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
editWe 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
editLet 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
editConsider 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
editReferences
edit- ^ Eric Holmgren, "Über Systeme von linearen partiellen Differentialgleichungen", Öfversigt af Kongl. Vetenskaps-Academien Förhandlinger, 58 (1901), 91–103.
- ^ 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.
- ^ François Treves, "Introduction to pseudodifferential and Fourier integral operators", vol. 1, Plenum Press, New York, 1980.