Vanishing contact structure problem and convergence of the viscosity solutions. Issue 9 (2nd September 2019)

• Record Type: Journal Article
• Authors: Chen, Qinbo; Cheng, Wei; Ishii, Hitoshi; Zhao, Kai
• Is Part Of: Communications in partial differential equations. Volume 44:Issue 9(2019)
• Page Start: 801
• Page End: 836
• Publication Date: 2019-09-02
• Subjects: Hamilton-Jacobi equation -- convergence -- vanishing discount -- vanishing contact structure -- Weak KAM theory -- Aubry-Mather theory; 35F21 -- 49L25 -- 37J50; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2019.1608561
• Languages: English
• ISSNs: 0360-5302

Abstract

Abstract: This article is devoted to studying 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 converge 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.