Electron. J. Diff. Equ., Vol. 2011 (2011), No. 132, pp. 1-30.

Numerical investigation of the smallest eigenvalues of the p-Laplace operator on planar domains

Jiri Horak

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 p0 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.

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Jirí Horák
Universität zu Köln, Mathematisches Institut
50923 Köln, Germany
email: jhorak@math.uni-koeln.de

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