Bulk-boundary eigenvalues for Bilaplacian problems

Davide Buoso; Carles Falcó; María del Mar González; Manuel Miranda

doi:10.3934/dcds.2022096

Article Contents

2023, Volume 43, Issue 3&4: 1175-1200. Doi: 10.3934/dcds.2022096

This issue Previous Article Cauchy-Dirichlet problems for the porous medium equation Next Article Controlled boundary explosions: Dynamics after blow-up for some semilinear problems with global controls

Bulk-boundary eigenvalues for Bilaplacian problems

1.
Dipartimento per lo Sviluppo Sostenibile e la Transizione Ecologica, Università degli Studi del Piemonte Orientale "A. Avogadro", Piazza Sant'Eusebio 5, 13100 Vercelli, Italy
2.
Mathematical Institute, University of Oxford, OX2 6GG Oxford, United Kingdom
3.
Departamento de Matemáticas, Universidad Autónoma de Madrid and ICMAT, Madrid 28049, Spain
4.
Institute of Cross-Disciplinary Physics and Complex Systems, IFISC (UIB-CSIC), 07122 Palma de Mallorca, Spain

^*Corresponding author: María del Mar González
^*Corresponding author: María del Mar González

Dedicated to Prof. Juan Luis V´azquez for his 75th birthday

Received: December 2021

Revised: May 2022

Early access: July 2022

Published: March 2023

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Abstract

We initiate the study of a bulk-boundary eigenvalue problem for the Bilaplacian with a particular third order boundary condition that arises from the study of dynamical boundary conditions for the Cahn-Hilliard equation. First we consider continuity properties under parameter variation (in which the parameter also affects the domain of definition of the operator). Then we look at the ball and the annulus geometries (together with the punctured ball), obtaining the eigenvalues as solutions of a precise equation involving special functions. An interesting outcome of our analysis in the annulus case is the presence of a bifurcation from the zero eigenvalue depending on the size of the annulus.

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Mathematics Subject Classification: 35A09, 35B30, 35G05, 35P15.

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Figure 1. Function $ \Psi(\lambda) $

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Figure 2. Comparison between function $ \frac{s J_{\ell+s}(r)}{J'_{\ell+s}(r)} $ and its limit, a cotangent function scaled and displaced. The black line is the identity function

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Figure 3. First four eigenvalues ($ k = 1,2,3,4 $) for $ \ell = 0,1,2,3 $ and different values of $ \gamma $ and $ a $ (in dimension $ n = 2 $)

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Figure 4. $ \det W_\ell(\lambda) $ as a function of $ \lambda $ for $ n = 2, \gamma = 0.4,\ell = 0,1,2 $ for the cases $ a = 0.19 $ and $ a = 0.21 $

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Figure 5. First four eigenvalues ($ k = 1,2,3,4 $) for $ \ell = 0,1,2,3 $ and different values of $ \gamma $ and $ a $ ($ n = 4 $)

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