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.

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:

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 $\Sigma$.

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.

See also

• Cauchy–Kowalevski theorem
• FBI transform