One dimensional systems with singular perturbations

Singular Perturbations of Finite Dimensional Gradient Flows

In this paper we give a description of the asymptotic behavior, as ε → 0, of the ε-gradient flow in the finite dimensional case. Under very general assumptions we prove that it converges to an evolution obtained by connecting some smooth branches of solutions to the equilibrium equation (slow dynamics) through some heteroclinic solutions of the gradient flow (fast dynamics).

Perturbations of One - Dimensional Schr

Singular Perturbations in Noisy Dynamical Systems

Consider a deterministic dynamical system in a domain containing a stable equilibrium, e.g., a particle in a potential well. The particle, independent of initial conditions, eventually reaches the bottom of the well. If however, the particle is subjected to white noise, due, e.g., to collisions with a population of smaller, lighter particles comprising the medium through which the particle trav...

Dispersion Estimates for One-dimensional Schrödinger Equations with Singular Potentials

We derive a dispersion estimate for one-dimensional perturbed radial Schrödinger operators. We also derive several new estimates for solutions of the underlying differential equation and investigate the behavior of the Jost function near the edge of the continuous spectrum.

Nonconvex Variational Problems with General Singular Perturbations

We study the effect of a general singular perturbation on a nonconvex variational problem with infinitely many solutions. Using a scaling argument and the theory of T-convergence of nonlinear functionals, we show that if the solutions of the perturbed problem converge in L1 as the perturbation parameter goes to zero, then the limit function satisfies a classical minimal surface problem. Introdu...