Multiple-valued functions in the sense of F. J. Almgren

(eng) A multiple-valued function is a "function" that assumes two or more distinct values in its range for at least one point in its domain. While these "functions" are not functions in the normal sense of being single-valued, the usage is so common that there is no way to dislodge it. This thesis is devoted to a particular class of multiple-valued functions: Q-valued functions. A Q-valued function is essentially a rule assigning Q unordered and not necessarily distinct points of R^n to each element of R^m. This object is one of the key ingredients of Almgren's 1700 pages proof that the singular set of an m-dimensional mass minimizing integral current in R^n has dimension at most m-2. We start by developing a decomposition theory and show for instance when a continuous Q-valued function can or cannot be seen as Q "glued" continuous classical functions. Then, the decomposition theory is used to prove intrinsically a Rademacher type theorem for Lipschitz Q-valued functions. A couple of Lipschitz extension theorems are also obtained for partially defined Lipschitz Q-valued functions. The second part is devoted to a Peano type result for a particular class of nonconvex-valued differential inclusions. To the best of the author's knowledge this is the first theorem, in the nonconvex case, where the existence of a continuously differentiable solution is proved under a mere continuity assumption on the corresponding multifunction. An application to a particular class of nonlinear differential equations is included. The third part is devoted to the calculus of variations in the multiple-valued framework. We define two different notions of Dirichlet nearly minimizing Q-valued functions, generalizing Dirichlet energy minimizers studied by Almgren. Hölder regularity is obtained for these nearly minimizers and we give some examples showing that the branching phenomena can be much worse in this context.