Variational equations on mixed Riemannian–Lorentzian metrics

Variational Equations on Mixed Riemannian-lorentzian Metrics

A class of elliptic-hyperbolic equations is placed in the context of a geometric variational theory, in which the change of type is viewed as a change in the character of an underlying metric. A fundamental example of a metric which changes in this way is the extended projective disc, which is Riemannian at ordinary points, Lorentzian at ideal points, and singular on the absolute. Harmonic fiel...

On Asymptotic Variational Wave Equations

We investigate the equation (ut + (f(u))x)x = f (u)(ux)/2 where f(u) is a given smooth function. Typically f(u) = u/2 or u/3. This equation models unidirectional and weakly nonlinear waves for the variational wave equation utt − c(u)(c(u)ux)x = 0 which models some liquid crystals with a natural sinusoidal c. The equation itself is also the Euler-Lagrange equation of a variational problem. Two n...

On Exponential Dichotomy of Variational Difference Equations

On Exponential Stability of Variational Difference Equations

We prove that a general system of variational difference equations is uniformly exponentially stable if and only if certain associated sets are of the second category. We also deduce necessary and sufficient conditions for uniform exponential stability of systems with uniformly bounded coefficients. We apply our results for the study of exponential stability of linear skew-product flows, genera...

Variational equations with constraints

In this paper we deal with a constrained variational equation associated with the usual weak formulation of an elliptic boundary value problem in the context of Banach spaces, which generalizes the classical results of existence and uniqueness. Furthermore, we give a precise estimation of the norm of the solution.