LOCAL SOLVABILITY OF PARTIAL DIFFERENTIAL EQUATIONS

The most primitive question one can ask concerning a partial differential equation i s i f there exists,a solution, at least locally, and not subjected to any additional condition. For ordinary differential equations, very satisfactory theorems stating local existence of solutions under very mild hypothesis of regularity had been known since long ago, and it came as a surprise when Hans Lewy discovered, in 1956, his now famous example of a first-order linear equation whose coefficients are polynomial of degree at most one, failing to have local solutions. Indeed, if f E COO ( R 3) is conveniently chosen, the equation (8x + iOy- (x + iy)8z)u = f (x, y, z) E R3 does not have distribution solutions in any open subset of IR 3 ( [L] ). For elliptic equations local solvability was known and Hormander had proved in i s thesis that linear operators of real principal type were locally solvable. Let us recall some definitions and notation. A linear partial differential operator in an open subset n of IR n has the form P(x, D)u = L a", (x)D" ' u, l " ' l �