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 $\alpha=\{\alpha_1,\dots,\alpha_n\}\in \N_0^n,$, with $\N_0$ 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 $\R^n\,$, 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 $P=\sum_{|\alpha|\le m}A_\alpha(x)\partial_x^\alpha\,$ 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,

$\mathop{\rm Char}P=\{(x,\xi)\subset T^*\R^n\backslash 0:\sigma_p(P)(x,\xi)=0\},\text{ with }\sigma_p(x,\xi)=\sum_{|\alpha|=m}i^{|\alpha|}A_\alpha(x)\xi^\alpha\,$

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

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

$\partial_t^m u=F(t,x,\partial_x^\alpha\,\partial_t^k u), \quad \alpha\in\N_0^n, \quad k\in\N_0, \quad |\alpha|+k\le m, \quad k\le m-1,$

with the Cauchy data

$\partial_t^k u|_{t=0}=\phi_k(x), \qquad 0\le k\le m-1,$

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 $(t,x)=(0,0)\in(\R\times\R^n)\,$.

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

$\partial_t^m u = F(t,x,\partial_x^\alpha\,\partial_t^k u) = \sum_{\alpha\in\N_0^n,0\le k\le m-1, |\alpha| + k\le m}A_{\alpha,k}(t,x) \, \partial_x^\alpha \, \partial_t^k u,\,$

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

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. 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.
3. François Treves, "Introduction to pseudodifferential and Fourier integral operators", vol. 1, Plenum Press, New York, 1980.