2023/05/31 by Anup Biswas, Biswas, Anup, Erwin Topp +1
Mathematics · #35J60 #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2305.19527
openalex publication_date 2023/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the existence-uniqueness of solution (u, λ) to the ergodic Hamilton-Jacobi equation (-Δ)s u + H(x, ∇ u) = f-λ in ℝd, and u≥ 0, where s∈ ((1)/(2), 1). We show that the critical λ=λ^*, defined as the infimum of all λ attaining a non-negative supersolution, attains a nonnegative solution u. Under suitable conditions, it is also shown that λ^* is the supremum of all λ for which a non-positive subsolution is possible. Moreover, uniqueness of the solution u, corresponding to λ^*, is also established. Furthermore, we provide a probabilistic characterization that determines the uniqueness of the pair (u, λ^*) in the class of all solution pair (u, λ) with u≥ 0. Our proof technique involves both analytic and probabilistic methods in combination with a new local Lipschitz estimate obtained in this article.