Maximization of the first eigenvalue in problems involving the bi-Laplacian. (English) Zbl 1238.35066
Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 12, e-Suppl., e800-e809 (2009).
Summary: This paper concerns maximization of the first eigenvalue in problems involving the bi-Laplacian under either Navier boundary conditions or Dirichlet boundary conditions. Physically, in the case of \(N=2\), our equation models the vibration of a nonhomogeneous plate \(\Omega \) which is either hinged or clamped along the boundary. Given several materials (with different densities) of total extension \(|\Omega |\), we investigate the location of these materials throughout \(\Omega \) so as to maximize the first eigenvalue in the vibration of the corresponding plate.
MSC:
| 35P05 | General topics in linear spectral theory for PDEs |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
| 74K20 | Plates |
| 74H45 | Vibrations in dynamical problems in solid mechanics |
References:
| [1] | Struwe, M., Variational Methods (1990), Springer-Verlag: Springer-Verlag Berlin, New York · Zbl 0746.49010 |
| [2] | Agmon, S.; Douglis, A.; Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Commun. Pure Appl. Math., 12, 623-727 (1959) · Zbl 0093.10401 |
| [3] | C. Anedda, F. Cuccu, G. Porru, Minimization of the first eigenvalue in problems involving the bi-Laplacian, Rev. Mat. Teoria Aplic. (in press); C. Anedda, F. Cuccu, G. Porru, Minimization of the first eigenvalue in problems involving the bi-Laplacian, Rev. Mat. Teoria Aplic. (in press) · Zbl 1338.35311 |
| [4] | Cox, S. J.; McLaughlin, J. R., Extremal eigenvalue problems for composite membranes, I, II, Appl. Math. Optim., 22, 153-167 (1990), 169-187 · Zbl 0709.73044 |
| [5] | Cuccu, F.; Porru, G., Optimization in eigenvalue problems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10, 51-58 (2003) · Zbl 1044.35036 |
| [6] | F. Cuccu, B. Emamizadeh, G. Porru, Optimization of the first eigenvalue in problems involving the \(p\)-Laplacian, Proc. Amer. Math. Soc. (in press); F. Cuccu, B. Emamizadeh, G. Porru, Optimization of the first eigenvalue in problems involving the \(p\)-Laplacian, Proc. Amer. Math. Soc. (in press) · Zbl 1163.35025 |
| [7] | Cuccu, F.; Emamizadeh, B.; Porru, G., Nonlinear elastic membranes involving the p-Laplacian operator, Electron. J. Differential Equations, 49, 10 (2006) · Zbl 1128.35333 |
| [8] | Cuccu, F.; Emamizadeh, B.; Porru, G., Optimization problems for an elastic plate, J. Math. Phys., 47, 8, 082901 (2006), 12 pp · Zbl 1112.74046 |
| [9] | Burton, G. R., Variational problems on classes of rearrangements and multiple configurations for steady vortices, Ann. Inst. Henri Poincaré, 6, 4, 295-319 (1989) · Zbl 0677.49005 |
| [10] | Burton, G. R.; McLeod, J. B., Maximisation and minimisation on classes of rearrangements, Proc. Roy. Soc. Edinburgh Sect. A, 119, 3-4, 287-300 (1991) · Zbl 0736.49006 |
| [11] | Auchmuty, G., Dual principles for eigenvalue problems, (Browder, F., Nonlinear Functional Analysis and its Applications (1986), A.M.S: A.M.S Providence, RI), 55-71 · Zbl 0594.49028 |
| [12] | B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, in: Lectures Notes in Mathematics, vol. 1150, Berlin, 1985; B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, in: Lectures Notes in Mathematics, vol. 1150, Berlin, 1985 · Zbl 0593.35002 |
| [13] | Grunau, H. C.; Sweers, G., Sign change for the Green function and for the first eigenfunction of equations of clamped-plate type, Arch. Ration. Mech. Anal., 150, 179-190 (1999) · Zbl 0973.74044 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
