The second eigenvalue of the fractional p-Laplacian

By Lorenzo Brasco, Enea Parini. 2014

256 citations recorded by the source corpus.

Abstract We consider the eigenvalue problem for the fractional p-Laplacian in an open bounded, possibly disconnected set Ω ⊂ ℝ n ${\Omega\subset\mathbb{R}^{n}}$ , under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfunctions, we show that the second eigenvalue λ 2 ⁢ ( Ω ) ${\lambda_{2}(\Omega)}$ is well-defined, and we characterize it by means of several equivalent variational formulations. In particular, we extend the mountain pass characterization of Cuesta, De Figueiredo and Gossez to the nonlocal and nonlinear setting. Finally, we consider the minimization problem inf ⁡ { λ 2 ⁢ ( Ω ) : | Ω | = c } . $\inf\{\lambda_{2}(\Omega):|\Omega|=c\}.$ We prove that, differently from the local case, an optimal shape does not exist, even among disconnected sets. A minimizing sequence is given by the union of two disjoint balls of volume c / 2 ${c/2}$ whose mutual distance tends to infinity.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/69525c2c4fad612a509d8e6d4cf0ab21f5aaaeb3

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Citations in the corpus

A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators (2021, background): “In particular, the counterpart of the Hong–Krahn–Szegö inequality for the fractional Laplacian and the fractional p-Laplacian was established in [15] and it presents significant differences with the classical case: in particular, the shape optimizer for the second eigenvalue of the fractional p-Lapl”
Bifurcation and multiplicity results for critical fractional p‐Laplacian problems (2014, background): “y a rapidly growing literature on problems involving these nonlocal operators. In particular, fractional p-eigenvalue problems have been studied in Brasco and Parini [6], Brasco, Parini and Squassina [5], Franzina and Palatucci [20], Iannizzotto and Squassina [28] and in Lindgren and Lindqvist [31].”
Local behavior of fractional $p$-minimizers (2015, unknown): “Also, for general existence results and other regularity issues, we refer to the very recent contributions in [15], and in [2], where the related fractional p-eigenvalue problem has been considered.”
Higher Hölder regularity for the fractional p-Laplacian in the superquadratic case (2017, background): “onclusion. A local L∞-bound for solutions with a right-hand side will be needed. The result below is [9, Theorem 3.8]. Just like for the previous result, the stronger assumption g∈ Ws,p(RN) stated in [9] is not needed. Theorem 3.2 (Local boundedness). Let 1 <p<∞ and 0 <s<1. Let E⋐ E′ ⊂ R”
Gradient regularity in mixed local and nonlocal problems (2022, methodology): “These techniques ultimately rely on those used in the nonlocal case [9–11, 30, 31, 44–47] for purely nonlocal operators.”
On Dirichlet problem for fractional p-Laplacian with singular non-linearity (2016, methodology): “Before proving our next result, let us recall some estimates or inequalities from [7].”
The Brezis–Nirenberg problem for the fractional p-Laplacian (2015, background): “Using Brasco and Parini [6, Lemma A.2] and testing the equation (−∆)sp Uε = U p∗s−1 ε with gε,δ(Uε) ∈ W s,p 0 (Ω) gives ‖Gε,δ(Uε)‖ p ≤ ˆ R2N (Uε(x)− Uε(y)) p−1 (gε,δ(Uε(x))− gε,δ(Uε(y))) |x− y|N+sp dxdy = ˆ RN Uε(x) p∗s−1 gε,δ(Uε(x)) dx = |Uε| p∗s p∗s + ˆ RN (gε,δ(Uε(x))− Uε(x))Uε(x) p∗s−1 dx.”
Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems (2021, background): “1 nonlocal case, see Di Nezza-Palatucci-Valdinoci [34], Lindgren-Lindqvist [54], Brasco-LindgrenParini [18], Brasco-Parini [19], Franzina-Palatucci [44], Nguyen-Dı́az-Quoc-Hung [58] and the references therein.”
A Hopf's lemma and a strong minimum principle for the fractional p-Laplacian (2016, background): “See for instance [10, 5, 6, 12, 21, 23, 24, 31, 19] and the references therein.”
Eigenvalues and minimizers for a non-standard growth non-local operator (2018, background): “This line of research was widely continued, only to mention a few works, we remit to [6, 7, 15] and references to these papers.”
Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: A unified approach via fractional De Giorgi classes (2016, background): “ Rn\BR(x0) |u(x)|p−1 |x− x0|n+sp dx 1 p−1 , for any ﬁxed x0 ∈ Rn and R > 0. The above scale-invariant quantity has been introduced in [26, 27] and also considered in several other papers, such as [9, 44]. Observe in particular that it is ﬁnite, provided u ∈ Lp−1 s (R n). Another functional space ”