2014/08/26 by C. De Coster, Serge Nicaise, De Coster, Colette +4 · 1 citation
Computer Science · Engineering · Mathematics · #35B65 #35K55 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Fluid Dynamics and Thin Films #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1408.6180
openalex publication_date 2014/08/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we are interested in the following fourth order eigenvalue\nproblem coming from the buckling of thin films on liquid substrates:\n n
begincases\n
Delta2 u+
kappa2 u=-
lambda
Delta u amp;
textin B1,
newline\n u=
partialr u= 0 amp;
texton
partial B1,\n
endcases where B1 is the unit ball in \ℝN.\n When \κ > 0 is small, we show that the first eigenvalue is simple and\nthe first eigenfunction, which gives the shape of the film for small\ndisplacements, is positive. However, when \κ increases, we establish that\nthe first eigenvalue is not always simple and the first eigenfunction may\nchange sign. More precisely, for any \κ \∈ (0,+\∞), we give the\nexact multiplicity of the first eigenvalue and the number of nodal regions of\nthe first eigenfunction.\n