Английская Википедия:Holmgren's uniqueness theorem

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Шаблон: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 <math>\alpha=\{\alpha_1,\dots,\alpha_n\}\in \N_0^n,</math>, with <math>\N_0</math> standing for the nonnegative integers; denote <math>|\alpha|=\alpha_1+\cdots+\alpha_n</math> and

<math>\partial_x^\alpha = \left(\frac{\partial}{\partial x_1}\right)^{\alpha_1} \cdots \left(\frac{\partial}{\partial x_n}\right)^{\alpha_n}</math>.

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

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

Let <math>P=\sum_{|\alpha|\le m}A_\alpha(x)\partial_x^\alpha</math> be a differential operator with real-analytic coefficients.

Assume that the hypersurface <math>\Sigma</math> is noncharacteristic with respect to <math>P</math> at every one of its points:

<math>\mathop{\rm Char}P\cap N^*\Sigma=\emptyset</math>.

Above,

<math>\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</math>

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

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

Holmgren's theorem

Let <math>u</math> be a distribution in <math>\Omega</math> such that <math>Pu=0</math> in <math>\Omega</math>. If <math>u</math> vanishes in <math>\Omega_{-}</math>, then it vanishes in an open neighborhood of <math>\Sigma</math>.

Relation to the Cauchy–Kowalevski theorem

Consider the problem

<math>\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,</math>

with the Cauchy data

<math>\partial_t^k u|_{t=0}=\phi_k(x), \qquad 0\le k\le m-1,</math>

Assume that <math>F(t,x,z)</math> is real-analytic with respect to all its arguments in the neighborhood of <math>t=0,x=0,z=0</math> and that <math>\phi_k(x)</math> are real-analytic in the neighborhood of <math>x=0</math>.

Theorem (Cauchy–Kowalevski)

There is a unique real-analytic solution <math>u(t,x)</math> in the neighborhood of <math>(t,x)=(0,0)\in(\R\times\R^n)</math>.

Note that the Cauchy–Kowalevski theorem does not exclude the existence of solutions which are not real-analytic.Шаблон:Citation needed

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

<math>\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,</math>

Holmgren's theorem states that the solution <math>u</math> is real-analytic and hence, by the Cauchy–Kowalevski theorem, is unique.

See also

• Cauchy–Kowalevski theorem
• FBI transform