Definition of the Subject
This entry aims to illustrate some applications of weak KAM theory to the analysis of Hamilton-Jacobi equations. The presentation focuses on two specific problems, namely, the existence of C 1 classical subsolutions for a class of stationary (i.e., independent of the time) Hamilton-Jacobi equations and the long-time behavior of viscosity solutions of an evolutive version of it.
The Hamiltonian is assumed to satisfy mild regularity conditions, under which the corresponding Hamilton equations cannot be written. Consequently, PDE techniques will be solely employed in the analysis, since the powerful tools of the Hamiltonian dynamics are not available.
Introduction
Given a continuous or more regular Hamiltonian H(x, p) defined on the cotangent bundle of a boundaryless manifold M, where x and p are the state and the momentum variable, respectively, and satisfying suitable convexity and coercivity assumptions, is considered the family of Hamilton-Jacobi equations
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Abbreviations
- Hamilton-Jacobi equations:
-
This class of first-order partial differential equations has a central relevance in several branches of mathematics, both from a theoretical and an application point of view. It is of primary importance in classical mechanics, Hamiltonian dynamics, Riemannian and Finsler geometry, and optimal control theory, as well. It furthermore appears in the classical limit of the Schrödinger equation. A connection with Hamilton’s equations, in the case where the Hamiltonian has sufficient regularity, is provided by the classical Hamilton-Jacobi method which shows that the graph of the differential of any regular, say C 1, global solution to the equation is an invariant subset for the corresponding Hamiltonian flow. The drawback of this approach is that such regular solutions do not exist in general, even for very regular Hamiltonians. See the next paragraph for more comments on this issue.
- Viscosity solutions:
-
As already pointed out, Hamilton-Jacobi equations do not have in general global classical solutions, i.e., everywhere differentiable functions satisfying the equation pointwise. The method of characteristics just yields local classical solutions. This explains the need of introducing weak solutions. The idea for defining those of viscosity type is to consider C 1 functions whose graph, up to an additive constant, touches that of the candidate solution at a point and then stay locally above (resp. below) it. These are the viscosity test functions, and it is required that the Hamiltonian satisfies suitable inequalities when its first-order argument is set equal to the differential of them at the first coordinate of the point of contact. Similarly, it is defined as the notion of viscosity sub- and supersolution. Clearly, a viscosity solution satisfies pointwise the equation at any differentiability points. A peculiarity of the definition is that a viscosity solution can admit no test function at some point, while the nonemptiness of both classes of test functions is equivalent to the solution being differentiable at the point. Nevertheless, powerful existence, uniqueness, and stability results hold in the framework of viscosity solution theory. The notion of viscosity solutions was introduced by Crandall and Lions at the beginning of the 1980s. We refer to Bardi and Capuzzo Dolcetta (1997), Barles (1994), and Koike (2004) for a comprehensive treatment of this topic.
- Semiconcave and semiconvex functions:
-
These are the appropriate regularity notions when working with viscosity solution techniques. The definition is given by requiring some inequalities, involving convex combinations of points, to hold. These functions possess viscosity test functions of one of the two types at any point. When the Hamiltonian enjoys coercivity properties ensuring that any viscosity solution is locally Lipschitz continuous, then a semiconcave or semiconvex function is the solution if and only if it is a classical solution almost everywhere, i.e., up to a set of zero Lebesgue measure.
- Metric approach:
-
This method applies to stationary Hamilton-Jacobi equations with the Hamiltonian only depending on the state and momentum variable. This consists of defining a length functional, on the set of Lipschitz-continuous curves, related to the corresponding sublevels of the Hamiltonian. The associated length distance, obtained by performing the infimum of the intrinsic length of curves joining two given points, plays a crucial role in the analysis of the equation and, in particular, enters in representation formulae for any viscosity solution. One important consequence is that only the sublevels of the Hamiltonian matter for determining such solutions. Accordingly, the convexity condition on the Hamiltonian can be relaxed, just requiring quasiconvexity, i.e., convexity of sublevels. Note that in this case the metric is of Finsler type and the sublevels are the unit cotangent balls of it.
- Critical equations:
-
To any Hamiltonian is associated a one-parameter family of Hamilton-Jacobi equations obtained by fixing a constant level of Hamiltonian. When studying such a family, one comes across a threshold value under which no subsolutions may exist. This is called the critical value and the same name is conferred to the corresponding equation. If the ground space is compact, then the critical equation is unique among those of the family for which viscosity solutions do exist. When, in particular, the underlying space is a torus or, in other terms, the Hamiltonian is \( {\mathbb{Z}}^N \) periodic, then such functions play the role of correctors in related homogenization problems.
- Aubry set:
-
The analysis of the critical equation shows that the obstruction for getting subsolutions at subcritical levels is concentrated on a special set of the ground space, in the sense that no critical subsolution can be strict around it. This is precisely the Aubry set. This is somehow compensated by the fact that critical subsolutions enjoy extra regularity properties on the Aubry set.
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Siconolfi, A. (2013). Hamilton-Jacobi Equations and Weak KAM Theory. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-3-642-27737-5_268-3
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