# Nodal properties of eigenfunctions of a generalized buckling problem on balls

Nodal properties of eigenfunctions of a generalized buckling problem on balls In this paper we are interested in the following fourth order eigenvalue problem coming from the buckling of thin films on liquid substrates: \begin{aligned} {\left\{ \begin{array}{ll} \Delta ^2 u+ \kappa ^2 u=-\lambda \Delta u &{}\text {in } B_1,\\ u=\partial _r u= 0 &{}\text {on } \partial B_1, \end{array}\right. } \end{aligned} Δ 2 u + κ 2 u = - λ Δ u in B 1 , u = ∂ r u = 0 on ∂ B 1 , where $$B_1$$ B 1 is the unit ball in  $${\mathbb R}^N$$ R N . When $$\kappa > 0$$ κ > 0 is small, we show that the first eigenvalue is simple and the first eigenfunction, which gives the shape of the film for small displacements, is positive. However, when $$\kappa$$ κ increases, we establish that the first eigenvalue is not always simple and the first eigenfunction may change sign. More precisely, for any $$\kappa \in \mathopen ]0,+\infty \mathclose [$$ κ ∈ ] 0 , + ∞ [ , we give the exact multiplicity of the first eigenvalue and the number of nodal regions of the first eigenfunction. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Nodal properties of eigenfunctions of a generalized buckling problem on balls

, Volume 19 (4) – Mar 20, 2015
33 pages

/lp/springer_journal/nodal-properties-of-eigenfunctions-of-a-generalized-buckling-problem-J6umNicAU5
Publisher
Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-015-0331-y
Publisher site
See Article on Publisher Site

### Abstract

In this paper we are interested in the following fourth order eigenvalue problem coming from the buckling of thin films on liquid substrates: \begin{aligned} {\left\{ \begin{array}{ll} \Delta ^2 u+ \kappa ^2 u=-\lambda \Delta u &{}\text {in } B_1,\\ u=\partial _r u= 0 &{}\text {on } \partial B_1, \end{array}\right. } \end{aligned} Δ 2 u + κ 2 u = - λ Δ u in B 1 , u = ∂ r u = 0 on ∂ B 1 , where $$B_1$$ B 1 is the unit ball in  $${\mathbb R}^N$$ R N . When $$\kappa > 0$$ κ > 0 is small, we show that the first eigenvalue is simple and the first eigenfunction, which gives the shape of the film for small displacements, is positive. However, when $$\kappa$$ κ increases, we establish that the first eigenvalue is not always simple and the first eigenfunction may change sign. More precisely, for any $$\kappa \in \mathopen ]0,+\infty \mathclose [$$ κ ∈ ] 0 , + ∞ [ , we give the exact multiplicity of the first eigenvalue and the number of nodal regions of the first eigenfunction.

### Journal

PositivitySpringer Journals

Published: Mar 20, 2015

### References

• The physics and physiology of lung surfactants
Zasadzinski, JA; Ding, J; Warriner, HE; Bringezu, AJ; Waring, AJ

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