Bases and comparison results for linear elliptic eigenproblems

This paper describes some results about the construction and comparison of sequences of eigenvalues and eigenvectors of a pair (a,m) of continuous symmetric bilinear forms on a real Hilbert space V. The results are used to describe the properties of some non-standard self-adjoint linear elliptic eigenproblems on H1(Ω) where Ω is a nice bounded region in Rn, N⩾2. These include eigenproblems with Robin type boundary conditions, Steklov eigenproblems and problems where the eigenvalue appears in both the equation and the boundary conditions. Different variational principles for the eigenvalues and eigenvectors are introduced and convex analysis is used. Both minimax and maximin characterizations of higher eigenvalues are described. Various orthogonal decompositions are described and criteria for the eigenfunctions to be orthogonal bases of specific subspaces are found. Comparison results for the eigenvalues of different pairs of bilinear forms are proved. Finally these results are used to obtain spectral formulae for weak solutions of parametrized linear systems.