Abstract
The singular set of a viscosity solution to a Hamilton–Jacobi equation is known to propagate, from any noncritical singular point, along singular characteristics which are curves satisfying certain differential inclusions. In the literature, different notions of singular characteristics were introduced. However, a general uniqueness criterion for singular characteristics, not restricted to mechanical systems or problems in one space dimension, is missing at the moment. In this paper, we prove that, for a Tonelli Hamiltonian on mathbb {R}^2, two different notions of singular characteristics coincide up to a bi-Lipschitz reparameterization. As a significant consequence, we obtain a uniqueness result for the class of singular characteristics that was introduced by Khanin and Sobolevski in the paper [On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations. Arch. Ration. Mech. Anal., 219(2):861–885, 2016].
Highlights
This paper is devoted to study the local propagation of singularities for viscosity solutions of the Hamilton–Jacobi equations
Cheng where H is a Tonelli Hamiltonian in (HJs) and H is of class C1 and strictly convex in the p-variable in (HJloc)
The local structure of singular characteristics was further investigated by the first author and Yu in [11], where singular characteristics were proved more regular near the starting point than the arcs constructed in [1]
Summary
The local structure of singular (generalized) characteristics was further investigated by the first author and Yu in [11], where singular characteristics were proved more regular near the starting point than the arcs constructed in [1] Such additional properties will be crucial for the analysis we develop in this paper. We proceed to recall their definition: given a semiconcave solution u of (HJloc), a Lipschitz singular curve x : [0, T ] → is called a strict singular characteristic from x ∈ Sing (u) if there exists a measurable selection p(t) ∈ D+u(x(t)) such that x(t) = Hp(x(t), p(t)) a.e. t ∈ [0, T ],. We give a detailed proof of the existence result for strict singular characteristics
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