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]
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.
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,
![{\displaystyle \mathop {\rm {Char}} P=\{(x,\xi )\subset T^{*}\mathbb {R} ^{n}\backslash 0:\sigma _{p}(P)(x,\xi )=0\},{\text{ with }}\sigma _{p}(x,\xi )=\sum _{|\alpha |=m}i^{|\alpha |}A_{\alpha }(x)\xi ^{\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f733a8e0063c750a77ae0046d0db06370647ea4f)
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
![{\displaystyle \partial _{t}^{m}u=F(t,x,\partial _{x}^{\alpha }\,\partial _{t}^{k}u),\quad \alpha \in \mathbb {N} _{0}^{n},\quad k\in \mathbb {N} _{0},\quad |\alpha |+k\leq m,\quad k\leq m-1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8b9b2495434831cacb71d33405b99f7821c4f3)
with the Cauchy data
![{\displaystyle \partial _{t}^{k}u|_{t=0}=\phi _{k}(x),\qquad 0\leq k\leq m-1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5d42ccba2c0d0be84b6ebac04bafaa227c34424)
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
![{\displaystyle \partial _{t}^{m}u=F(t,x,\partial _{x}^{\alpha }\,\partial _{t}^{k}u)=\sum _{\alpha \in \mathbb {N} _{0}^{n},0\leq k\leq m-1,|\alpha |+k\leq m}A_{\alpha ,k}(t,x)\,\partial _{x}^{\alpha }\,\partial _{t}^{k}u,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e00ab6f1752df66307ed2ff43277b9d922867593)
Holmgren's theorem states that the solution
is real-analytic and hence, by the Cauchy–Kowalevski theorem, is unique.
See also
References
- ^ 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.