A computer-assisted method for excluding eigenvalues of an elliptic operator linearized at a solution of a nonlinear problem

We propose a computer-assisted method for excluding eigenvalues of an elliptic operator linearized at a solution of a nonlinear problem. The method works in both the one-dimensional and the two-dimensional case. We begin by finding an approximate solution to a nonlinear problem, and we then enclose the solution by using Nakao’s numerical verification method. Instead of considering directly the eigenvalues for the elliptic operator linearized at the verified solution, we linearize the operator at the approximate solution. We present a theorem that allows us to determine under which conditions and in which disks there will be no eigenvalues. Thus, if any of those disks are contained in the enclosed area, we can exclude those eigenvalues. Next, we construct various computable criteria that allow us to use a computer program to find these disks. Finally, we use our results to determine which eigenvalues to exclude for the operator linearized at the verified solution. We present some verified results.