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.
[1] We will use the multi-index notation: Let
standing for the nonnegative integers; denote
and Holmgren's theorem in its simpler form could be stated as follows: This statement, with "analytic" replaced by "smooth", is Hermann Weyl's classical lemma on elliptic regularity:[2] This statement can be proved using Sobolev spaces.
be a connected open neighborhood in
be an analytic hypersurface in
, such that there are two open subsets
, nonempty and connected, not intersecting
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
= { ( x , ξ ) ∈
ξ
The classical formulation of Holmgren's theorem is as follows: Consider the problem with the Cauchy data Assume that
is real-analytic with respect to all its arguments in the neighborhood of
are real-analytic in the neighborhood of
Note that the Cauchy–Kowalevski theorem does not exclude the existence of solutions which are not real-analytic.
[citation needed] 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.