The vanishing discount problem and viscosity Mather measures. Part 1: The problem on a torus

Abstract

Abstract. We develop a variational approach to the vanishing discount problem for fully nonlinear, degenerate elliptic, partial differential equations. Under mild assumptions, we introduce viscosity Mather measures for such partial differential equations, which are natural extensions of the Mather measures. Using the viscosity Mather measures, we prove that the whole family of solutions v λ of the discount problem with the factor λ > 0 converges to a solution of the ergodic problem as λ → 0.

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

“…Our procedure is close in spirit to Evans interpretation of Mather theory in terms of complementarity problems, see [8], [9], and also [11]. We think that this alternative approach is interesting per se and can handle to extend the asymptotic result to more general setting, for instance in the case of fully nonlinear second order equations (see [13] for such generalizations).…”

“…The idea of performing some duality between generalized Lagrangians and measures, in order to study the asymptotic of solution to discounted equations, has been introduced in [13]. The authors however use as duality tool the Sion minimax Theorem, while we instead employ a separation result for convex subsets in locally convex Hausdorff space, see Appendix B.…”

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].…”

On uniqueness sets of additive eigenvalue problems and applications

“…To be more precise, [10, Lemma 6.12] deals with the special case A(x) = a(x)I n where a ∈ C 2 (T n , [0, ∞)) and I n is the identity matrix of size n. For general diffusion matrix A satisfying (H3), we perform first inf-sup convolutions, and then normal convolution of a solution w of (VE). See also [9] for a form of (4.2) in fully nonlinear, degenerate elliptic PDE settings.…”