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

TL;DR

This paper develops variational principles for self-adjoint operator functions from second-order evolution equations, providing new min-max characterizations of eigenvalues and illustrating with a damped beam example.

Contribution

It introduces a novel variational framework for self-adjoint operator functions derived from second-order systems, including generalized Rayleigh functionals and eigenvalue characterizations.

Findings

Established variational principles for eigenvalues above the essential spectrum.

Defined a generalized Rayleigh functional for the operator family.

Applied the theory to a damped beam equation example.

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.