-homogeneous −Δ 1 , by means of power law approximation and Γ -convergence. M.E. Amendola et al. / Some remarks about dimension reduction for −Δ 1Definition 1.2 (Γ -convergence for a sequence of functionals). Let {J n } be a sequence of functionals defined on X with values in R. The functional J : X → R is said to be the Γ -liminf (resp. Γ -limsup) of {J n } with respect to the metric d if for every u ∈ X J(u) = inf liminf n→∞ J n (u n ): u n ∈ X, u n → u in X resp. limsup n→∞ .Thus we writeMoreover, the functional J is said to be the Γ -limit of {J n } ifand we may writeWe refer to [10] for a detailed treatment of Γ -convergence. For what concerns the Γ -convergence of families of functionals we will recall the results contained in [3, Section 2.1].

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Convergence of the eigenvalues of the $$p$$ -Laplace operator as $$p$$ goes to 1

Cited by 28 publications

References 22 publications

Asymptotic behavior of extremals for fractional Sobolev inequalities associated with singular problems

Asymptotic behavior of extremals for fractional Sobolev inequalities associated with singular problems

On multiplicity of eigenvalues and symmetry of eigenfunctions of the $p$-Laplacian

Some remarks about dimension reduction for −Δ1

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