Article
Mathematics, Applied
Cui Chen, Ya-Nan Wang, Jun Yan
Summary: In this paper, we investigate the viscosity solutions of the non-autonomous Hamilton-Jacobi equation on a compact Riemannian manifold, showing convergence to a time-periodic viscosity solution.
SCIENCE CHINA-MATHEMATICS
(2021)
Article
Mathematics, Applied
Yves Achdou, Claude Le Bris
Summary: We study the homogenization problem for a certain class of stationary Hamilton-Jacobi equations. By perturbing a periodic Hamiltonian near the origin, we prove that the limiting problem consists of a Hamilton-Jacobi equation outside the origin, with the same effective Hamiltonian as in periodic homogenization, and a Dirichlet condition at the origin to account for the perturbation. Various comments and extensions are provided.
COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics
William M. Feldman, Jean-Baptiste Fermanian, Bruno Ziliotto
Summary: The article provides an example of the failure of homogenization for a viscous Hamilton-Jacobi equation with a non-convex Hamiltonian.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics
Atilla Yilmaz
Summary: In this study, homogenization of a class of viscous Hamilton-Jacobi equations in the stationary and ergodic setting in one space dimension is proven. The approach involves showing the existence of a unique sublinear corrector with certain properties for directions outside of a bounded interval. The effective Hamiltonian is determined to be coercive, equal to beta within a specific interval, and strictly monotone elsewhere.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
William Cooperman
Summary: We studied a problem concerning the Hamilton-Jacobian equation, where the Hamiltonian is periodic, coercive, and convex. By combining the representation formula from optimal control theory and a theorem by Alexander, we obtained a homogenized rate that is close to optimal and holds in all dimensions.
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2022)
Article
Mathematics, Applied
Benjamin Seeger
Summary: This study focuses on the homogenization of nonlinear, first-order equations with highly oscillatory mixing spatio-temporal dependence. The results show that the homogenized equations are stochastic Hamilton-Jacobi equations with deterministic, spatially homogenous Hamiltonians driven by white noise in time. Additionally, the paper proves some general regularity and path stability results for stochastic Hamilton-Jacobi equations, which are essential for proving homogenization results and are of independent interest.
COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics
Kaizhi Wang, Lin Wang, Jun Yan
Summary: The paper provides necessary and sufficient conditions for the existence of viscosity solutions of nonlinear first order PDEs, proving compactness of the set of solutions. Furthermore, it explores the long-term behavior of viscosity solutions for Cauchy problems using weak KAM theory and dynamic methods.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Statistics & Probability
William Cooperman
Summary: We prove a quantitative rate of homogenization for the G equation in a random environment with finite range of dependence, using ideas from percolation theory. The proof bootstraps a result of Cardaliaguet-Souganidis, who proved qualitative homogenization in a more general ergodic environment.
PROBABILITY THEORY AND RELATED FIELDS
(2023)
Article
Mathematical & Computational Biology
N. El Khatib, N. Forcadel, M. Zaydan
Summary: This paper establishes a rigorous connection between microscopic and macroscopic pedestrian models at a convergent junction. By injecting the microscopic model into a non-local PDE, the study demonstrates the local uniform convergence of the viscosity solution of the non-local PDE towards the solution of the macroscopic model.
MATHEMATICAL MODELLING OF NATURAL PHENOMENA
(2022)
Article
Mathematics, Applied
Kaizhi Wang, Jun Yan, Kai Zhao
Summary: This article discusses the existence and multiplicity of nontrivial time periodic viscosity solutions to a contact Hamilton-Jacobi equation, and investigates the long time behavior of these solutions. It is found that for a certain class of initial data, the corresponding viscosity solutions converge to asymptotic time periodic viscosity solutions. The article also analyzes a bifurcation phenomenon for a parameter-dependent Hamilton-Jacobi equation.
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
(2023)
Article
Mathematics, Applied
Jiahui Hong, Wei Cheng, Shengqing Hu, Kai Zhao
Summary: In this paper, various representation formulas for the viscosity solutions of the contact type Hamilton-Jacobi equations are discussed by utilizing Herglotz' variational principle.
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Ioana Ciotir, Rim Fayad, Nicolas Forcadel, Antoine Tonnoir
Summary: This work introduces a non-local Hamilton-Jacobi model for traffic flow and proves the existence and uniqueness of its solution, as well as a numerical scheme with estimated error. It also presents some numerical illustrations.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Andrea Davini, Lin Wang
Summary: This paper discusses the convergence behavior of the solutions of a critical equation as lambda approaches 0 in different directions. It focuses on the asymptotics of the solutions and presents an example where the equation may have a unique solution under certain conditions.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2021)
Article
Mathematics, Applied
Carlos Esteve-Yague, Enrique Zuazua
Summary: We prove the differentiability of the viscosity solution to a Hamilton-Jacobi equation with a smooth convex Hamiltonian, and explicitly compute the directional Gateaux derivatives almost everywhere in R-N using the optimality system of the associated optimal control problem. Furthermore, we show that these directional Gateaux derivatives correspond to the unique duality solution to the linear transport equation with discontinuous coefficient in the one-dimensional case in space and in the quadratic case in any space dimension. These results are motivated by an optimal inverse-design problem and allow for the derivation of necessary first-order optimality conditions and the implementation of gradient-based methods for numerical approximation.
