Vanishing contact structure problem and convergence of the viscosity solutions

Abstract

This paper is devoted to study the vanishing contact structure problem which is a generalization of the vanishing discount problem. Let H λ (x, p, u) be a family of Hamiltonians of contact type with parameter λ > 0 and converges to G(x, p). For the contact type Hamilton-Jacobi equation with respect to H λ , we prove that, under mild assumptions, the associated viscosity solution u λ converges to a specific viscosity solution u 0 of the vanished contact equation. As applications, we give some convergence results for the nonlinear vanishing discount problem.

Aubry–Mather Theory for Contact Hamiltonian Systems

“…The vanishing discount problem for discounted Hamilton-Jacobi equations has been widely studied by using dynamical and PDE approaches, see [19,23,24,[26][27][28]37,38]. Recently, some new progresses have been made on the vanishing contact structure problem (see [11,12,46,47]) and the vanishing discount problem from the negative direction [20]. Consider the HJ equation 7) it was shown in [20] that if H is a C 3 Tonelli Hamiltonian satisfying H(x, 0) ≤ c(H), then the minimal viscosity solution converges uniformly, as λ → 0 − , to the same viscosity solution of H(x, Du) = c(H) like the convergence as λ → 0 + established in [19].…”

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

“…-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]).…”

The vanishing discount problem for Hamilton–Jacobi equations in the Euclidean space

“…The function P 0 in R 2N given by P 0 (x, y) = min z∈A [S 0 (x, z) + S 0 (z, y)] for x, y ∈ R N is called the Peierls barrier. See [17,Proposition 3.7.2] and [7,15,19,20]. Proof.…”