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 (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 $\alpha =\{\alpha _{1},\dots ,\alpha _{n}\}\in \mathbb {N} _{0}^{n},$ , with $\mathbb {N} _{0}$ standing for the nonnegative integers; denote $|\alpha |=\alpha _{1}+\cdots +\alpha _{n}$ and

\partial _{x}^{\alpha }=\left({\frac {\partial }{\partial x_{1}}}\right)^{\alpha _{1}}\cdots \left({\frac {\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)∂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 $\mathbb {R} ^{n}\,$ , 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 $P=\sum _{|\alpha |\leq m}A_{\alpha }(x)\partial _{x}^{\alpha }\,$ 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:

{\mathop {\rm {Char}}}P\cap N^{*}\Sigma =\emptyset

Above,

{\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 }\,

the principal symbol of $P\,$ . $N^{*}\Sigma \,$ is a conormal bundle to $\Sigma \,$ , defined as $N^{*}\Sigma =\{(x,\xi )\in T^{*}\mathbb {R} ^{n}:x\in \Sigma ,\,\xi |_{T_{x}\Sigma }=0\}\,$ .

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 _{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,

with the Cauchy data

\partial _{t}^{k}u|_{t=0}=\phi _{k}(x),\qquad 0\leq k\leq 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 $(t,x)=(0,0)\in (\mathbb {R} \times \mathbb {R} ^{n})\,$ .

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 _{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,\,

Holmgren's theorem states that the solution $u\,$ is real-analytic and hence, by the Cauchy–Kowalevski theorem, is unique.

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. 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.

This article is issued from Wikipedia - version of the 11/26/2013. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.

Holmgren's uniqueness theorem

Simple form of Holmgren's theorem

Classical form

Relation to the Cauchy–Kowalevski theorem

See also

References