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2022)
Article
Mathematics, Applied
Cui Chen, Jiahui Hong, Kai Zhao
Summary: This paper investigates the global propagation of singularities of the viscosity solution to the discounted Hamilton-Jacobi equation. By reducing the original equation to a time-dependent evolutionary Hamilton-Jacobi equation, it is shown that singularities propagate along specific characteristics and time can extend indefinitely. The use of a-compactness of the Euclidean space and the local Lipschitz issue are key technical difficulties in studying the global results.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2022)
Article
Mathematics, Applied
Cyril Imbert, Tianling Jin, Luis Silvestre
ADVANCES IN NONLINEAR ANALYSIS
(2019)
Article
Mathematics
Isabeau Birindelli, Giulio Galise, Hitoshi Ishii
REVISTA MATEMATICA IBEROAMERICANA
(2020)
Article
Mathematics, Applied
Cyril Imbert, Luis Silvestre
JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
(2020)
Article
Mathematics, Applied
Giulio Galise, Alessandro Iacopetti, Fabiana Leoni, Filomena Pacella
ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE
(2020)
Article
Mathematics
Isabeau Birindelli, Giulio Galise, Hitoshi Ishii
Summary: In this study, we investigate the positivity sets of nonnegative supersolutions of fully nonlinear elliptic equations and establish the strong maximum principle under certain geometric assumptions. Geometric characterizations of the positivity sets of nonnegative supersolutions were obtained, contributing to a better understanding of the behavior of these solutions in open subsets of R-N.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics
Giulio Galise, Alessandro Iacopetti, Fabiana Leoni
JOURNAL OF DIFFERENTIAL EQUATIONS
(2020)
Article
Mathematics, Applied
Cyril Imbert, Rana Tarhini, Francois Vigneron
INTERFACES AND FREE BOUNDARIES
(2020)
Article
Mathematics, Applied
Cyril Imbert, Luis Silvestre
Summary: We establish interior Schauder estimates for kinetic equations with integrodifferential diffusion by studying equations of a specific form and obtaining a priori estimate for the function f in a properly scaled Holder space under suitable conditions on the diffusion operator's kernel.
Article
Mathematics
Jessica Guerand, Cyril Imbert
Summary: This article discusses kinetic Fokker-Planck equations with essentially bounded coefficients. A weak Harnack inequality for nonnegative super-solutions is derived by considering their log-transform and adapting an argument introduced by S. N. Kruzkov (1963). This result is based on a new weak Poincare inequality that shares similarities with the one introduced by W. Wang and L. Zhang in a series of works on ultraparabolic equations (2009, 2011, 2017). This functional inequality is combined with a classical covering argument recently adapted by L. Silvestre and the second author (2020) to kinetic equations.
JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU
(2023)
Article
Mathematics
Isabeau Birindelli, Giulio Galise, Andrei Rodriguez-Paredes
Summary: We provide sufficient conditions for the existence and uniqueness of solutions for degenerate elliptic equations with a nonlinear gradient term in bounded uniformly convex domains, depending on the size of the domain, forcing term, and the properties of the nonlinear gradient term H. The results apply to a wide class of equations, including linear degenerate operators, weighted partial trace operators, and the homogeneous Monge-Ampere operator.
JOURNAL OF CONVEX ANALYSIS
(2021)
Article
Mathematics
I. Birindelli, G. Galise, H. Ishii
Summary: This study investigates the Dirichlet problem for truncated Laplacians on a bounded convex domain, establishing necessary and sufficient conditions for the existence of solutions. The strict convexity of the domain is shown to be sufficient for solvability, with applications to eigenvalue problems and principal eigenfunctions. Additionally, nonexistence results are derived for certain cases involving nearly flat boundary portions.
MATHEMATISCHE ANNALEN
(2021)
Article
Mathematics
Cyril Imbert, Luis Silvestre
EMS SURVEYS IN MATHEMATICAL SCIENCES
(2020)
Article
Mathematics
Cyril Imbert, Clement Mouhot, Luis Silvestre
JOURNAL DE L ECOLE POLYTECHNIQUE-MATHEMATIQUES
(2020)
Article
Mathematics, Interdisciplinary Applications
Isabeau Birindelli, Giulio Galise
MATHEMATICS IN ENGINEERING
(2020)
Article
Mathematics, Applied
Cyril Imbert, Clement Mouhot, Luis Silvestre
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2020)