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Bases and comparison results for linear elliptic eigenproblems. (English) Zbl 1238.35069

Summary: 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 \(H^{1}(\varOmega )\) where \(\varOmega \) is a nice bounded region in \(\mathbb R^n, N\geqslant 2\). 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.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
49R05 Variational methods for eigenvalues of operators
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
15A63 Quadratic and bilinear forms, inner products
47A75 Eigenvalue problems for linear operators
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