文档介绍：Weak KAM from a PDE point of
view: viscosity solutions of the
Hamilton-Jacobi equation and
Aubry set
Albert FATHI
Course CANPDE 17-19 February 2011
1
Contents
1 The different forms of the Hamilton-Jacobi Equation 2
2 Viscosity Solutions 4
3 Lower and upper differentials 9
4 Criteria for viscosity solutions 14
5 Coercive Hamiltonians 15
6 Stability 16
7 Uniqueness 17
8 Construction of viscosity solutions 20
9 Strict subsolutions 22
10 Quasi-convexity and viscosity subsolutions 23
11 The viscosity semi-distance 28
12 The projected Aubry set 31
13 The representation formula 34
14 Tonelli Hamiltonians and Lagrangians 35
2
We will mainly introduce the notion of viscosity solutions for the Hamilton-Jacobi
equation which is a first-order PDE. There also is an extensive literature on viscosity
solutions of second-order PDE’s, we do not touch this topic at all, see for example [CIL92].
The notion of viscosity solution is due to Crandall and Lions, see [CL83]. There are two
excellent books on the subject one by by Guy Barles [Bar94] and another one by Martino
Bardi and Italo Capuzzo-Dolceta [BCD97]. A first introduction to viscosity solutions can
be found in Craig Evans’ book [Eva98]. Our treatment has been extremely influenced
by the content of these three books. Although many things are standard, we will do the
theory on general manifolds since this is the right setting for weak KAM theory. This
is probably the first time that a general introduction on viscosity solutions on manifolds
appears in print. Whatever is not in the standard es from joint work with
Antonio Siconolfi, see [FS04] and [FS05]. Of course, our treatment follows some of the
unpublished notes [Fat08]. We hardly touch the dynamical implications of the theory,
and refer the reader to Patrick Bernard’panion notes [Ber11]
We would like to apologize for the small number of references. In a work of this size,
to give a fair and large set of references in the subject is nowa