Fractional eigenvalues

By Erik Lindgren, Peter Lindqvist. 2012

379 citations recorded by the source corpus.

We study the non-local eigenvalue problem $$\begin{aligned} 2\, \int \limits _{\mathbb{R }^n}\frac{|u(y)-u(x)|^{p-2}\bigl (u(y)-u(x)\bigr )}{|y-x|^{\alpha p}}\,dy +\lambda |u(x)|^{p-2}u(x)=0 \end{aligned}$$for large values of $$p$$ and derive the limit equation as $$p\rightarrow \infty $$. Its viscosity solutions have many interesting properties and the eigenvalues exhibit a strange behaviour.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/3b2f3d8980252cfead33fce7ac7171e97e18e591

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

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.”
Existence of solutions for perturbed fractional p-Laplacian equations (2016, result): “As for some recent results on the fractional p-Laplacian, we refer to for example [18,19,22] and the references therein.”
The second eigenvalue of the fractional p-Laplacian (2014, background): “the set σs,p(Ω) = {λ ∈ R : λ is an (s, p)-eigenvalue} is a closed set (see [21]).”
Convexity properties of Dirichlet integrals and Picone-type inequalities (2014, background): “the reader is referred to [8, 21] for a complete account on the case q = p.”
A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators (2021, background): “For the second eigenvalue, however, the situation in the nonlocal case is quite different from the classical one, since in general nonlocal energy functionals are deeply influenced by the mutual position of the different connected components of the domain, see [50].”
Local behavior of fractional $p$-minimizers (2015, neutral): “Secondly, the analysis in the papers [3] and [18] considers the special case when p is suitably large – thus falling in the Morrey embedding case when concerning regularity.”
Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case (2015, background): “This nonlinear and nonlocal eigenvalue problem has been first introduced in [21].”
On a new class of fractional partial differential equations (2014, background): “As mentioned in the introduction, this is not to be confused with existing notions of the fractional p-Laplace’s equation from the real interpolation theory, see for example [5, 10, 21].”
Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian (2015, neutral): “We already know that the first eigenvalue λ1 of (−Δ)p defined as λ1 = inf u∈W0\{0} ‖u‖pW0 ‖u‖pLp(Ω) lies in (0, ∞), see [16,25].”
Fractional Sobolev spaces with variable exponents and fractional p(x)-Laplacians (2017, background): “In the constant exponent case it is known as the fractional p-Laplacian, see [2, 3, 5, 7, 8, 9, 11, 12, 15, 16, 17] and references therein.”
On Dirichlet problem for fractional p-Laplacian with singular non-linearity (2016, methodology): “In [32, 38], it was shown that this minimizer is achieved by a unique positive and bounded function φ1.”