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Qualitative properties of solutions of Hamilton-Jacobi equations (after A. Fathi, A. Siconolfi, P. Bernard). (Propriétés qualitatives des solutions des équations de Hamilton-Jacobi (d’après A. Fathi, A. Siconolfi, P. Bernard).) (French) Zbl 1162.35003

Séminaire Bourbaki. Volume 2006/2007. Exposés 967–981. Paris: Société Mathématique de France (ISBN 978-2-85629-253-2/pbk). Astérisque 317, 269-294, Exp. No. 975 (2008).
This work is an introduction to the theorem of A. Fathi and A. Siconolfi [Invent. Math. 155, No. 2, 363–388 (2004; Zbl 1061.58008), Calc. Var. Partial Differ. Equ. 22, No. 2, 185–228 (2005; Zbl 1065.35092)] and shows its impact on the qualitative properties of Hamilton-Jacobi equations. The first section is dedicated to the definition of the viscosity solutions, leading to a theorem of existence (Lions, Papanicolaou, Varardhan) for \[ H(x,Du)=c,\quad x\in \mathbb T^N. \tag{1} \] The second section introduces the Lax-Oleinik formulas like an explicit solution of the Cauchy problem \[ u_t+H(x,Du)=0,\;t>0,\;x\in \mathbb T^N\qquad u(0,x)=u_0 (x),\tag{2} \] and identifies the main properties. These first two sections are not original, their merit is to summarize in a few pages two different results for (2). The short section 3 provides the very simple \(C^{1,1}\) regularity properties, crucial for the future. Section 4 analyzes the theorem of Fathi-Siconolfi, and Section 5 presents several applications and additions: sets of uniqueness for the stationary equation (1), existence of invariant regions for the system Hamiltonian and additional regularity.
For the entire collection see [Zbl 1151.00015].

MSC:

35F30 Boundary value problems for nonlinear first-order PDEs
37J50 Action-minimizing orbits and measures (MSC2010)
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35B65 Smoothness and regularity of solutions to PDEs
35C15 Integral representations of solutions to PDEs
35D40 Viscosity solutions to PDEs
70H20 Hamilton-Jacobi equations in mechanics
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