Analytical and numerical results for the Fučı́k spectrum of the Laplacian

摘要

Nontrivial solutions of Δu+μu+−νu−=0 in Ω⊂Rn with zero Dirichlet condition on ∂Ω are studied. The collection of the pairs (μ,ν) is the so called Fučı́k spectrum σF. A new variational formulation for parts of σF is presented and analyzed. Based on this formulation a minimization algorithm for the computation of a part of σF is developed. Alternatively, an approach using an implicit function argument is discussed analytically and, via Newton's method, also numerically. By combining the variational minimization method with Newton's method a new bifurcation result is first observed numerically and then proved rigorously. By replacing the variational minimization method by the Mountain Pass Algorithm higher curves in σF are found numerically. Several numerical results are discussed including a further example of the previously recorded phenomenon of crossing of Fučı́k curves originating from different eigenvalues.