## António Caetano

## A primer on function spaces of variable smoothness and integrability

After explaining the main ideas behind the definition of the Besov spaces and the Triebel-Lizorkin spaces, which encompass many classical function spaces used when solving integro-differential equations, we move to the consideration of corresponding spaces with variable parameters and point out at least the recent structure result which says that any function in the latter spaces can be written as a linear combination of so-called atomic functions.

The last part is joint work with A. Almeida.

The last part is joint work with A. Almeida.

Sandrina Santos

## An overview on variational and topological methods for elliptic PDEs

There has been a considerable mathematical interest in boundary value problems (BVPs) involving nonlinear elliptic partial differential equations (PDEs) of second order, mostly due to their applications in modelling physical and mechanical phenomena, game theory, population dynamics and probability. A starting point of such studies is to prove the existence and multiplicity of solutions and, if possible, provide their precise sign information. Currently, the main techniques for obtaining multiplicity results are variational methods (e.g. critical point theory) and topological methods (e.g. degree theory and Morse theory). In this talk, we will give a general overview on these topics and briefly discuss two Dirichlet BVPs involving the p-Laplacian operator: a nonlinear equation problem and a semilinear eigenvalue inclusion problem.