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An Introduction to the Theory of Viscosity Solutions for First-Order Hamilton–Jacobi Equations and Applications

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Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2074))

Abstract

In this course, we first present an elementary introduction to the concept of viscosity solutions for first-order Hamilton–Jacobi Equations: definition, stability and comparison results (in the continuous and discontinuous frameworks), boundary conditions in the viscosity sense, Perron’s method, Barron–Jensen solutions … etc. We use a running example on exit time control problems to illustrate the different notions and results. In a second part, we consider the large time behavior of periodic solutions of Hamilton–Jacobi Equations: we describe recents results obtained by using partial differential equations type arguments. This part is complementary of the course of H. Ishii which presents the dynamical system approach (“weak KAM approach”).

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Notes

  1. 1.

    Here we use the notation Du for the full gradient of u in space and time but, in general, we will use it for the gradient in space of u.

  2. 2.

    Here we have also used a less important (but simplifying) property, namely the commutation with constants: for any \(c\,\in \,\mathbb{R}\), S, x, t and for any function u( ⋅), G(S, x, t, u( ⋅) + c) = G(S, x, t, u( ⋅)) + c.

  3. 3.

    In Biton [19], a non-trivial counterexample to the uniqueness for (17) is given in a situation where the Cauchy–Lipschitz Theorem cannot be applied to (18).

  4. 4.

    This is a key point: the compactness of the domain (periodicity) plays a crucial role here since local uniform convergence is the same as global uniform convergence.

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  1. Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 7350), Fédération Denis Poisson (FR CNRS 2964), Université de Tours, Parc de Grandmont, 37200, Tours, France

    Guy Barles

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Barles, G. (2013). An Introduction to the Theory of Viscosity Solutions for First-Order Hamilton–Jacobi Equations and Applications. In: Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications. Lecture Notes in Mathematics(), vol 2074. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36433-4_2

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