Differential equations which have no solutions

Abstract

In Chapter III we proved that a differential equation with constant coefficients can be solved for an arbitrary right-hand side f, at least in relatively compact open subsets of the open set where f is defined. It was discovered rather recently by H. Lewy [1] that the situation is completely different when the coefficients are variable. In fact, he proved the existence of functions f ∈ C^∞ (R₃) such that the equation

$$-i{{D}_{1}}u+{{D}_{2}}u-2\left( {{x}_{1}}+i{{x}_{2}} \right){{D}_{3}}u=f$$

(6.0.1)

does not have any (distribution) solution u in any open non-void subset of R₃. In section 6.1 we shall give an extension of this example due to Hörmander [10], [11] by proving a necessary condition for a differential equation P(x, D)u = f to have a solution locally for every f ∈ C^∞. In Chapter VIII we shall see that a strengthened form of this condition is also sufficient to imply local existence of solutions for every f, provided that there are no multiple real characteristics.