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 (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 ⊂ Rⁿ, 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–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

References