Holmgren's uniqueness theorem


In the theory of partial differential equations, Holmgren's uniqueness theorem, or simply Holmgren's theorem, named after the Swedish mathematician Erik Albert Holmgren, 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:
This statement, with "analytic" replaced by "smooth", is Hermann Weyl's classical lemma on elliptic regularity:
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:

Relation to the Cauchy–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.
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.