Functional Analysis and Applications Seminar 

11/04/2012, 15:00 — 16:00 — Room P3.10, Mathematics Building
Catarina Carvalho, Instituto Superior Técnico, UTL and CEAF

Layer potentials $C^*$-algebras of conical domains

In boundary problems for elliptic systems, namely through the method of layer potentials, one is often led to study invertibility of integral operators on the boundary. If the domain is sufficiently regular, classic Fredholm theory applies. On singular domains, however, the relevant operators are no longer compact. The main aim of this talk is to give a suitable replacement of classic Fredholm theory in the setting of domains with conical singularities. The key idea is to use the theory of pseudodifferential operators on Lie groupoids. In that respect, to a conical domain $\Omega$ we first associate a boundary groupoid $\mathcal{G}$ over a desingularization of $\partial \Omega$ and define the so-called layer potentials $C^\ast$-algebra, which turns out to be a good replacement for the ideal of compact operators. We use a representation of $\Psi(\mathcal{G})$ as bounded operators on suitable Sobolev spaces with weight at $\partial \Omega$ to give Fredholm criteria, reducing to ellipticity and invertibility of indicial operators on cones at each singularity.

The talk is based on joint work with Victor Nistor and Yu Qiao.