Differential equations which have no solutions

Part of the Die Grundlehren der Mathematischen Wissenschaften book series (GL, volume 116)


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 fC (R3) such that the equation
$$-i{{D}_{1}}u+{{D}_{2}}u-2\left( {{x}_{1}}+i{{x}_{2}} \right){{D}_{3}}u=f$$
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 fC. 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.


Differential Operator Variable Coefficient Order Differential Operator Linear Partial Differential Operator Real Symmetric Matrix 
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Copyright information

© Springer-Verlag OHG, Berlin · Göttingen · Heidelberg 1963

Authors and Affiliations

  1. 1.University of Stockholm and at Stanford UniversitySweden

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