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.[1]

Simple form of Holmgren's theorem

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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 Aα(x)∂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:[2]

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

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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  .[3]

Relation to the Cauchy–Kowalevski theorem

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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.[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.

See also

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References

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  1. ^ Eric Holmgren, "Über Systeme von linearen partiellen Differentialgleichungen", Öfversigt af Kongl. Vetenskaps-Academien Förhandlinger, 58 (1901), 91–103.
  2. ^ Stroock, W. (2008). "Weyl's lemma, one of many". Groups and analysis. London Math. Soc. Lecture Note Ser. Vol. 354. Cambridge: Cambridge Univ. Press. pp. 164–173. MR 2528466.
  3. ^ François Treves, "Introduction to pseudodifferential and Fourier integral operators", vol. 1, Plenum Press, New York, 1980.