## Optimization of the first eigenvalue in problems involving the $p$-Laplacian

HTML articles powered by AMS MathViewer

- by Fabrizio Cuccu, Behrouz Emamizadeh and Giovanni Porru PDF
- Proc. Amer. Math. Soc.
**137**(2009), 1677-1687 Request permission

## Abstract:

This paper concerns minimization and maximization of the first eigenvalue in problems involving the $p$-Laplacian, under homogeneous Dirichlet boundary conditions. Physically, in the case of $N=2$ and $p$ close to $2$, our equation models the vibration of a nonhomogeneous membrane $\Omega$ which is fixed along the boundary. Given several materials (with different densities) of total extension $|\Omega |$, we investigate the location of these material inside $\Omega$ so as to minimize or maximize the first mode in the vibration of the membrane.## References

- Aomar Anane,
*Simplicité et isolation de la première valeur propre du $p$-laplacien avec poids*, C. R. Acad. Sci. Paris Sér. I Math.**305**(1987), no. 16, 725–728 (French, with English summary). MR**920052** - Giles Auchmuty,
*Dual variational principles for eigenvalue problems*, Nonlinear functional analysis and its applications, Part 1 (Berkeley, Calif., 1983) Proc. Sympos. Pure Math., vol. 45, Amer. Math. Soc., Providence, RI, 1986, pp. 55–71. MR**843549** - John E. Brothers and William P. Ziemer,
*Minimal rearrangements of Sobolev functions*, J. Reine Angew. Math.**384**(1988), 153–179. MR**929981** - G. R. Burton,
*Rearrangements of functions, maximization of convex functionals, and vortex rings*, Math. Ann.**276**(1987), no. 2, 225–253. MR**870963**, DOI 10.1007/BF01450739 - G. R. Burton,
*Variational problems on classes of rearrangements and multiple configurations for steady vortices*, Ann. Inst. H. Poincaré Anal. Non Linéaire**6**(1989), no. 4, 295–319 (English, with French summary). MR**998605**, DOI 10.1016/S0294-1449(16)30320-1 - G. R. Burton and J. B. McLeod,
*Maximisation and minimisation on classes of rearrangements*, Proc. Roy. Soc. Edinburgh Sect. A**119**(1991), no. 3-4, 287–300. MR**1135975**, DOI 10.1017/S0308210500014840 - S. Chanillo, D. Grieser, M. Imai, K. Kurata, and I. Ohnishi,
*Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes*, Comm. Math. Phys.**214**(2000), no. 2, 315–337. MR**1796024**, DOI 10.1007/PL00005534 - Fabrizio Cuccu, Kanhaiya Jha, and Giovanni Porru,
*Geometric properties of solutions to maximization problems*, Electron. J. Differential Equations (2003), No. 71, 8. MR**1993779** - Steven J. Cox and Joyce R. McLaughlin,
*Extremal eigenvalue problems for composite membranes. I, II*, Appl. Math. Optim.**22**(1990), no. 2, 153–167, 169–187. MR**1055658**, DOI 10.1007/BF01447325 - F. Cuccu and G. Porcu,
*Existence of solutions in two optimization problems*, C. R. Acad. Bulgare Sci.**54**(2001), no. 9, 33–38. MR**1879204** - J. García-Melián and J. Sabina de Lis,
*Maximum and comparison principles for operators involving the $p$-Laplacian*, J. Math. Anal. Appl.**218**(1998), no. 1, 49–65. MR**1601841**, DOI 10.1006/jmaa.1997.5732 - David Gilbarg and Neil S. Trudinger,
*Elliptic partial differential equations of second order*, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR**737190**, DOI 10.1007/978-3-642-61798-0 - Antoine Henrot,
*Extremum problems for eigenvalues of elliptic operators*, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006. MR**2251558**, DOI 10.1007/3-7643-7706-2 - Bernd Kawohl, Marcello Lucia, and S. Prashanth,
*Simplicity of the principal eigenvalue for indefinite quasilinear problems*, Adv. Differential Equations**12**(2007), no. 4, 407–434. MR**2305874** - Jonas Nycander and Behrouz Emamizadeh,
*Variational problem for vortices attached to seamounts*, Nonlinear Anal.**55**(2003), no. 1-2, 15–24. MR**2001628**, DOI 10.1016/S0362-546X(03)00207-4 - Wacław Pielichowski,
*The optimization of eigenvalue problems involving the $p$-Laplacian*, Univ. Iagel. Acta Math.**42**(2004), 109–122. MR**2157626** - Michael Struwe,
*Variational methods*, Springer-Verlag, Berlin, 1990. Applications to nonlinear partial differential equations and Hamiltonian systems. MR**1078018**, DOI 10.1007/978-3-662-02624-3 - Peter Tolksdorf,
*Regularity for a more general class of quasilinear elliptic equations*, J. Differential Equations**51**(1984), no. 1, 126–150. MR**727034**, DOI 10.1016/0022-0396(84)90105-0 - J. L. Vázquez,
*A strong maximum principle for some quasilinear elliptic equations*, Appl. Math. Optim.**12**(1984), no. 3, 191–202. MR**768629**, DOI 10.1007/BF01449041

## Additional Information

**Fabrizio Cuccu**- Affiliation: Dipartimento di Matematica e Informatica, Universitá di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy
- MR Author ID: 689288
- Email: fcuccu@unica.it
**Behrouz Emamizadeh**- Affiliation: Dipartimento di Matematica e Informatica, Universitá di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy
- Email: porru@unica.it
**Giovanni Porru**- Affiliation: Department of Mathematics, The Petroleum Institute, P. O. Box 2533, Abu Dhabi, United Arab Emirates
- Email: bemamizadeh@pi.ac.ae
- Received by editor(s): June 28, 2007
- Published electronically: December 11, 2008
- Communicated by: Chuu-Lian Terng
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**137**(2009), 1677-1687 - MSC (2000): Primary 35P15, 47A75
- DOI: https://doi.org/10.1090/S0002-9939-08-09769-4
- MathSciNet review: 2470826