Legendre functions and the theory of characteristics

We devise a framework encompassing the classical theory of characteristics and the theory valid in the convex case recently obtained by R.T. Rockafellar and P. Wolenski. It relies on a notion of transform introduced by I. Ekeland. It involves a class of functions called Ekeland functions which is large enough to encompass convex functions, concave functions and linear-quadratic functions, as well as the class of classical Legendre functions. We also introduce a class of functions called (generalized) Legendre functions which is not as large as the class of Ekeland functions but has better reciprocity properties. It is obtained by an extension procedure and it enables one to recover the usual FenchelLegendre transform of convex functions. Key words: characteristics, Ekeland function, Ekeland transform, Hamilton-Jacobi equations, Legendre function, Legendre transform, subdi¤erential Mathematics Subject Classi…cation: 49J52, 49J53, 90C26, 53D05, 37K25 Dedicated to Ivar Ekeland on his sixtieth birthday