Some Asymptotic Problems on the Theory of Viscosity Solutions of Hamilton-Jacobi Equations

Download or read book Some Asymptotic Problems on the Theory of Viscosity Solutions of Hamilton-Jacobi Equations written by Son Nguyen Thai Tu and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Viscosity solutions arise naturally in many fields of study from engineering, physics, and operations research to economics. The study of viscosity solutions on its own has uncovered many new and interesting research problems, including the study of the asymptotic behavior of solutions with respect to the changing of parameters. In this dissertation, I present some new problems following the line of the asymptotic behavior of solutions. Each of the problems is related to the other through the old underlying theme of optimal control theory, yet presents many new problems on their own that are yet to be studied.The first direction is on homogenization of Hamilton-Jacobi equations. Using deep analysis of the dynamics of minimizers corresponding to the solution, I established in [113] the optimal rate of convergence under the multi-scale setting in one dimension, which could not be obtained by the previous pure PDEs technique. The second direction concerns various asymptotic problems for equations with state-constraint. In [75], my co-authors and I established some first quantitative results on the rate of convergence of the solution to the Hamilton-Jacobi equations with state-constraint on a nested domain setting. Utilizing the weak KAM theory, in [114], I established qualitatively various convergence results for the vanishing discount procedure with changing domains together with a new description of the regularity of the additive eigenvalues with respect to domain perturbation. Lastly, in [61], my co-author and I established the rate of convergence for the vanishing viscosity procedure, concerning the viscous state-constraint viscosity (large) solution that blows on the boundary of the underlying domain. This is the first-rate established for blow-up solutions in the literature as far as we know.