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

Abstract

Variational principles are proved for self-adjoint operator functions arising from variational evolution equations of the form $$\langle\ddot{z}(t),y \rangle + \mathfrak{d}[\dot{z} (t), y] + \mathfrak{a}_0 [z(t),y] = 0.$$ 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 $$\mathfrak{t}(\lambda)[x,y] := \lambda^2\langle x,y\rangle + \lambda\mathfrak{d}[x,y] + \mathfrak{a}_0[x,y],$$ 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.Comment: to appear in Operators and Matrice