16,350 research outputs found
Weak KAM theory for general Hamilton-Jacobi equations II: the fundamental solution under Lipschitz conditions
We consider the following evolutionary Hamilton-Jacobi equation with initial
condition: \begin{equation*} \begin{cases}
\partial_tu(x,t)+H(x,u(x,t),\partial_xu(x,t))=0,\\ u(x,0)=\phi(x), \end{cases}
\end{equation*} where . Under some assumptions on
the convexity of with respect to and the uniform Lipschitz of
with respect to , we establish a variational principle and
provide an intrinsic relation between viscosity solutions and certain minimal
characteristics. By introducing an implicitly defined {\it fundamental
solution}, we obtain a variational representation formula of the viscosity
solution of the evolutionary Hamilton-Jacobi equation. Moreover, we discuss the
large time behavior of the viscosity solution of the evolutionary
Hamilton-Jacobi equation and provide a dynamical representation formula of the
viscosity solution of the stationary Hamilton-Jacobi equation with strictly
increasing with respect to
A Dynamical Approach to Viscosity Solutions of Hamilton-Jacobi Equations
In this paper, we consider the following Hamilton-Jacobi equation with
initial condition: \begin{equation*} \begin{cases}
\partial_tu(x,t)+H(x,t,u(x,t),\partial_xu(x,t))=0, u(x,0)=\phi(x). \end{cases}
\end{equation*} Under some assumptions on the convexity of w.r.t.
, we develop a dynamical approach to viscosity solutions and show that there
exists an intrinsic connection between viscosity solutions and certain minimal
characteristics.Comment: This paper has been withdrawn by the author due to a crucial error in
Lemma 3.
Weak KAM theory for general Hamilton-Jacobi equations III: the variational principle under Osgood conditions
We consider the following evolutionary Hamilton-Jacobi equation with initial
condition: \begin{equation*} \begin{cases}
\partial_tu(x,t)+H(x,u(x,t),\partial_xu(x,t))=0,\\ u(x,0)=\phi(x), \end{cases}
\end{equation*} where . Under some assumptions on
the convexity of with respect to and the Osgood growth of
with respect to , we establish an implicitly variational
principle and provide an intrinsic relation between viscosity solutions and
certain minimal characteristics. Moreover, we obtain a representation formula
of the viscosity solution of the evolutionary Hamilton-Jacobi equation
Variational principle for contact Tonelli Hamiltonian systems
We establish an implicit variational principle for the equations of the
contact flow generated by the Hamiltonian with respect to the
contact 1-form under Tonelli and Osgood growth assumptions. It
is the first step to generalize Mather's global variational method from the
Hamiltonian dynamics to the contact Hamiltonian dynamics.Comment: arXiv admin note: text overlap with arXiv:1408.379
Aubry-Mather and weak KAM theories for contact Hamiltonian systems. Part 1: Strictly increasing case
This paper is concerned with the study of Aubry-Mather and weak KAM theories
for contact Hamiltonian systems with Hamiltonians defined on
, satisfying Tonelli conditions with respect to and
, where
is a connected, closed and smooth manifold. First, we show the uniqueness
of the backward weak KAM solutions of the corresponding Hamilton-Jacobi
equation. Using the unique backward weak KAM solution , we prove the
existence of the maximal forward weak KAM solution . Next, we analyse
Aubry set for the contact Hamiltonian system showing that it is the
intersection of two Legendrian pseudographs and , and that
the projection induces a bi-Lipschitz
homeomorphism from Aubry set
onto the projected Aubry set . At last, we introduce the notion of
barrier functions and study their interesting properties along calibrated
curves. Our analysis is based on a recent method by [43,44].Comment: 34 page
Large N_c Expansion in Chiral Quark Model of Mesons
We study SU(3)_L\timesSU(3)_R chiral quark model of mesons up to the next
to leading order of expansion. Composite vector and axial-vector mesons
resonances are introduced via non-linear realization of chiral SU(3) and vector
meson dominant. Effects of one-loop graphs of pseudoscalar, vector and
axial-vector mesons is calculated systematically and the significant results
are obtained. We also investigate correction of quark-gluon coupling and
relationship between chiral quark model and QCD sum rules. Up to powers four of
derivatives, chiral effective lagrangian of mesons is derived and evaluated to
the next to leading order of . Low energy limit of the model is
examined. Ten low energy coupling constants in ChPT are
obtained and agree with ChPT well.Comment: 49 pages, latex file, 6 eps figure
Noncommutative QED and Muon Anomalous Magnetic Moment
The muon anomalous value, , is calculated up to one-loop
level in noncommutative QED. We argue that relativistic muon in E821 experiment
nearly always stays at the lowest Landau level. So that spatial coordinates of
muon do not commute each other. Using parameters of E821 experiment, KG
and muon energy 3.09GeV/c, we obtain the noncommutativity correction to
is about , which significantly makes standard model
prediction close to experiment.Comment: revtex, 6 page, 5 figure
Weak KAM theory for general Hamilton-Jacobi equations I: the solution semigroup under proper conditions
We consider the following evolutionary Hamilton-Jacobi equation with initial
condition: \begin{equation*} \begin{cases}
\partial_tu(x,t)+H(x,u(x,t),\partial_xu(x,t))=0,\\ u(x,0)=\phi(x). \end{cases}
\end{equation*} Under some assumptions on with respect to and
, we provide a variational principle on the evolutionary Hamilton-Jacobi
equation. By introducing an implicitly defined solution semigroup, we extend
Fathi's weak KAM theory to certain more general cases, in which explicitly
depends on the unknown function . As an application, we show the viscosity
solution of the evolutionary Hamilton-Jacobi equation with initial condition
tends asymptotically to the weak KAM solution of the following stationary
Hamilton-Jacobi equation: \begin{equation*} H(x,u(x),\partial_xu(x))=0.
\end{equation*}.Comment: This is a revised version of arXiv:1312.160
Rigorous Effective Field Theory Study on Pion Form Factor
We study cross section and phase shift of
scattering below 1GeV in framework of chiral constituent quark model.
The results including all order contribution of the chiral perturbation
expansion and all one-loop effects of pseudoscalar mesons, but without any
adjust parameters. Width of predicted by the model strongly depends on
transition momentum-square . We show that the mass pamameter of
-meson in its propagator is very different from its physical mass due to
momentum-dependent width of . The mass difference between and
are predicted successfully. The rigorous theoretical prediction on
cross section and the phase shift in
scattering agree with data excellentlly.Comment: revtex file, 10 pages, 4 eps figure
A representation formula of viscosity solutions to weakly coupled systems of Hamilton-Jacobi equations with applications to regularizing effect
Based on a fixed point argument, we give a {\it dynamical representation} of
the viscosity solution to Cauchy problem of certain weakly coupled systems of
Hamilton-Jacobi equations with continuous initial datum. Using this formula, we
obtain some regularity results related to the viscosity solution, including a
partial extension of Lions' regularizing effect \cite{L} to the case of weakly
coupled systems
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