Assume that A is a bounded self-adjoint operator in a Hilbert space H. Then, the variational principle max v |(Au, v)|2 (Av, v) = (Au, u) (*) holds if and only if A ≥ 0, that is, if (Av, v) ≥ 0 for all v ∈ H. We define the quotient on the left-hand side in (*) to be zero if (Av, v) = 0. As an application of this principle a variational principle for the electrical capacitance of a conductor of ...

We show that the minimal speed for the existence of monotonic fronts of the equation ut = (u )xx + f(u) with f(0) = f(1) = 0, m > 1 and f > 0 in (0, 1), derives from a variational principle. The variational principle allows to calculate, in principle, the exact speed for general f . The case m = 1 when f (0) = 0 is included as an extension of the results.

We consider the localized entropy of a point w ∈ R which is computed by considering only those (n, ε)-separated sets whose statistical sums with respect to an m-dimensional potential Φ are ”close” to a given value w. Previously, a local version of the variational principle was established for systems on non-Besicovitch compact metric spaces. We extend this result to all compact metric spaces.

We reexamine the problem of having nonconservative equations of motion arise from the use of a variational principle. In particular, a formalism is developed that allows the inclusion of fractional derivatives. This is done within the Lagrangian framework by treating the action as a Volterra series. It is then possible to derive two equations of motion, one of these is an advanced equation and ...

nonequilibrium statistical mechanics we focus on nonequilibrium corrections Δs to entropy and energy of the fluid in terms of the nonequilibrium density distribution function, f. We also evaluate coefficients of wave model of heat such as: relaxation time, propagation speed and thermal inertia. With these data a quadratic Lagrangian and a variational principle of Hamilton's type follows for a f...