OPTIMIZATION OF THE FIRST EIGENVALUE IN PROBLEMS THE p -LAPLACIAN

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 Ω which is fixed along the boundary. Given several materials (with different densities) of total extension |Ω|, we investigate the location of these material inside Ω so as to minimize or maximize the first mode in the vibration of the membrane.

[1]  J.-L. Lions,et al.  Simplicit'e et isolation de la premi`ere valeur propre du p-Laplacien avec poids , 1987 .

[2]  Wacław Pielichowski.  The optimization of eigenvalue problems involving the -Laplacian. , 2004 .

[3]  G. R. Burton,et al.  Variational problems on classes of rearrangements and multiple configurations for steady vortices , 1989 .

[4]  P. Tolksdorf,et al.  Regularity for a more general class of quasilinear elliptic equations , 1984 .

[5]  Steven J. Cox,et al.  Extremal eigenvalue problems for composite membranes, I , 1990 .

[6]  G. R. Burton,et al.  Rearrangements of functions, maximization of convex functionals, and vortex rings , 1987 .

[7]  P. Rosenbloom,et al.  Rearrangements of functions , 1986 .

[8]  GEOMETRIC PROPERTIES OF SOLUTIONS TO MAXIMIZATION PROBLEMS , 2003 .

[9]  G. Burton,et al.  Maximisation and minimisation on classes of rearrangements , 1991, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[10]  D. Grieser,et al.  Symmetry Breaking and Other Phenomena in the Optimization of Eigenvalues for Composite Membranes , 1999, math/9912116.

[11]  William P. Ziemer,et al.  Minimal rearrangements of Sobolev functions. , 1987 .

[12]  Dual variational principles for eigenvalue problems , 1986 .

[13]  Marcello Lucia,et al.  Simplicity of the principal eigenvalue for indefinite quasilinear problems , 2007, Advances in Differential Equations.

[14]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[15]  Steven J. Cox,et al.  Extremal eigenvalue problems for composite membranes, II , 1990 .

[16]  Antoine Henrot,et al.  Extremum Problems for Eigenvalues of Elliptic Operators , 2006 .

[17]  B. Emamizadeh,et al.  Variational problem for vortices attached to seamounts , 2003 .

[18]  Juan Luis Vázquez,et al.  A Strong Maximum Principle for some quasilinear elliptic equations , 1984 .