Solvability of Differential Equations II

In the 50's, the consensus was that all linear PDEs were solvable. Therefore it came as a surprise Hans Lewy in 1957 presented a non-vanishing complex vector field that is not locally solvable. Actually, the vector field is the tangential Cauchy-Riemann operator on the boundary of a strictly pseudoconvex domain. Hormander then proved in 1960 that almost all linear partial differential equations are not locally solvable. After a rapid development in the 60's Nirenberg and Treves formulated their famous conjecture in 1970: that condition (PSI) is necessary and sufficient for the local solvability of differential equations of principal type. Principal type essentially means simple characteristics, and condition (Psi) only involves the sign changes of the imaginary part of the highest order terms along the bicharacteristics of the real part. The Nirenberg-Treves conjecture was finally proved in 2006. We shall present the background, the main results, some examples and generalizations to systems of differential equations. We shall also study the propagation of singularities for operators of principal type satifying the more restrictive condition (P), for which we have complete results.