Eigenvalues of polyharmonic operators on variable domains
ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 4, pp. 1225-1235.

We consider a class of eigenvalue problems for polyharmonic operators, including Dirichlet and buckling-type eigenvalue problems. We prove an analyticity result for the dependence of the symmetric functions of the eigenvalues upon domain perturbations and compute Hadamard-type formulas for the Frechét differentials. We also consider isovolumetric domain perturbations and characterize the corresponding critical domains for the symmetric functions of the eigenvalues. Finally, we prove that balls are critical domains.

DOI : 10.1051/cocv/2013054
Classification : 35J40, 35B20, 35P15
Mots clés : polyharmonic operators, eigenvalues, domain perturbation
@article{COCV_2013__19_4_1225_0,
     author = {Buoso, Davide and Lamberti, Pier Domenico},
     title = {Eigenvalues of polyharmonic operators on variable domains},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1225--1235},
     publisher = {EDP-Sciences},
     volume = {19},
     number = {4},
     year = {2013},
     doi = {10.1051/cocv/2013054},
     mrnumber = {3182687},
     zbl = {1291.35144},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2013054/}
}
TY  - JOUR
AU  - Buoso, Davide
AU  - Lamberti, Pier Domenico
TI  - Eigenvalues of polyharmonic operators on variable domains
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2013
SP  - 1225
EP  - 1235
VL  - 19
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2013054/
DO  - 10.1051/cocv/2013054
LA  - en
ID  - COCV_2013__19_4_1225_0
ER  - 
%0 Journal Article
%A Buoso, Davide
%A Lamberti, Pier Domenico
%T Eigenvalues of polyharmonic operators on variable domains
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2013
%P 1225-1235
%V 19
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2013054/
%R 10.1051/cocv/2013054
%G en
%F COCV_2013__19_4_1225_0
Buoso, Davide; Lamberti, Pier Domenico. Eigenvalues of polyharmonic operators on variable domains. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 4, pp. 1225-1235. doi : 10.1051/cocv/2013054. http://www.numdam.org/articles/10.1051/cocv/2013054/

[1] S. Agmon, Lectures on elliptic boundary value problems. Van Nostrand Mathematical Studies, No. 2 D. Van Nostrand Co., Inc. Princeton, N.J., Toronto London (1965). | MR | Zbl

[2] M.S. Ashbaugh and R.D. Benguria, On Rayleigh's conjecture for the clamped plate and its generalization to three dimensions. Duke Math. J. 78 (1995) 1-17. | MR | Zbl

[3] M.S. Ashbaugh and D. Bucur, On the isoperimetric inequality for the buckling of a clamped plate. Z. Angew. Math. Phys. 54 (2003) 756-770. | MR | Zbl

[4] D. Bucur and G. Buttazzo, Variational methods in some shape optimization problems. Appunti dei Corsi Tenuti da Docenti della Scuola, Notes of Courses Given by Teachers at the School. Scuola Normale Superiore, Pisa (2002). | MR | Zbl

[5] V. Burenkov and P.D. Lamberti, Sharp spectral stability estimates via the Lebesgue measure of domains for higher order elliptic operators. Rev. Mat. Comput. 25 (2012) 435-457. | MR | Zbl

[6] V. Burenkov and P.D. Lamberti, Spectral stability of higher order uniformly elliptic operators. Sobolev spaces in mathematics. II, in vol. 9 of Int. Math. Ser. (NY). Springer, New York (2009) 69-102. | MR | Zbl

[7] V. Burenkov, P.D. Lamberti and M. Lanza De Cristoforis, Spectral stability of nonnegative selfadjoint operators. Sovrem. Mat. Fundam. Napravl. 15 (2006) 76-111, in Russian. English transl. in J. Math. Sci. (NY) 149 (2008) 1417-1452. | MR

[8] G. Buttazzo and G. Dal Maso, An existence result for a class of shape optimization problems. Arch. Rational Mech. Anal. 122 (1993) 183-195. | MR | Zbl

