Subanalytic sets in the calculus of variation

Sammanfattning: The purpose of this thesis is to develop a theory for certain extreme value problems, depending analytically on some set of parameters. In particular it aims at a description of the singularities of the extreme value functions. In this context, a central role is played by the concept of subanalytic functions (i.e. functions whose graphs are subanalytic in the sense of Hironaka). The content of this paper naturally divides into three parts. After some preliminary material in the three first chapters, the first main topic comes up in Chapter IV, where it is proved that singular supports of subanalytic functions are subanalytic. As a consequence, it follows that the singular set of a subanalytic set is again subanalytic. Chapter V is then devoted to finding sufficient conditions on an extreme value problem for its extreme value function to be subanalytic. In Chapter VI finally, two applications of the general theory are presented. The first is to prove that cut loci on analytic Riemannian manifolds are always stratifiable and triangulable. The second concerns the theory of phase transitions in a certain model in statistical mechanics (the so called Van der Waals limit). In particular, the piecewise analytic behaviour of the free energy as a function of temperature (and density) is explained in this context.