Holmgren's uniqueness theorem explained

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

\alpha = \{\alpha _{1,...,\alpha} _{n\}\in} \N ^{n,} _{0} ,with \N _{0} standing for the nonnegative integers;denote |\alpha| = \alpha _{1+ … +\alpha} _{n} and\partial ^{\alpha} _{x} = \left( (\partial)/(\partial x _{1}) \right) ^{\alpha _{1}} … \left( (\partial)/(\partial x _{n}) \right) ^{\alpha _{n}} .

Holmgren's theorem in its simpler form could be stated as follows:

Assume that P = ∑_|α| ≤m A_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

\Omega be a connected open neighborhood in \R ^{n} , and let \Sigma be an analytic hypersurface in \Omega , such that there are two open subsets \Omega _{+} and \Omega _{-} in \Omega , nonempty and connected, not intersecting \Sigma nor each other, such that \Omega = \Omega _{-} \cup\Sigma\cup\Omega _{+} .

Let

P = \sum _{|\alpha|\le} A _{\alpha(x)\partial} ^{\alpha} ^{x} be a differential operator with real-analytic coefficients.

Assume that the hypersurface

\Sigma is noncharacteristic with respect to P at every one of its points:\rmChar P\cap N ^{*\Sigma=\emptyset} .

Above,

\rmChar P = \{(x,\xi)\subset T ^{*\R} ^{n\backslash} 0 :\sigma _{p(P)(x,\xi)=0\},with\sigma} _{p(x,\xi)=\sum} _{|\alpha|=m} i ^{|\alpha|} A ^{\alpha} _{\alpha(x)\xi}

the principal symbol of

P .N ^{*\Sigma} is a conormal bundle to \Sigma , defined asN ^{*\Sigma=\{(x,\xi)\in} T ^{*\R} ^{n:x\in\Sigma,\xi|} ^{T _{x\Sigma}} =0 \} .

The classical formulation of Holmgren's theorem is as follows:

Holmgren's theorem

Let

u be a distribution in \Omega such that Pu =0 in \Omega . If u vanishes in \Omega _{-} , then it vanishes in an open neighborhood of \Sigma .^[3]

Relation to the Cauchy - Kowalevski theorem

Consider the problem

\partial ^{m} _{t} u = F(t,x,\partial ^{k} _{t} u), \alpha\in\N ^{n, k\in\N} _{0 , |\alpha| + k\le} m, k\le m -1 ,

with the Cauchy data

\partial ^{k} _{t} u| _{t=0} = \phi _{k(x),} 0 \le k\le m -1 ,

Assume that

F(t,x,z) is real-analytic with respect to all its arguments in the neighborhood of t =0 ,x =0 ,z =0 and that \phi _{k(x)} are real-analytic in the neighborhood of x =0 .

Theorem (Cauchy - Kowalevski)

There is a unique real-analytic solution

u(t,x) in the neighborhood of (t,x) = ( 0 , 0 )\in(\R x \R ^{n)} .

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

F(t,x,z) is polynomial of order one in z , so that\partial ^{m} _{t} u = F(t,x,\partial ^{k} _{t} u) = \sum _{\alpha\in\N ^{n, 0 \le} _{0} k\le m -1 , |\alpha| + k\le m} A _{\alpha,k} (t,x) \partial ^{\alpha} _{x} \partial ^{k} _{t} u,

Holmgren's theorem states that the solution

u is real-analytic and hence, by the Cauchy - Kowalevski theorem, is unique.

See also

• Cauchy - Kowalevski theorem
• FBI transform

References

1. Eric Holmgren, "Über Systeme von linearen partiellen Differentialgleichungen", Öfversigt af Kongl. Vetenskaps-Academien Förhandlinger, 58 (1901), 91–103.
2. Book: Stroock, W.. 2528466. Weyl's lemma, one of many. Groups and analysis. 164 - 173. London Math. Soc. Lecture Note Ser.. 354. Cambridge Univ. Press. Cambridge. 2008.
3. [François Treves]