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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.
The large time behavior of solutions to Cauchy problem for viscous Hamilton-Jacobi equation is classified. The large time asymptotics are given by very singular self-similar solutions on one hand and by self-similar viscosity solutions on the other hand
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
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
The main purpose of this paper is to study the global propagation of singularities of viscosity solution to discounted Hamilton-Jacobi equation begin{equation}label{eq:discount 1}tag{HJ$_lambda$} lambda v(x)+H( x, Dv(x) )=0 , quad xin mathbb{R}^n.
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