2023/01/25 by Qinbo Chen, Albert Fathi, Chen, Qinbo +5 · 4 citations
Medicine · Physics and Astronomy · Mathematics · #Mathematical and Theoretical Epidemiology and Ecology Models #Quantum chaos and dynamical systems #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2301.10478
Given a continuous Hamiltonian H : (x,p,u) ↦ H(x,p,u) defined on T^*M × \mathbb R , where M is a closed connected manifold, we study viscosity solutions, uλ: M→ \mathbb R, of discounted equations: H(x, dx uλ, λuλ(x))=c in M, where λ>0 is called a discount factor and c is the critical value of H(⋅, ⋅ , 0). When H is convex and superlinear in p and non--decreasing in u, under an additional non--degeneracy condition, we obtain existence and uniqueness (with comparison principles) results of solutions and we prove that the family of solutions (uλ)λ>0 converges to a specific solution u0 of H(x, dx u0, 0)=c in M. Our degeneracy condition requires H to be increasing (in u) on localized regions linked to the support of Mather measures, whereas usual similar results are obtained for Hamiltonians that are everywhere increasing in u.