The eigenvalue problem for the p-Laplace operator with p>1 on planar domains
with the 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