Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 132, pp. 1-30.
Title: Numerical investigation of the smallest eigenvalues of the
p-Laplace operator on planar domains
Author: Jiri Horak (Univ. zu Koln, Germany)
Abstract:
The eigenvalue problem for the p-Laplace operator with p>1 on
planar domains with zero Dirichlet boundary condition is
considered. The Constrained Descent Method and the Constrained
Mountain Pass Algorithm are used in the Sobolev space setting to
numerically investigate the dependence of the two smallest
eigenvalues on p. Computations are conducted for values of p
between 1.1 and 10. Symmetry properties of the second eigenfunction
are also examined numerically. While for the disk an odd symmetry
about the nodal line dividing the disk in halves is maintained for
all the considered values of p, for rectangles and triangles
symmetry changes as p varies. Based on the numerical evidence the
change of symmetry in this case occurs at a certain value p_0
which depends on the domain.
Submitted September 22, 2011. Published October 13, 2011.
Math Subject Classifications: 35J92, 49M30, 49R05.
Key Words: p-Laplace operator; eigenvalue; mountain pass algorithm; symmetry.