Variational principles for self-adjoint operator functions arising from second order systems

Jacob, Birgit and Langer, Matthias and Trunk, Carsten (2016) Variational principles for self-adjoint operator functions arising from second order systems. Operators and Matrices, 10 (3). pp. 501-531. ISSN 1846-3886

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    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.