Lasry-Lions, Lax-Oleinik and generalized characteristics

Abstract

Abstract. In the recent works [9] and [12], an intrinsic approach of the propagation of singularities along the generalized characteristics was obtained, even in global case, by a procedure of sup-convolution with the kernel the fundamental solutions of the associated Hamilton-Jacobi equations.In the present paper, we exploit the relations among Lasry-Lions regularization, Lax-Oleinik operators (or inf/sup-convolution) and generalized characteristics, which are discussed in the context of the variational setting of Tonelli Hamiltonian dynamics, such as Mather theory and weak KAM theory.

Herglotz’ Generalized Variational Principle and Contact Type Hamilton-Jacobi Equations

“…If E = ∅, then a = 0 and the proof is the same. This leads to (11) and (12). ⊓ ⊔ In view to Lemma 1, we conclude that inf ξ ∈A J(ξ ) is bounded below.…”

Lasry–Lions approximations for discounted Hamilton–Jacobi equations

“…H(x, Du(x)) = 0, when 0 is Mañé's critical value. Comparing to the results in [14], there exists a unique q x ∈ D + u(x) such that…”

“…-Ilmanen's lemma on insertion of C 1,1 functions between a semiconvex function less than a semiconcave function ( [3] and [17]). -In [14], the authors also obtained the limiting behavior of the derivatives of the approximating sequence and the relation between the regular and singular dynamics of the associated Hamiltonian dynamical systems. -There also exists a connection to the theory of generalized characteristics by the recent work on global propagation of singularities of the viscosity solutions of Hamilton-Jacobi equations ( [9]), and [6,7,10] for more about the singularities propagation of weak KAM solutions.…”

Singularities of solutions of Hamilton-Jacobi equations

“…In a weak KAM context, this method was also widely used as an interaction of the positive-negative Lax-Oleinik operators ([Ber07, Ber10,Ber12,FZ10]). The relation between Lasry-Lions regularization and generalized characteristics was also studied in [CC16,CCZ18]. This method was applied to minimal homoclinic orbits with respect to the Aubry set ([CC15]).…”