Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T21:33:13.289Z Has data issue: false hasContentIssue false
  • English
  • Français

Eigenvalues of polyharmonic operators on variabledomains

Published online by Cambridge University Press:  06 September 2013

Davide Buoso  and
Pier Domenico Lamberti
Davide Buoso
Affiliation:
Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste, 63, 35126 Padova, Italy. [email protected]; [email protected]
Pier Domenico Lamberti
Affiliation:
Dipartimento di Matematica, Università degli Studi di Padova, Via Trieste, 63, 35126 Padova, Italy. [email protected]; [email protected]
Get access

Abstract

We consider a class of eigenvalue problems for polyharmonic operators, includingDirichlet and buckling-type eigenvalue problems. We prove an analyticity result for thedependence of the symmetric functions of the eigenvalues upon domain perturbations andcompute Hadamard-type formulas for the Frechét differentials. We also considerisovolumetric domain perturbations and characterize the corresponding critical domains forthe symmetric functions of the eigenvalues. Finally, we prove that balls are criticaldomains.

Keywords

Polyharmonic operatorseigenvaluesdomain perturbation
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 4 , October 2013 , pp. 1225 - 1235
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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).
Ashbaugh, M.S. and Benguria, R.D., On Rayleigh’s conjecture for the clamped plate and its generalization to three dimensions. Duke Math. J. 78 (1995) 1–17. Google Scholar
Ashbaugh, M.S. and Bucur, D., On the isoperimetric inequality for the buckling of a clamped plate. Z. Angew. Math. Phys. 54 (2003) 756–770. Google Scholar
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).
Burenkov, V. and Lamberti, P.D., Sharp spectral stability estimates via the Lebesgue measure of domains for higher order elliptic operators. Rev. Mat. Comput. 25 (2012) 435–457. Google Scholar
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.
Burenkov, V., Lamberti, P.D. and Lanza de Cristoforis, M., 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. Google Scholar
Buttazzo, G. and Dal Maso, G., An existence result for a class of shape optimization problems. Arch. Rational Mech. Anal. 122 (1993) 183–195. Google Scholar
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.
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.
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).
Grinfeld, P., Hadamard’s formula inside and out. J. Optim. Theory Appl. 146 (2010) 654–690. Google Scholar
J.K. Hale, Eigenvalues and perturbed domains. Ten mathematical essays on approximation in analysis and topology. Elsevier B.V., Amsterdam (2005) 95–123.
A. Henrot, Extremum problems for eigenvalues of elliptic operators. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2006).
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).
D. Henry, Topics in Nonlinear Analysis, in vol. 192 of Trabalho de Mathemãtica. Universidade de Brasilia, Departamento de Matemãtica-IE (1982).
B. Kawohl, Some nonconvex shape optimization problems. Optimal shape design (Tróia, 1998). In vol. 746 of Lect. Notes Math. Springer, Berlin (2000) 49–02.
S. Kesavan, Symmetrization and applications. In vol. 3 of Ser. Anal. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2006).
Lamberti, P.D. and Lanza de Cristoforis, M., Critical points of the symmetric functions of the eigenvalues of the Laplace operator and overdetermined problems. J. Math. Soc. Japan 58 (2006) 231–245. Google Scholar
Lamberti, P.D. and Lanza de Cristoforis, M., 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. Google Scholar
Lamberti, P.D. and Lanza de Cristoforis, M., 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. Google Scholar
Mohr, E., 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. Google Scholar
Nadirashvili, N.S., Rayleigh’s conjecture on the principal frequency of the clamped plate. Arch. Rational Mech. Anal. 129 (1995) 1–10. Google Scholar
Ortega, J.H. and Zuazua, E., Generic simplicity of the spectrum and stabilization for a plate equation. SIAM J. Control Optim. 39 (2001) 1585–1614. Google Scholar
Oudet, E., Numerical minimization of eigenmodes of a membrane with respect to the domain. ESAIM: COCV 10 (2004) 315–330. Google Scholar
G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics. In vol. 27 of Ann. Math. Studies. Princeton University Press, Princeton, NJ (1951).
Szegö, G., On membranes and plates. Proc. Nat. Acad. Sci. USA 36 (1950) 210–216. Google ScholarPubMed
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).
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.