Variational problems with singular perturbation

In this paper, we construct the local minimum of a certain variational problem which we take in the form $\mathrm{inf}\int_\Omega\left\{\frac{\epsilon}{2}kg^2|\nabla w|^2+\frac{1}{4\epsilon}f^2g^4(1-w^2)^2\right\}\,\mathrm{d}x$, where $\epsilon$ is a small positive parameter and $\Omega\subset\mathbb{R}^n$ is a convex bounded domain with smooth boundary. Here $f,g,k\in C^3(\Omega)$ are strictly positive functions in the closure of the domain $\bar{\Omega}$. If we take the inf over all functions $H^1(\Omega)$, we obtain the (unique) positive solution of the partial differential equation with Neumann boundary conditions (respectively Dirichlet boundary conditions). We wish to restrict the inf to the local (not global) minimum so that we consider solutions of this Neumann problem which take both signs in $\Omega$ and which vanish on $(n-1)$ dimensional hypersurfaces $\Gamma_\epsilon\subset\Omega$. By using a $\Gamma$-convergence method, we find the structure of the limit solutions as $\epsilon\to0$ in terms of the weighted geodesics of the domain $\Omega$

Orbital Evolution of Scattered Planets

A simple dynamical model is employed to study the possible orbital evolution of scattered planets and phase plane analysis is used to classify the parameter space and solutions. Our results reconfirm that there is always an increase in eccentricity when the planet was scattered to migrate outward when the initial eccentricity is zero. Applying our study on the Solar System and considering the existence of the Kuiper Belt, this conclusion implies that Neptune was dynamically coupled with the Kuiper Belt in the early phase of the Solar System, which is consistent with the simulational model in Thommes, Duncan & Levison (1999).Comment: AAS Latex file, 21 pages, accepted by Ap