## Eugénio Rocha

## Multiplicity results for elliptic problems in adapted Sobolev spaces

We discuss the existence and multiplicity of solutions of some classes of variational elliptic problems defined in Sobolev spaces, which are adapted to a monotone operator (with further required properties). Such framework may include local and nonlocal operators, as the p-Laplacian and the fractional Laplacian. We also point out the relevance of best constants and embedding in multiplicity results.

Luís Castro

## Operator theory for problems of wave diffraction by arbitrary wedges

The problem of plane wave diffraction by a wedge sector having arbitrary aperture angle has a very long and interesting research background. We may recognize significant research on this topic for more than one century. Despite this fact, up to now no clear unified approach was implemented to treat such a problem from a rigourous mathematical way and in a consequent appropriate Sobolev space setting. In the present talk, we will consider the corresponding boundary value problems for the Helmholtz equation, with complex wave number, admitting Dirichlet and Neumann boundary conditions. We use operator theory to study those problems. In fact, the main ideas will be based on a convenient combination of potential representation formulas associated with (weighted) Mellin pseudo-differential operators in appropriate Sobolev spaces, and a detailed Fredholm analysis. Thus, we prove that the problems have unique solutions (with continuous dependence on the data), which are represented by the single and double layer potentials, where the densities are solutions of derived pseudo-differential equations on the half-line.

The talk is based on joint research with D. Kapanadze (A. Razmadze Mathematical Institute, Tbilisi State University, Tbilisi, Georgia).

The talk is based on joint research with D. Kapanadze (A. Razmadze Mathematical Institute, Tbilisi State University, Tbilisi, Georgia).