Partial differential equations Existence theory Theorem

Holmgren's uniqueness theorem

Builds on Linear PDE Characteristic variety Analytic function

Statement

Let $\Omega \subseteq \R^n$ be open, and let $P = \sum_{\abs{\alpha} \le m} a_\alpha(x) \, \partial^\alpha$ be a linear differential operator of order $m$ whose coefficients $a_\alpha$ are real-analytic on $\Omega$ . Let $S \subseteq \Omega$ be a real-analytic hypersurface and let $x_0 \in S$ be a point at which $S$ is noncharacteristic for $P$ . Let $U \subseteq \Omega$ be an open neighbourhood of $x_0$ , and let $u \in C^m(U)$ satisfy $P u = 0$ in $U$ and $\partial^\alpha u = 0$ on $S \cap U$ for all $\abs{\alpha} \le m - 1$ . Then $u = 0$ in some neighbourhood of $x_0$ .

Remarks

Equivalently, two $C^m$ solutions of $P u = f$ with the same Cauchy data on $S$ coincide near $x_0$ ; here $f$ need not be analytic. The proof is by duality: one solves the adjoint equation $P^{*} v = p$ for polynomials $p$ , with zero Cauchy data on a nearby analytic hypersurface, using the Cauchy–Kovalevskaya theorem. Integration by parts then shows that $\int u \, p = 0$ on a lens-shaped region for every polynomial $p$ , and the Weierstrass approximation theorem gives $u = 0$ there. Example: for the wave equation $\partial_t^2 u - \partial_x^2 u = 0$ , the line $t = 0$ is noncharacteristic, so a $C^2$ solution with $u(0, x) = \partial_t u(0, x) = 0$ vanishes near the line. The analyticity of the coefficients cannot simply be dropped: there are linear operators with smooth coefficients for which the Cauchy problem has nonzero solutions with zero data. Hörmander extended the theorem to distribution solutions and to $C^1$ hypersurfaces.

Going deeper

For a student

The theorem is a unique continuation result across noncharacteristic hypersurfaces. It applies to the Laplace, heat and wave equations, since their coefficients are constant. For the wave equation it is a step towards domain of dependence arguments. For the Laplace equation it shows that the Cauchy problem has at most one solution, even though it is not well posed. The noncharacteristic hypothesis matters: for the heat operator the line $t = 0$ is characteristic, and Tychonoff's example gives a nonzero solution of the heat equation with zero initial values.

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Tag	`1VVKF0Y19B`
URI	`m2://1VVKF0Y19B`
Kind	statement, theorem
Writing	W0 stub
MSC	35A02, 35A20
Status	This copy of the site has no build data. Run `m2 status 1VVKF0Y19B`.
About	Linear PDE
Uses	Linear PDE, Characteristic variety, Analytic function
Same as	Wikidata Q5883744 Wikipedia Holmgren's uniqueness theorem