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 (18731943), is a uniqueness result for linear partial differential equations with real analytic coefficients.[1]

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 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

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 CauchyKowalevski 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 (CauchyKowalevski)
There is a unique real-analytic solution in the neighborhood of .

Note that the CauchyKowalevski 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 CauchyKowalevski theorem, is unique.

See also

References

  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. 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.
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