An enclosure method of eigenvalues for the elliptic operator linearized at an exact solution of nonlinear problems

We consider eigenvalue enclosing of the elliptic operator which is linearized at an exact solution of certain nonlinear elliptic equation. This problem is important in the mathematically rigorous analysis of the stability or bifurcation of some solutions for nonlinear problems. We formulate such a kind of eigenvalue problem as the nonlinear system which contains both linearized eigenvalue problem and the original nonlinear equation. We also consider the indices of eigenvalues, especially the first eigenvalue of such a problem. In these enclosing procedures, the finite-dimensional verified computations for linear and nonlinear system of equations play an essential role. A numerical example is presented.