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Register a new userWeak KAM theory for general Hamilton-Jacobi equations III: the variational principle under Osgood conditions
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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 assumptions on the convexity of $H(x,u,p)$ with respect to $p$ and the Osgood growth of $H(x,u,p)$ with respect to $u$, 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.
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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_aomega)=H(x,p;omega)$. We consider for an initial condition $fin C^0 ( {mathbb R}^n)$, the family of variational solutions of the stochastic Hamilton-Jacobi equations $$left{ begin{aligned} frac{partial u^{ varepsilon }}{partial t}(t,x;omega)+Hleft (frac{x}{ varepsilon } , frac{partial u^varepsilon }{partial x}(t,x;omega);omega right )=0 & u^varepsilon (0,x;omega)=f(x)& end{aligned} right .$$ Under some coercivity assumptions on $p$ -- but without any convexity assumption -- we prove that for a.e. $omega in Omega$ we have $C^0-lim u^{varepsilon}(t,x;omega)=v(t,x)$ where $v$ is the variational solution of the homogenized equation $$left{ begin{aligned} frac{partial v}{partial t}(x)+{overline H}left (frac{partial v }{partial x}(x) right )=0 & v (0,x)=f(x)& end{aligned} right.$$
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.
For a convex, coercive continuous Hamiltonian on a compact closed Riemannian manifold $M$, we construct a unique forward weak KAM solution of [ H(x, d_x u)=c(H) ] by a vanishing discount approach, where $c(H)$ is the Ma~ne critical value. We also discuss the dynamical significance of such a special solution.
If $U:[0,+infty[times M$ is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation $$partial_tU+ H(x,partial_xU)=0,$$ where $M$ is a not necessarily compact manifold, and $H$ is a Tonelli Hamiltonian, we prove the set $Sigma(U)$, of points where $U$ is not differentiable, is locally contractible. Moreover, we study the homotopy type of $Sigma(U)$. We also give an application to the singularities of a distance function to a closed subset of a complete Riemannian manifold.
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 balanced split sets, identifying the cases where the only balanced split locus is the singular locus, and the cases where this doesnt hold. This clarifies the relationship between viscosity solutions and the more classical approach of characteristics and shocks.
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