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January  2016, 15(1): 261-286. doi: 10.3934/cpaa.2016.15.261

Blow-up scaling and global behaviour of solutions of the bi-Laplace equation via pencil operators

Pablo Álvarez-Caudevilla 1, and  V. A. Galaktionov 2,

1. 

Universidad Carlos III de Madrid, Av. Universidad 30, 28911-Leganés
2. 

Department of Mathematical Sciences, University of Bath, Bath BA2 7AY

Received  March 2015 Revised  July 2015 Published  December 2015

As the main problem, the bi-Laplace equation \begin{eqnarray} \Delta^2 u=0 \quad (\Delta=D_x^2+D_y^2) \end{eqnarray} in a bounded domain $\Omega \subset R^2$, with inhomogeneous Dirichlet or Navier-type conditions on the smooth boundary $\partial \Omega$ is considered. In addition, there is a finite collection of curves \begin{eqnarray} \Gamma = \Gamma_1\cup...\cup\Gamma_m \subset \Omega, \end{eqnarray} on which we assume homogeneous Dirichlet conditions $u=0$, focusing at the origin $0 \in \Omega$ (the analysis would be similar for any other point). This makes the above elliptic problem overdetermined. Possible types of the behaviour of solution $u(x,y)$ at the tip $0$ of such admissible multiple cracks, being a singularity point, are described, on the basis of blow-up scaling techniques and spectral theory of pencils of non self-adjoint operators. Typical types of admissible cracks are shown to be governed by nodal sets of a countable family of harmonic polynomials, which are now represented as pencil eigenfunctions, instead of their classical representation via a standard Sturm--Liouville problem. Eventually, for a fixed admissible crack formation at the origin, this allows us to describe all boundary data, which can generate such a blow-up crack structure. In particular, it is shown how the co-dimension of this data set increases with the number of asymptotically straight-line cracks focusing at 0.
Keywords: Bi- and Laplace equations, harmonic polynomials, higher-order equations, pencil of non self-adjoint operators, nodal sets..
Mathematics Subject Classification: 31A30, 35A20, 35C11, 35G1.
Citation: Pablo Álvarez-Caudevilla, V. A. Galaktionov. Blow-up scaling and global behaviour of solutions of the bi-Laplace equation via pencil operators. Communications on Pure & Applied Analysis, 2016, 15 (1) : 261-286. doi: 10.3934/cpaa.2016.15.261
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V. Adamyan and V. Pivovarchik, On spectra of some classes of quadratic operator pencils, Contributions to operator theory in spaces with an indefinite metric (Vienna, 1995), Oper. Theory Adv. Appl., 106, Birkhäuser, Basel, (1998), 23-36. doi: 10.1007/978-3-0348-8812-7_2.  Google Scholar

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P. Álvarez-Caudevilla and V. A. Galaktionov, The $p$-Laplace equation in domains with multiple crack section via pencil operators, Advances Nonlinear Studies, 15 (2015) no. 1, 91-116. Google Scholar

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V. A. Galaktionov, On extensions of Hardy's inequalities, Comm. Cont. Math., 7 (2005), 97-120. doi: 10.1142/S0219199705001659.  Google Scholar

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V. A. Kondrat'ev, Boundary value problems for parabolic equations in closed regions, Trans. Moscow Math. Soc., 15 (1966), 400-451.  Google Scholar

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V. A. Kondrat'ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc., 16 (1967), 209-292.  Google Scholar

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M. Krein and H. Langer, On some mathematical principles in the linear theory of damped oscillations of continua. I, II, Int. Equat. Oper. Theory, 1 (1978), 364-399, 539-566. doi: 10.1007/BF01682844.  Google Scholar

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A. Lemenant, On the homogeneity of global minimizers for the Mumford-Shah functional when $K$ is a smooth cone, Rend. Sem. Mat. Univ. Padova, 122 (2009), 129-159. doi: 10.4171/RSMUP/122-9.  Google Scholar

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A. S. Markus, Introduction to Spectral Theory of Polynomial Operator Pencils, Translated from the Russian by H. H. McFaden. Translation edited by Ben Silver. With an appendix by M. V. Keldysh. Transl. of Math. Mon., 71, Amer. Math. Soc., Providence, RI, 1988.  Google Scholar

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C. Sturm, Mémoire sur une classe d'équations à différences partielles, J. Math. Pures Appl., 1 (1836), 373-444. Google Scholar

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Acad. Press, New York-London, 1975.  Google Scholar

[2]

V. Adamyan and V. Pivovarchik, On spectra of some classes of quadratic operator pencils, Contributions to operator theory in spaces with an indefinite metric (Vienna, 1995), Oper. Theory Adv. Appl., 106, Birkhäuser, Basel, (1998), 23-36. doi: 10.1007/978-3-0348-8812-7_2.  Google Scholar

[3]

P. Álvarez-Caudevilla and V. A. Galaktionov, The $p$-Laplace equation in domains with multiple crack section via pencil operators, Advances Nonlinear Studies, 15 (2015) no. 1, 91-116. Google Scholar

[4]

J. W. Dettman, Mathematical Methods in Physics and Engineering, Mc-Graw-Hill, New York, 1969.  Google Scholar

[5]

Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev and S. I. Pohozaev, Global solutions of higher-order semilinear parabolic equations in the supercritical range, Adv. Differ. Equat., 9 (2004), 1009-1038.  Google Scholar

[6]

D. Funaro, Polynomial Approximation of Differential Equations, Springer-Verlag, Berlin/Tokyo, 1992.  Google Scholar

[7]

V. A. Galaktionov, Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications, Chapman \& Hall/CRC, Boca Raton-Washington D.C., Florida, 2004. doi: 10.1201/9780203998069.  Google Scholar

[8]

V. A. Galaktionov, On extensions of Hardy's inequalities, Comm. Cont. Math., 7 (2005), 97-120. doi: 10.1142/S0219199705001659.  Google Scholar

[9]

A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Nauka, Moscow, 1976.  Google Scholar

[10]

V. A. Kondrat'ev, Boundary value problems for parabolic equations in closed regions, Trans. Moscow Math. Soc., 15 (1966), 400-451.  Google Scholar

[11]

V. A. Kondrat'ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc., 16 (1967), 209-292.  Google Scholar

[12]

M. Krein and H. Langer, On some mathematical principles in the linear theory of damped oscillations of continua. I, II, Int. Equat. Oper. Theory, 1 (1978), 364-399, 539-566. doi: 10.1007/BF01682844.  Google Scholar

[13]

A. Lemenant, On the homogeneity of global minimizers for the Mumford-Shah functional when $K$ is a smooth cone, Rend. Sem. Mat. Univ. Padova, 122 (2009), 129-159. doi: 10.4171/RSMUP/122-9.  Google Scholar

[14]

A. S. Markus, Introduction to Spectral Theory of Polynomial Operator Pencils, Translated from the Russian by H. H. McFaden. Translation edited by Ben Silver. With an appendix by M. V. Keldysh. Transl. of Math. Mon., 71, Amer. Math. Soc., Providence, RI, 1988.  Google Scholar

[15]

C. Sturm, Mémoire sur une classe d'équations à différences partielles, J. Math. Pures Appl., 1 (1836), 373-444. Google Scholar

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