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 $$\alpha=\{\alpha_1,\dots,\alpha_n\}\in \N_0^n,$$, 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 = &sum;undefined A&alpha;(x)&part;$&alpha; x$ is an elliptic partial differential operator with real-analytic coefficients. If Pu is real-analytic in a connected open neighborhood &Omega; &sub; 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 &Omega;, then u is also smooth in &Omega;.

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 $$\Sigma$$.

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 \quad k\le m-1,$$
 * \alpha|+k\le m,

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.