“Holmgren's uniqueness theorem”的意思、由来-开放百科全书

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Holmgren's uniqueness theorem

释义

Simple form of Holmgren's theorem
Classical form
Relation to the Cauchy–Kowalevski theorem
See also
References

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

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)∂{{su|p=α|b=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 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.

References

1. ^Eric Holmgren, "Über Systeme von linearen partiellen Differentialgleichungen", Öfversigt af Kongl. Vetenskaps-Academien Förhandlinger, 58 (1901), 91–103.
2. ^{{cite book|mr=2528466|last=Stroock|first = W.|chapter=Weyl's lemma, one of many|title=Groups and analysis|pages=164–173|series=London Math. Soc. Lecture Note Ser.|volume=354|publisher=Cambridge Univ. Press|location=Cambridge|year=2008}}
3. ^François Treves,"Introduction to pseudodifferential and Fourier integral operators", vol. 1, Plenum Press, New York, 1980.

2 : Partial differential equations|Theorems in analysis

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Simple form of Holmgren's theorem

Classical form

Relation to the Cauchy–Kowalevski theorem

See also

References