On the vanishing contact structure for viscosity solutions of contact type Hamilton-Jacobi equations I: Cauchy problem

Abstract

We study the representation formulae for the fundamental solutions and viscosity solutions of the Hamilton-Jacobi equations of contact type. We also obtain a vanishing contact structure result for relevant Cauchy problems which can be regarded as an extension to the vanishing discount problem.

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

“…-We can derive the generalized Euler-Lagrange equations in a modern and rigorous way which does not appear in both [41,42]; -There should be an extension of the main results of this paper under much more general conditions (like Osgood type conditions) to guarantee the existence and uniqueness of the solutions of the associated Carathéodory equation (9). -Along this line, the quatitative semiconcavity and convexity estimate of the associated fundamental solutions have been obtained in [7] recently, which is useful for the intrinsic study of the global propagation of singularities of the viscosity solutions of (5) and (6) ( [8,4,5,6]); -When the Lagrangian has the form L(x, v) − λ u, by solving the associated Carathéodory equation (9) directly, one gets the representation formula for the associated viscosity solutions immediately ( [19,40,43,13]). The representation formula bridges the PDE aspects of the problem with the dynamical ones; -Consider a family of Lagrangians in the form {L(x, v) + ∑ k i=1 a i j u i }, a problem of Herglotz' variational principle in the vector form is closely connected to certain stochastic model of weakly coupled Hamilton-Jacobi equations (see, for instance, [20,23,38]).…”

“…-The solutions of the equations (1) determine a family of contact transformations, see [30,11,21,28]; -The generalized variational principle gives a variational description of energynonconservative processes even when F in (1) is independent of t. -If F has the form F = −λ u + L(x, v), then the relevant problems are closely connected to the Hamilton-Jacobi equations with discount factors (see, for instance, [19,18,9,34,35,37,29,36]). As an extension to nonlinear discounted problems, various examples are discussed in [14,43]. -Even for a energy-nonconservative process which can be described with the generalized variational principle, one can systematically derive conserved quantities as Noether's theorems such as [26,27]; -The generalized variational principle provides a link between the mathematical structure of control and optimal control theories and contact transformation (see [25]); -There are some interesting connections between contact transformations and equilibrium thermodynamics (see, for instance, [39]).…”

Vanishing contact structure problem and convergence of the viscosity solutions

“…In recent works [34,36,35], certain implicit variational principle is applied to give representation formula for the viscosity solutions or weak KAM solutions of (HJ λ ). An alternative approach following Herglotz' generalized variational principle is also obtained from the Lagrangian formalism [5], which is later used to obtain a vanishing contact structure result on relavent Cauchy problem for evolutionary equations [39]. But, these results are established under C 2 Tonelli conditions, so they are no longer applicable to our settings in this paper.…”

Lasry–Lions approximations for discounted Hamilton–Jacobi equations

“…To answer Problem above, a possible systematic approach will be based on the recent works [21] and [22]. Moreover, one can understand such a problem as follows ( [11] and [23]): We suppose L is a function of C 2 class and it satisfies the following conditions: (L3) There exists K > 0 such that where the infinmum is taken over of ξ ∈ Γ t x,y and u ξ : [0, t] → R n is a absolutely continuous curve determined by (3.6). In the case of discounted equations, L(x, u, v) = −λu + L(x, v).…”

“…In this paper, we extend the Lasry-Lions regularization procedure to the viscosity solution of the discounted Hamilton-Jacobi equation λu λ (x) + H(x, Du λ (x)) = 0, x ∈ R n with a discount factor λ > 0. The associated dynamical system is dissipative system and it is a very special kind of contact type Hamiltonian systems (see, for instance, [21], [22], [11] and [23]). In fact, by defining a new Hamiltonian H λ (t, x, p) = e λt H(x, e −λt p), this equation can be reduced to a time-dependent evolutionary Hamilton-Jacobi equation…”