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Bulk-boundary eigenvalues for Bilaplacian problems

  • *Corresponding author: María del Mar González

    *Corresponding author: María del Mar González 

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

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  • 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.

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


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

    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

    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 $)

    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 $

    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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