Local solvability of the quaternionic inhomogeneous Beltrami equation

Presenter: Paula Cerejeiras, University of Aveiro, Portugal

Joint work with J. B. Reyes, A.  G. Adán, U. Kähler

In recent years one observe a growing interest in study of Dirac operators with non-constant coefficients. This is motivated either for practical applications (e.g. study of electromagnetic potentials, boundary value problems on inhomogeneous media, etc.) as well as for pure scientific curiosity. One of the principal problems arising from dealing with operators with non-constant coefficients is the question of its local solvability. While for Dirac operators with constant coefficients we always have a solution, this is not necessarily true if the coefficients are variable. In this talk we discuss the local solvability of the inhomogeneous Beltrami equations in quaternionic analysis. Such equations can always be rewritten as a Dirac equation with non-constant coefficients therefore, providing a perfect case study. We give an example of a Beltrami equation with no distributional solution and deduce a condition for local solvability.