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 <(z) over dot(t), y > vertical bar partial derivative[(z) over dot(t), y] vertical bar a(0)[z(t), y] = 0. Here a(0) and partial derivative 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(lambda)[x, y] := lambda(2)< x, y > +lambda partial derivative[x, y] + a(0) [x, y], where lambda is an element of C and x, y are in the domain of the form 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 A by a min-max and a max-min variational principle. The obtained results are illustrated with a damped beam equation.