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

Variational principles are proved for self-adjoint operator functions arising from variational evolution equations of the form $$⟨\ddot{z}(t),y⟩+\mathfrak{d}[\dot{z}(t),y]+{\mathfrak{a}}_0[z(t),y]=0.$$ Here ${\mathfrak{a}}_0$ and $\mathfrak{d}$ 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 $\mathcal{A}$, the forms $$\mathfrak{t}(\lambda )[x,y]:={\lambda }^2⟨x,y⟩+\lambda \mathfrak{d}[x,y]+{\mathfrak{a}}_0[x,y],$$ where $\lambda \in \mathbb{C}$ and $x,y$ are in the domain of the form ${\mathfrak{a}}_0$, and a corresponding operator family $T(\lambda )$. Using form methods we define a generalized Rayleigh functional and characterize the eigenvalues above the essential spectrum of $\mathcal{A}$ by a min-max and a max-min variational principle. The obtained results are illustrated with a damped beam equation.