[9] R. Dalmasso, Un problème de symétrie pour une équation biharmonique. (French). A problem of symmetry for a biharmonic equation. In vol. 11 of Ann. Fac. Sci. Toulouse Math. (1990) 45-53. | EuDML | Numdam | MR | Zbl

[10] D. Daners, Domain perturbation for linear and semi-linear boundary value problems. Handbook of differential equations: stationary partial differential equations. Handb. Differ. Equ. VI. Elsevier/North-Holland, Amsterdam (2008) 1-81. | MR

[11] F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic boundary value problems. Positivity preserving and nonlinear higher order elliptic equations in bounded domains. Lect. Notes Math. Springer Verlag, Berlin (2010). | MR

[12] P. Grinfeld, Hadamard's formula inside and out. J. Optim. Theory Appl. 146 (2010) 654-690. | MR

[13] J.K. Hale, Eigenvalues and perturbed domains. Ten mathematical essays on approximation in analysis and topology. Elsevier B.V., Amsterdam (2005) 95-123. | MR

[14] A. Henrot, Extremum problems for eigenvalues of elliptic operators. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2006). | MR

[15] D. Henry, Perturbation of the boundary in boundary-value problems of partial differential equations. With editorial assistance from Jack Hale and Antonio Luiz Pereira. In vol. 318 of London Math. Soc. Lect. Note Ser. Cambridge University Press, Cambridge (2005). | MR

[16] D. Henry, Topics in Nonlinear Analysis, in vol. 192 of Trabalho de Mathemãtica. Universidade de Brasilia, Departamento de Matemãtica-IE (1982).

[17] B. Kawohl, Some nonconvex shape optimization problems. Optimal shape design (Tróia, 1998). In vol. 746 of Lect. Notes Math. Springer, Berlin (2000) 49-02. | MR

[18] S. Kesavan, Symmetrization and applications. In vol. 3 of Ser. Anal. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2006). | MR

[19] P.D. Lamberti and M. Lanza De Cristoforis, Critical points of the symmetric functions of the eigenvalues of the Laplace operator and overdetermined problems. J. Math. Soc. Japan 58 (2006) 231-245. | MR

[20] P.D. Lamberti and M. Lanza De Cristoforis, A real analyticity result for symmetric functions of the eigenvalues of a domain dependent Dirichlet problem for the Laplace operator. J. Nonlinear Convex Anal. 5 (2004) 19-42. | MR

[21] P.D. Lamberti and M. Lanza De Cristoforis, An analyticity result for the dependence of multiple eigenvalues and eigenspaces of the Laplace operator upon perturbation of the domain. Glasg. Math. J. 44 (2002) 29-43. | MR

[22] E. Mohr, Über die Rayleighsche Vermutung: unter allen Platten von gegebener Fläche und konstanter Dichte und Elastizität hat die kreisförmige den tiefsten Grundton. Ann. Mat. Pura Appl. 104 (1975) 85-122. | MR

[23] N.S. Nadirashvili, Rayleigh's conjecture on the principal frequency of the clamped plate. Arch. Rational Mech. Anal. 129 (1995) 1-10. | MR

[24] J.H. Ortega and E. Zuazua, Generic simplicity of the spectrum and stabilization for a plate equation. SIAM J. Control Optim. 39 (2001) 1585-1614. | MR

[25] E. Oudet, Numerical minimization of eigenmodes of a membrane with respect to the domain. ESAIM: COCV 10 (2004) 315-330. | Numdam | MR

[26] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics. In vol. 27 of Ann. Math. Studies. Princeton University Press, Princeton, NJ (1951).

[27] G. Szegö, On membranes and plates. Proc. Nat. Acad. Sci. USA 36 (1950) 210-216. | MR

[28] N.B. Willms, An isoperimetric inequality for the buckling of a clamped plate, Lecture at the Oberwolfach meeting on “Qualitiative properties of PDE” organized, edited by H. Berestycki, B. Kawohl and G. Talenti (1995).

[29] S.A. Wolf and J.B. Keller, Range of the first two eigenvalues of the Laplacian. In vol. 447 of Proc. Roy. Soc. London Ser. A (1994) 397-412. | MR

Cité par Sources :