We study the asymptotic behavior of the solutions to a family of discounted Hamilton-Jacobi equations, posed in RN, when the discount factor goes to zero. The ambient space being noncompact, we introduce an assumption implying that the Aubry set is compact and there is no degeneracy at infinity. Our approach is to deal not with a single Hamiltonian and Lagrangian but with the whole space of generalized Lagrangians, and then to define via duality minimizing measures associated with both the corresponding ergodic and discounted equations. The asymptotic result follows from the convergence properties of these measures concerning the narrow topology. We use as duality tool a separation theorem in locally convex Hausdorff spaces, and we use the strict topology in the space of the bounded generalized Lagrangians as well.

The vanishing discount problem for Hamilton–Jacobi equations in the Euclidean space / Ishii, H.; Siconolfi, A.. - In: COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0360-5302. - 45:6(2020), pp. 525-560. [10.1080/03605302.2019.1710845]

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

Siconolfi A.
2020

Abstract

We study the asymptotic behavior of the solutions to a family of discounted Hamilton-Jacobi equations, posed in RN, when the discount factor goes to zero. The ambient space being noncompact, we introduce an assumption implying that the Aubry set is compact and there is no degeneracy at infinity. Our approach is to deal not with a single Hamiltonian and Lagrangian but with the whole space of generalized Lagrangians, and then to define via duality minimizing measures associated with both the corresponding ergodic and discounted equations. The asymptotic result follows from the convergence properties of these measures concerning the narrow topology. We use as duality tool a separation theorem in locally convex Hausdorff spaces, and we use the strict topology in the space of the bounded generalized Lagrangians as well.
2020
35B40; 35F21; 37J99; 49L25; ergodic problem; Hamilton–Jacobi equations; vanishing discount problem
01 Pubblicazione su rivista::01a Articolo in rivista
The vanishing discount problem for Hamilton–Jacobi equations in the Euclidean space / Ishii, H.; Siconolfi, A.. - In: COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0360-5302. - 45:6(2020), pp. 525-560. [10.1080/03605302.2019.1710845]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1471830
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