A variational description of dissipative processes

The Roussopoulos variational formalism is applied to equations with a first-order time derivative. The resulting functional is a very general variational characterization of dissipative processes in that it admits trial functions which do not satisfy the equation of interest, do not satisfy the boundary conditions, and are not continuous. It is shown that the previous functionals presented in the literature for this type of problem are special cases of this more general functional. The use of this variational principle to estimate a rather general class of characteristics of interest is discussed. It is also pointed out that this general functional is not unique and arguments are given to deal with this non-uniqueness. The variational description of the Sturm-Liouville equation is considered in this same generality. This leads to a generalization (to a more complete class of admissible trial functions) of the classical Rayleigh quotient for estimating eigenvalues.