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

摘要: The eigenvalue problem for the p-Laplace operator with p>1 on planar domains zero Dirichlet boundary condition is considered. Constrained Descent Method and Mountain Pass Algorithm are used in Sobolev space setting to numerically investigate dependence of two smallest eigenvalues p. Computations conducted values p between 1.1 10. Symmetry properties second eigenfunction also examined numerically. While disk an odd symmetry about nodal line dividing halves maintained all considered p, rectangles triangles changes as varies. Based numerical evidence change this case occurs at a certain value p_0 which depends domain.