×

The second eigenvalue of the fractional \(p\)-Laplacian. (English) Zbl 1349.35263

For a bounded, open set \(\Omega\subset\mathbb R^N\) the authors consider the nonlinear eigenvalue problem \[ (- \Delta_p)^su=\lambda |u|^{p-2}u\text{ in }\Omega,\quad u=0\text{ in }\mathbb R^N\setminus\Omega, \] where \(0<s<1\) and \(1<p<\infty\) and the operator \((-\Delta_p)^s\) is defined as \[ (-\Delta_p)^su(x):=2\lim_{\delta\to 0+}\int_{\{y\in\mathbb R^N:|y-x|\geq\delta\}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+sp}}dy, \] called by the authors fractional \(p\)-Laplacian. The main aim of the paper is to study the second eigenvalue \(\lambda_2(\Omega)\) of the problem. It is shown that the second eigenvalue in fact exists (in particular, there is no accumulation of eigenvalues to the first eigenvalue \(\lambda_1(\Omega)\)). A mountain pass variational characterization of \(\lambda_2(\Omega)\) is proved. Moreover, the estimate \[ \lambda_2 (\Omega)>\left(\frac{2|B|}{|\Omega|}\right)^{\frac{sp}{N}}\lambda_1(B) \] is proven, where \(B\) is an arbitrary \(N\)-dimensional ball, and the authors show sharpness of the above estimate by verifying that for \(\Omega\) being the union of two disjoint balls of equal volume whose mutual distance tends to infinity, the eigenvalue \(\lambda_2 (\Omega)\) converges to the given bound. Consequently, for \(c>0\) the shape optimization problem of finding a domain \(\Omega_0\) such that \(\lambda_2 (\Omega_0) = \inf \{\lambda_2 (\Omega) : |\Omega| = c\}\) does not admit a solution.

MSC:

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
35R09 Integro-partial differential equations
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Brasco L. and Franzina G., An anisotropic eigenvalue problem of Stekloff type and weighted Wulff inequalities, NoDEA Nonlinear Differential Equations Appl. 20 (2013), 1795-1830. · Zbl 1292.35192
[2] Brasco L. and Franzina G., On the Hong-Krahn-Szego inequality for the p-Laplace operator, Manuscripta Math. 141 (2013), 537-557. · Zbl 1317.35149
[3] Brasco L. and Franzina G., Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Math. J. 37 (2014), 769-799. · Zbl 1315.47054
[4] Brasco L., Lindgren E. and Parini E., The fractional Cheeger problem, Interfaces Free Bound. 16 (2014), 419-458. · Zbl 1301.49115
[5] Caffarelli L. and Silvestre L., An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245-1260. · Zbl 1143.26002
[6] Cuesta M., Minimax theorems on \({C^{1}}\) manifolds via Ekeland variational principle, Abstr. Appl. Anal. 13 (2003), 757-768. · Zbl 1072.58004
[7] Cuesta M., De Figueiredo D. G. and Gossez J.-P., The beginning of the Fučik spectrum for the p-Laplacian, J. Differential Equations 159 (1999), 212-238. · Zbl 0947.35068
[8] Di Castro A., Kuusi T. and Palatucci G., Local behavior of fractional p-minimizers, preprint (2014), ; to appear in Ann. Inst. H. Poincaré Anal. Non Linéaire. · Zbl 1355.35192
[9] Di Castro A., Kuusi T. and Palatucci G., Nonlocal Harnack inequalities, J. Funct. Anal. 267 (2014), 1807-1836. · Zbl 1302.35082
[10] Di Nezza E., Palatucci G. and Valdinoci E., Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521-573. · Zbl 1252.46023
[11] Drábek P. and Robinson S. B., Resonance problems for the p-Laplacian, J. Funct. Anal. 169 (1999), 189-200. · Zbl 0940.35087
[12] Dyda B., A fractional order Hardy inequality, Illinois J. Math. 48 (2004), 575-588. · Zbl 1068.26014
[13] Franzina G. and Palatucci G., Fractional p-eigenvalues, Riv. Mat. Univ. Parma (N.S.) 5 (2014), 315-328.
[14] Franzina G. and Valdinoci E., Geometric analysis of fractional phase transition interfaces, Geometric Properties for Parabolic and Elliptic PDEs, Springer INdAM Ser. 2, Springer, New York (2013), 117-130. · Zbl 1302.35402
[15] Goyal S. and Sreenadh K., On the Fučik spectrum of non-local elliptic operators, NoDEA Nonlinear Differential Equations Appl. 21 (2014), 567-588. · Zbl 1296.35106
[16] Hong I., On an inequality concerning the eigenvalue problem of membrane, Kōdai Math. Semin. Rep. 6 (1954), 113-114. · Zbl 0057.08805
[17] Iannizzotto A. and Squassina M., Weyl-type laws for fractional p-eigenvalue problems, Asymptot. Anal. 88 (2014), 233-245. · Zbl 1296.35103
[18] Kassmann M., A priori estimates for integro-differential operators with measurable kernels, Calc. Var. Partial Differential Equations 34 (2009), 1-21. · Zbl 1158.35019
[19] Krahn E., Über Minimaleigenschaften der Kugel in drei und mehr Dimensionen, Acta Comm. Univ. Dorpat. A9 (1926), 1-44. · JFM 52.0510.03
[20] Kuusi T., Mingione G. and Sire Y., Nonlocal equations with measure data, Comm. Math. Phys. 337 (2015), 1317-1368. · Zbl 1323.45007
[21] Lindgren E. and Lindqvist P., Fractional eigenvalues, Calc. Var. Partial Differential Equations 49 (2014), 795-826. · Zbl 1292.35193
[22] Lindqvist P., Notes on the p-Laplace equation, Report 102, University of Jyvaskyla, Department of Mathematics and Statistics, Jyvaskyla, 2006. · Zbl 1256.35017
[23] Maz’ya V. and Shaposhnikova T., On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal. 195 (2002), 230-238. · Zbl 1028.46050
[24] Pólya G., On the characteristic frequencies of a symmetric membrane, Math. Z. 63 (1955), 331-337. · Zbl 0065.08703
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.