Uniqueness for linear partial differential equations with real analytic coefficients
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 formula_1,
with formula_2 standing for the nonnegative integers;
denote formula_3 and
 formula_4.
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 formula_5 be a connected open neighborhood in formula_6, and let formula_7 be an analytic hypersurface in formula_5, such that there are two open subsets formula_9 and formula_10 in formula_5, nonempty and connected, not intersecting formula_7 nor each other, such that formula_13.
Let formula_14
the principal symbol of formula_15.
formula_16 is a conormal bundle to formula_7, defined as
formula_18.
The classical formulation of Holmgren's theorem is as follows:
Holmgren's theorem
"Let formula_19 be a distribution in formula_5 such that formula_21 in formula_5. If formula_19 vanishes in formula_10, then it vanishes in an open neighborhood of formula_7."
Relation to the Cauchy–Kowalevski theorem.
Consider the problem
formula_26
with the Cauchy data
formula_27
Assume that formula_28 is real-analytic with respect to all its arguments in the neighborhood of formula_29
and that formula_30 are real-analytic in the neighborhood of formula_31.
Theorem (Cauchy–Kowalevski)
"There is a unique real-analytic solution formula_32 in the neighborhood of formula_33".
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 formula_28 is polynomial of order one in formula_35, so that
formula_36
Holmgren's theorem states that the solution formula_19 is real-analytic and hence, by the Cauchy–Kowalevski theorem, is unique.