Abstract:
LetSdenote the general second order linear differential operator on the complex plane.
It is a well-known fact that the Dirichlet boundary value problem forSon a bounded domainΩwith a smooth boundary, is not always well-posed even whenSis elliptical.
This phenomenon led to a homotopic classification of elliptic systems. B. Bojarski, a pioneer in this subject, showed that the family of elliptic operators forms an open set in\doubc \sp 6with exactly six components.
It follows from his classification that the only components where the Dirichlet problem may be well-posed are the ones represented by the Laplacian or its complex conjugate...
Citation:
Habre, S. S. (1993). Second-order linear elliptic systems on the complex plane (Doctoral dissertation, Syracuse University Syracuse).