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EnglishWe consider the eigenvalue problem for the fractional p-Laplacian in an open bounded, possibly disconnected set ω ⊂ Rn, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfunctions, we show that the second eigenvalue λ2(ω) 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. Infλ(ω)=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 whose mutual distance tends to infinity.
| Data di pubblicazione: | 2016 | |
| Titolo: | The second eigenvalue of the fractional p-Laplacian | |
| Autori: | Brasco, Lorenzo; Parini, Enea | |
| Rivista: | ADVANCES IN CALCULUS OF VARIATIONS | |
| Keywords: | Caccioppoli estimates; Nonlocal eigenvalue problems; quasilinear nonlocal operators; spectral optimization; Analysis; Applied Mathematics | |
| Abstract in inglese: | We consider the eigenvalue problem for the fractional p-Laplacian in an open bounded, possibly disconnected set ω ⊂ Rn, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfunctions, we show that the second eigenvalue λ2(ω) 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. Infλ(ω)=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 whose mutual distance tends to infinity. | |
| Digital Object Identifier (DOI): | 10.1515/acv-2015-0007 | |
| Handle: | http://hdl.handle.net/11392/2357700 | |
| Appare nelle tipologie: | 03.1 Articolo su rivista |
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http://hdl.handle.net/11392/2357700
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