Holmgren's uniqueness theorem
Introduction In the theory of partial differential equations, '''Holmgren's uniqueness theorem''', or simply '''Holmgren's theorem''', named after the Swedish mathematician Erik Albert Holmgren (1873–1943), is a uniqueness result for linear partial differential equations with real analytic coefficients.
Simple form of Holmgren's theorem We will use the multi-index notation:Let ,with  standing for the nonnegative integers;denote  and: .Holmgren's theorem in its simpler form could be stated as follows::Assume that ''P'' = ∑|''α''| ≤''m'' ''A''''α''(x)∂  is an elliptic partial differential operator with real-analytic coefficients. If ''Pu'' is real-analytic in a connected open neighborhood ''Ω'' ⊂ '''R'''''n'', then ''u'' is also real-analytic.This statement, with "analytic" replaced by "smooth", is Hermann Weyl's classical lemma on elliptic regularity::If ''P'' is an elliptic differential operator and ''Pu'' is smooth in ''Ω'', then ''u'' is also smooth in ''Ω''.This statement can be proved using Sobolev spaces.
Classical form Let  be a connected open neighborhood in , and let  be an analytic hypersurface in , such that there are two open subsets  and  in , nonempty and connected, not intersecting  nor each other, such that .Let  be a differential operator with real-analytic coefficients.Assume that the hypersurface  is noncharacteristic with respect to  at every one of its points::.Above,: the principal symbol of . is a conormal bundle to , defined as.The classical formulation of Holmgren's theorem is as follows::'''Holmgren's theorem''':''Let  be a distribution in  such that  in . If  vanishes in , then it vanishes in an open neighborhood of .''
Relation to the Cauchy&ndash;Kowalevski theorem Consider the problem:with the Cauchy data:Assume that  is real-analytic with respect to all its arguments in the neighborhood of and that  are real-analytic in the neighborhood of . :'''Theorem''' (Cauchy–Kowalevski):''There is a unique real-analytic solution  in the neighborhood of ''.Note that the Cauchy–Kowalevski theorem does not exclude the existence of solutions which are not real-analytic.On the other hand, in the case when  is polynomial of order one in , so that:Holmgren's theorem states that the solution  is real-analytic and hence, by the Cauchy–Kowalevski theorem, is unique.
See also * Cauchy–Kowalevski theorem* FBI transform
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