Joint Colloquium: Ajiev -- Homogeneous right inverses and metric projections

 ___________________JOINT COLLOQUIUM______________________ 

Speaker: Dr Sergey Ajiev, (UNSW) 

Title: Homogeneous right inverses and metric projections.  

Date: Friday, August 21, 2009 Time: 2:00 pm Venue: Room 275 Carslaw Building, University
of Sydney 

Preceded by lunch, meeting at 12:30 p.m., outside foyer, second-floor Carslaw building.  

Abstract: 

Some problems, for example, in PDE can be reduced to the invertibility of a closed
operator with a non-trivial and non-complemented kernel defined, for example, on a
Sobolev space.  We construct homogeneous (non-linear) right-inverse operators and
establish explicit estimates for the exponents of their Hölder regularity and the
corresponding Hölder seminorms in both abstract and particular settings.  In the setting
of Banach spaces, our right inverses provide optimal solution for the equation Af=g.  

One deals with various types of Besov and Lizorkin-Triebel spaces of functions on an
arbitrary open subset of an Euclidean space defined in terms of local approximations,
differences, wavelet expansions, or a functional calculus and also with their duals,
subspaces, quotients and a wide class of "independently generated spaces".  

We also discuss the relations with the classical problem of the regularity of the metric
projection map and establish the global regularity of this map in a quantitative
manner.  Attention is paid to the sharpness of some results, natural generalisations and
the limitations of the linear and Lipschitz settings.