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Fractional eigenvalues

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Abstract

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.

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Notes

  1. These spaces are also known as Aronszajn, Gagliardo or Slobodeckij spaces

  2. The name “principal frequency” is synonymous.

  3. In the linear case this integral operator has been treated as the principal value of a singular integral.

  4. The idea is obvious in the case \(p=2\).

  5. Note added in proof  In the recent work “A note on positive eigenfunctions and hidden convexity”, Archiv der Mathematik, 2012, Volume 99, Issue 4, pp 367-374, by L. Brasco and G. Franzina, the range for \(\alpha \) in Theorem 14 and Theorem 16 has been widened.

  6. Note added in proof  In the recent work “A note on positive eigenfunctions and hidden convexity”, Archiv der Mathematik, 2012, Volume 99, Issue 4, pp 367-374, by L. Brasco and G. Franzina, the range for \(\alpha \) in Theorem 14 and Theorem 16 has been widened.

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Acknowledgments

We thank Evgenia Malinnikova for helping us to verify an inequality. We thank the referees for a careful reading of the manuscript and for drawing our attention to the article [9].

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Authors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491, Trondheim, Norway

    Erik Lindgren & Peter Lindqvist

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  1. Erik Lindgren
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  2. Peter Lindqvist
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Correspondence to Erik Lindgren.

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Communicated by Y. Giga.

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Lindgren, E., Lindqvist, P. Fractional eigenvalues. Calc. Var. 49, 795–826 (2014). https://doi.org/10.1007/s00526-013-0600-1

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