Catarina Carvalho
(Instituto Superior Técnico, UTL and CEAF)
" Layer potentials C*-algebras of
conical domains "
Abstract. 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 Ω we first associate a boundary groupoid G
over a desingularization of ∂Ω and define the so-called layer
potentials C*-algebra, which turns out to be a good replacement for the
ideal of compact operators. We use a representation of Ѱ(G) as bounded
operators on suitable Sobolev spaces with weight at ∂Ω 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.
