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 $\alpha=\\in \N_0^n,$,
with $\N_0$ standing for the nonnegative integers;
denote $, \alpha, =\alpha_1+\cdots+\alpha_n$ and

: $\partial_x^\alpha = \left(\frac\right)^ \cdots \left(\frac\right)^$.

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'' ⊂ R^''n'', 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 $\R^n$, 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 $P=\sum_A_\alpha(x)\partial_x^\alpha$ 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:

:$\mathopP\cap N^*\Sigma=\emptyset$.

Above,

: $\mathopP=\,\text\sigma_p(x,\xi)=\sum_i^A_\alpha(x)\xi^\alpha$

the principal symbol of $P$.
$N^*\Sigma$ is a conormal bundle to $\Sigma$, defined as
$N^*\Sigma=\$.

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 $\Sigma$.'' François Treves,
"Introduction to pseudodifferential and Fourier integral operators", vol. 1, Plenum Press, New York, 1980.

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

:$\partial_t^k u, _=\phi_k(x), \qquad 0\le k\le m-1,$

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

:$\partial_t^m u = F(t,x,\partial_x^\alpha\,\partial_t^k u) = \sum_A_{\alpha,k}(t,x) \, \partial_x^\alpha \, \partial_t^k u,$

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

Partial differential equations
Theorems in analysis
Uniqueness theorems