B. Jacob, M. Langer and C. Trunk,
Variational principles for self-adjoint operator functions arising from second order systems,
Oper. Matrices 10 (2016), 501–531
Abstract:
Variational principles are proved for self-adjoint operator functions arising from
variational evolution equations of the form
⟨z''(t),y⟩ + 𝔡[z'(t), y]
+ 𝔞0[z(t), y] = 0.
Here 𝔞0 and 𝔡 are densely defined, symmetric and positive
sesquilinear forms on a Hilbert space H.
We associate with the variational evolution equation an equivalent Cauchy problem
corresponding to a block operator matrix A, the forms
𝔱(λ)[x, y]
:= λ2⟨x, y⟩
+ λ𝔡[x, y]
+ 𝔞0[x, y],
where λ ∈ ℂ and x, y are in the domain of the
form 𝔞0, and a corresponding operator family T(λ).
Using form methods we define a generalized Rayleigh functional and characterize
the eigenvalues above the essential spectrum of A by a min-max
and a max-min variational principle.
The obtained results are illustrated with a damped beam equation.