On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations Associated with Nonlinear Boundary Conditions

In this article, we study the large time behavior of solutions of first-order Hamilton-Jacobi Equations, set in a bounded domain with nonlinear Neumann boundary conditions, including the case of dynamical boundary conditions. We establish general convergence results for viscosity solutions of these Cauchy-Neumann problems by using two fairly different methods : the first one relies only on partial differential equations methods, which provides results even when the Hamiltonians are not convex, and the second one is an optimal control/dynamical system approach, named the "weak KAM approach" which requires the convexity of Hamiltonians and gives formulas for asymptotic solutions based on Aubry-Mather sets

Time-Dependent Variational Approach to the Non-Abelian Pure Gauge Theory

The time-dependent variational approach to the pure Yang-Mills gauge theory, especially a color su(3) gauge theory, is formulated in the functional Schroedinger picture with a Gaussian wave functional approximation. The equations of motion for the quantum gauge fields are formulated in the Liouville-von Neumann form. This variational approach is applied in order to derive the transport coefficients, such as the shear viscosity, for the pure gluonic matter by using the linear response theory. As a result, the contribution to the shear viscosity from the quantum gluons is zero up to the lowest order of the coupling g in the quantum gluonic matter.Comment: 19 pages, no figures, using PTPTeX.cl

user's guide to viscosity solutions of second order partial differential equations

The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions.Comment: 67 page

Optimal control of the propagation of a graph in inhomogeneous media

We study an optimal control problem for viscosity solutions of a Hamilton–Jacobi equation describing the propagation of a one-dimensional graph with the control being the speed function. The existence of an optimal control is proved together with an approximate controllability result in the $H^{-1}$-norm. We prove convergence of a discrete optimal control problem based on a monotone finite difference scheme and describe some numerical results

An approximation scheme for an Eikonal Equation with discontinuous coefficient

We consider the stationary Hamilton-Jacobi equation where the dynamics can vanish at some points, the cost function is strictly positive and is allowed to be discontinuous. More precisely, we consider special class of discontinuities for which the notion of viscosity solution is well-suited. We propose a semi-Lagrangian scheme for the numerical approximation of the viscosity solution in the sense of Ishii and we study its properties. We also prove an a-priori error estimate for the scheme in an integral norm. The last section contains some applications to control and image processing problems

Baryon form factors in a diquark-quark bound state description

Nucleon form factors are calculated in a relativistic diquark--quark picture based on the Nambu--Jona-Lasinio model. The nucleon wave function is obtained in a static approximation to the quark exchange interaction between the valence quark and the diquark. We evaluate the valence quark and $0^+$--diquark contribution to the nucleon electromagnetic and weak currents. We find reasonable electric charge radii, magnetic moments as in the additive diquark model, and $g_A \approx 1$. We discuss the dependence on the model parameters.Comment: 10 pages, latex, 1 postscript figure included (uuencoded

Nuclear Force from Monte Carlo Simulations of Lattice Quantum Chromodynamics

The nuclear force acting between protons and neutrons is studied in the Monte Carlo simulations of the fundamental theory of the strong interaction, the quantum chromodynamics defined on the hypercubic space-time lattice. After a brief summary of the empirical nucleon-nucleon (NN) potentials which can fit the NN scattering experiments in high precision, we outline the basic formulation to derive the potential between the extended objects such as the nucleons composed of quarks. The equal-time Bethe-Salpeter amplitude is a key ingredient for defining the NN potential on the lattice. We show the results of the numerical simulations on a $32^4$ lattice with the lattice spacing $a \simeq 0.137$fm (lattice volume (4.4 fm)$^4$) in the quenched approximation. The calculation was carried out using the massively parallel computer Blue Gene/L at KEK. We found that the calculated NN potential at low energy has basic features expected from the empirical NN potentials; attraction at long and medium distances and the repulsive core at short distance. Various future directions along this line of research are also summarized.Comment: 13 pages, 4 figures, version accepted for publication in "Computational Science & Discovery" (IOP

Electron Depletion Due to Bias of a T-Shaped Field-Effect Transistor

A T-shaped field-effect transistor, made out of a pair of two-dimensional electron gases, is modeled and studied. A simple numerical model is developed to study the electron distribution vs. applied gate voltage for different gate lengths. The model is then improved to account for depletion and the width of the two-dimensional electron gases. The results are then compared to the experimental ones and to some approximate analytical calculations and are found to be in good agreement with them.Comment: 16 pages, LaTex (RevTex), 8 fig

Convergence of nonlocal threshold dynamics approximations to front propagation

In this note we prove that appropriately scaled threshold dynamics-type algorithms corresponding to the fractional Laplacian of order $\alpha \in (0,2)$ converge to moving fronts. When $\alpha \geqq 1$ the resulting interface moves by weighted mean curvature, while for $\alpha <1$ the normal velocity is nonlocal of ``fractional-type.'' The results easily extend to general nonlocal anisotropic threshold dynamics schemes.Comment: 19 page

High-energy magnetic excitations in overdoped La2−x_{2-x}Srx_{x}CuO4_{4} studied by neutron and resonant inelastic X-ray scattering

We have performed neutron inelastic scattering and resonant inelastic X-ray scattering (RIXS) at the Cu-$L_3$ edge to study high-energy magnetic excitations at energy transfers of more than 100 meV for overdoped La$_{2-x}$Sr$_{x}$CuO$_{4}$ with $x=0.25$ ($T_c=15$ K) and $x=0.30$ (non-superconducting) using identical single crystal samples for the two techniques. From constant-energy slices of neutron scattering cross-sections, we have identified magnetic excitations up to ~250 meV for $x=0.25$. Although the width in the momentum direction is large, the peak positions along the (pi, pi) direction agree with the dispersion relation of the spin-wave in the non-doped La$_{2}$CuO$_{4}$ (LCO), which is consistent with the previous RIXS results of cuprate superconductors. Using RIXS at the Cu-$L_3$ edge, we have measured the dispersion relations of the so-called paramagnon mode along both (pi, pi) and (pi, 0) directions. Although in both directions the neutron and RIXS data connect with each other and the paramagnon along (pi, 0) agrees well with the LCO spin-wave dispersion, the paramagnon in the (pi, pi) direction probed by RIXS appears to be less dispersive and the excitation energy is lower than the spin-wave of LCO near (pi/2, pi/2). Thus, our results indicate consistency between neutron inelastic scattering and RIXS, and elucidate the entire magnetic excitation in the (pi, pi) direction by the complementary use of two probes. The polarization dependence of the RIXS profiles indicates that appreciable charge excitations exist in the same energy range of magnetic excitations, reflecting the itinerant character of the overdoped sample. A possible anisotropy in the charge excitation intensity might explain the apparent differences in the paramagnon dispersion in the (pi, pi) direction as detected by the X-ray scattering.Comment: 7 pages, 7 figure