# Universal inequality and upper bounds of eigenvalues for non-integer poly-Laplacian on a bounded domain

Universal inequality and upper bounds of eigenvalues for non-integer poly-Laplacian on a bounded... In this paper we first study the universal inequality which is related to the eigenvalues of the fractional Laplacian $$(-\Delta )^s|_{\Omega }$$ ( - Δ ) s | Ω for $$s>0$$ s > 0 and $$s\in \mathbb {Q}_+$$ s ∈ Q + . Here $$\Omega \subset \mathbb {R}^n$$ Ω ⊂ R n is a bounded open domain, and $$\mathbb {Q}_+$$ Q + is the set of all positive rational numbers. Secondly, if $$s\in \mathbb {Q}_+$$ s ∈ Q + and $$s\ge 1$$ s ≥ 1 (in this case, the operator is also called the non-integer poly-Laplacian), then by this universal inequality and the variant of Chebyshev sum inequality, we can deduce the so-called Yang type inequality for the corresponding eigenvalue problem, which is the extension to the case of poly-Laplacian operators. Finally, we can get the upper bounds of the corresponding eigenvalues from the Yang type inequality. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

# Universal inequality and upper bounds of eigenvalues for non-integer poly-Laplacian on a bounded domain

, Volume 56 (5) – Aug 23, 2017
12 pages

/lp/springer_journal/universal-inequality-and-upper-bounds-of-eigenvalues-for-non-integer-JuUm0aVBpb
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
D.O.I.
10.1007/s00526-017-1220-y
Publisher site
See Article on Publisher Site

### Abstract

In this paper we first study the universal inequality which is related to the eigenvalues of the fractional Laplacian $$(-\Delta )^s|_{\Omega }$$ ( - Δ ) s | Ω for $$s>0$$ s > 0 and $$s\in \mathbb {Q}_+$$ s ∈ Q + . Here $$\Omega \subset \mathbb {R}^n$$ Ω ⊂ R n is a bounded open domain, and $$\mathbb {Q}_+$$ Q + is the set of all positive rational numbers. Secondly, if $$s\in \mathbb {Q}_+$$ s ∈ Q + and $$s\ge 1$$ s ≥ 1 (in this case, the operator is also called the non-integer poly-Laplacian), then by this universal inequality and the variant of Chebyshev sum inequality, we can deduce the so-called Yang type inequality for the corresponding eigenvalue problem, which is the extension to the case of poly-Laplacian operators. Finally, we can get the upper bounds of the corresponding eigenvalues from the Yang type inequality.

### Journal

Calculus of Variations and Partial Differential EquationsSpringer Journals

Published: Aug 23, 2017

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