Holmgren's uniqueness theorem

Template:Short description 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

$\underset{x}{\overset{\alpha }{\partial }}={\left(\frac{\partial }{\partial x_1}\right)}^{{\alpha }_1}\cdots {\left(\frac{\partial }{\partial x_n}\right)}^{{\alpha }_n}$.

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

Assume that P = ∑_|α| ≤m A_α(x)∂Template:Su 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 $\mathrm{\Omega }$ be a connected open neighborhood in ${ℝ}^n$, and let $\mathrm{\Sigma }$ be an analytic hypersurface in $\mathrm{\Omega }$, such that there are two open subsets ${\mathrm{\Omega }}_{+}$ and ${\mathrm{\Omega }}_{-}$ in $\mathrm{\Omega }$, nonempty and connected, not intersecting $\mathrm{\Sigma }$ nor each other, such that $\mathrm{\Omega }={\mathrm{\Omega }}_{-}\cup \mathrm{\Sigma }\cup {\mathrm{\Omega }}_{+}$.

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

Assume that the hypersurface $\mathrm{\Sigma }$ is noncharacteristic with respect to $P$ at every one of its points:

$\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{r}P\cap N^{*}\mathrm{\Sigma }=\mathrm{\varnothing }$.

Above,

$\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{r}P=\{(x,\xi )\subset T^{*}{ℝ}^n\mathrm{\setminus }0:{\sigma }_p(P)(x,\xi )=0\},\text{ with }{\sigma }_p(x,\xi )=\sum _{|\alpha |=m}{i}^{|\alpha |}{A}_{\alpha }(x){\xi }^{\alpha }$

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

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

Holmgren's theorem

Let $u$ be a distribution in $\mathrm{\Omega }$ such that $Pu=0$ in $\mathrm{\Omega }$. If $u$ vanishes in ${\mathrm{\Omega }}_{-}$, then it vanishes in an open neighborhood of $\mathrm{\Sigma }$.

Relation to the Cauchy–Kowalevski theorem

Consider the problem

$\underset{t}{\overset{m}{\partial }}u=F(t,x,\underset{x}{\overset{\alpha }{\partial }}\underset{t}{\overset{k}{\partial }}u),\alpha \in {\mathbb{N}}_0^n,k\in {\mathbb{N}}_0,|\alpha |+k\le m,k\le m-1,$

with the Cauchy data

$\underset{t}{\overset{k}{\partial }}u{|}_{t=0}={\varphi }_k(x),0\le k\le m-1,$

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 ${\varphi }_k(x)$ 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.Template:Citation needed

On the other hand, in the case when $F(t,x,z)$ is polynomial of order one in $z$, so that

$\underset{t}{\overset{m}{\partial }}u=F(t,x,\underset{x}{\overset{\alpha }{\partial }}\underset{t}{\overset{k}{\partial }}u)=\sum _{\alpha \in {\mathbb{N}}_0^n,0\le k\le m-1,|\alpha |+k\le m}{A}_{\alpha ,k}(t,x)\underset{x}{\overset{\alpha }{\partial }}\underset{t}{\overset{k}{\partial }}u,$

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