ﻻ يوجد ملخص باللغة العربية
We study the Hamilton-Jacobi equations $H(x,Du,u)=0$ in $M$ and $partial u/partial t +H(x,D_xu,u)=0$ in $Mtimes(0,infty)$, where the Hamiltonian $H=H(x,p,u)$ depends Lipschitz continuously on the variable $u$. In the framework of the semicontinuous viscosity solutions due to Barron-Jensen, we establish the comparison principle, existence theorem, and representation formula as value functions for extended real-valued, lower semicontinuous solutions for the Cauchy problem. We also establish some results on the long-time behavior of solutions for the Cauchy problem and classification of solutions for the stationary problem.
Let $(Omega, mu)$ be a probability space endowed with an ergodic action, $tau$ of $( {mathbb R} ^n, +)$. Let $H(x,p; omega)=H_omega(x,p)$ be a smooth Hamiltonian on $T^* {mathbb R} ^n$ parametrized by $omegain Omega$ and such that $ H(a+x,p;tau_aomeg
We prove that if a sequence of pairs of smooth commuting Hamiltonians converge in the $C^0$ topology to a pair of smooth Hamiltonians, these commute. This allows us define the notion of commuting continuous Hamiltonians. As an application we extend s
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 $phi(x)in C(M,mathbb{R})$. Under some
We consider the specified stochastic homogenization of first order evolutive Hamilton-Jacobi equations on a very simple junction, i.e the real line with a junction at the origin. Far from the origin, we assume that the considered hamiltonian is close
We study the singular locus of solutions to Hamilton-Jacobi equations with a Hamiltonian independent of $u$. In a previous paper, we proved that the singular locus is what we call a balanced split locus. In this paper, we find and classify all balanc