The Solvability of Differential Equations

Abstract

It was a great surprise when 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 pseudocon- vex domain. Hormander proved in 1960 that almost all linear partial differential equations are not locally solvable. This also has connections with the spectral instability of non-selfadjoint semiclassical operators. Nirenberg and Treves formulated their well-known conjecture in 1970: that condition (�) is necessary and sufficient for the local solvability of differential equations of principal type. Principal type essentially means simple character- istics, and condition (�) 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 ideas of the proof and some open problems.