A new method for large time behavior of degenerate viscous Hamilton--Jacobi equations with convex Hamiltonians

We investigate large-time asymptotics for viscous Hamilton--Jacobi equations with possibly degenerate diffusion terms. We establish new results on the convergence, which are the first general ones concerning equations which are neither uniformly parabolic nor first order. Our method is based on the nonlinear adjoint method and the derivation of new estimates on long time averaging effects. It also extends to the case of weakly coupled systems

Mott transition in cuprate high-temperature superconductors

In this study, we investigate the metal-insulator transition of charge transfer type in high-temperature cuprates. We first show that we must introduce a new band parameter in the three-band d-p model to reproduce the Fermi surface of high temperature cuprates such as BSCCO, YBCO and Hg1201. We present a new wave function of a Mott insulator based on the improved Gutzwiller function, and show that there is a transition from a metal to a charge-transfer insulator for such parameters by using the variational Monte Carlo method. This transition occurs when the level difference $\Delta_{dp}\equiv \epsilon_p-\epsilon_d$ between d and p orbitals reaches a critical value $(\Delta_{dp})_c$. The energy gain $\Delta E$, measured from the limit of large $\Delta_{dp}$, is proportional to $1/\Delta_{dp}$ for $\Delta_{dp}>(\Delta_{dp})_c$: $\Delta E\propto -t_{dp}^2/\Delta_{dp}$. We obtain $(\Delta_{dp})_c\simeq 2t_{dp}$ using the realistic band parameters

A PDE approach to large-time asymptotics for boundary-value problems for nonconvex Hamilton-Jacobi Equations

We investigate the large-time behavior of three types of initial-boundary value problems for Hamilton-Jacobi Equations with nonconvex Hamiltonians. We consider the Neumann or oblique boundary condition, the state constraint boundary condition and Dirichlet boundary condition. We establish general convergence results for viscosity solutions to asymptotic solutions as time goes to infinity via an approach based on PDE techniques. These results are obtained not only under general conditions on the Hamiltonians but also under weak conditions on the domain and the oblique direction of reflection in the Neumann case