Alcune generalizzazioni del Teorema di Poincaré-Miranda

Let f be a continuous vector field on a set D, defined as the product of n closed intervals, with values in R^n. If on each couple of opposite faces of D the field f points either always inward or always outwards, Poincaré-Miranda’s Theorem states the existence of a point in D where f is zero. In this talk I will present some generalizations of this result, obtained in joint work with A. Fonda. We begin introducing weaker boundary conditions and presenting a first generalization for domains diffeomorphic to the product of two convex bodies. Then we extend the result to domains with a more general structure, i.e. truncated convex bodies. This will allow us to deal also with more complex structures, where the topological degree is different from ±1. Such situations are tipical, as we will see, of the gradient of a functional in a neighbourhood of a degenerate multi-saddle point.