Approximation of the second eigenvalue of the [formula omitted]-Laplace operator in symmetric domains

Abstract

A novel approach to approximate the second eigenfunction and the second eigenvalue of the p-Laplace operator is suggested for some symmetric domains. In the case of the Dirichlet boundary condition, the algorithm has the restriction that the positive and the negative part of the second eigenfunction have equal Lp-norm, however, in the case of the Neumann boundary condition, the algorithm does not have such restriction so we implement it to perform bi-clustering. Some interesting estimates for the iterative method are obtained. At the end of the paper, we present various examples and computational tests, which support that the suggested iterative algorithm is well-defined and effective.