Sala Seminari (Dip. Matematica).
In this talk we will introduce the Willmore energy of surfaces in the three-dimensional euclidean space, which is the surface integral of the squared mean curvature. For a smooth closed embedded planar curve, we will consider the minimization of the Willmore energy among immersed surfaces of a prescribed genus having the given curve as boundary. Such problem can be seen as a generalization of the classical Plateau-Douglas problem, which is immediately trivial in the case of planar boundary curves. Exploiting the conformal properties of the functional and tools from the theory of varifolds with boundary, we will see that the problem does not reduce to a minimal surfaces problem and we will present some recent explicit results both of existence and non-existence of minimizers, depending on the prescribed boundary curve.