The second eigenvalue of the fractional $p-$Laplacian

Abstract

We consider the eigenvalue problem for the {\it fractional $p-$Laplacian} in an open bounded, possibly disconnected set $\Omega \subset {\mathbb{R}}^n$, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfuctions, we show that the second eigenvalue ${\lambda }_2(\Omega )$ 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 \[ \inf \{λ_2(Ω)\,:\,|Ω|=c\}. \] 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.

38 pages. The test function used in the proof of Theorem 3.1 needed to be slightly modified, in order to be admissible for $1<p<2$. We fixed this issue

References in corpus (2)

• Nonlocal equations with measure data
• On The Fučik Spectrum Of Non-Local Elliptic Operators

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