Holmgren's uniqueness theorem

We will use the multi-index notation:

Let $\backslash alpha=\backslash \{\backslash alpha\_1,\backslash dots,\backslash alpha\_n\backslash \}\backslash in\; \backslash N\_0^n,$,

with $\backslash N\_0$ standing for the nonnegative integers;

denote $|\backslash alpha|=\backslash alpha\_1+\backslash cdots+\backslash alpha\_n$ and

: $\backslash partial\_x^\backslash alpha\; =\; \backslash left(\backslash frac\{\backslash partial\}\{\backslash partial\; x\_1\}\backslash right)^\{\backslash alpha\_1\}\; \backslash cdots\; \backslash left(\backslash frac\{\backslash partial\}\{\backslash partial\; x\_n\}\backslash right)^\{\backslash alpha\_n\}$.

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

:Assume that P = ∑_|α| ≤m A_α(x)∂{{su|p=α|b=x}} 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:{{cite book|mr=2528466|last=Stroock|first = W.|chapter=Weyl's lemma, one of many|title=Groups and analysis|pages=164–173|series=London Math. Soc. Lecture Note Ser.|volume=354|publisher=Cambridge Univ. Press|location=Cambridge|year=2008}}

: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.

Let $\backslash Omega$ be a connected open neighborhood in $\backslash R^n$, and let $\backslash Sigma$ be an analytic hypersurface in $\backslash Omega$, such that there are two open subsets $\backslash Omega\_\{+\}$ and $\backslash Omega\_\{-\}$ in $\backslash Omega$, nonempty and connected, not intersecting $\backslash Sigma$ nor each other, such that $\backslash Omega=\backslash Omega\_\{-\}\backslash cup\backslash Sigma\backslash cup\backslash Omega\_\{+\}$.

Let $P=\backslash sum\_$

the principal symbol of $P$.

$N^*\backslash Sigma$ is a conormal bundle to $\backslash Sigma$, defined as

$N^*\backslash Sigma=\backslash \{(x,\backslash xi)\backslash in\; T^*\backslash R^n:x\backslash in\backslash Sigma,\backslash ,\backslash xi|\_\{T\_x\backslash Sigma\}=0\backslash \}$.

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

:Holmgren's theorem

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

"Introduction to pseudodifferential and Fourier integral operators", vol. 1, Plenum Press, New York, 1980.

Consider the problem

:$\backslash partial\_t^m\; u=F(t,x,\backslash partial\_x^\backslash alpha\backslash ,\backslash partial\_t^k\; u),$

\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

:$\backslash partial\_t^k\; u|\_\{t=0\}=\backslash phi\_k(x),\; \backslash qquad\; 0\backslash le\; k\backslash le\; m-1,$

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 $\backslash phi\_k(x)$ 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 $(t,x)=(0,0)\backslash in(\backslash R\backslash times\backslash R^n)$.

Note that the Cauchy–Kowalevski theorem does not exclude the existence of solutions which are not real-analytic.{{Citation needed|date=December 2021}}

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

:$\backslash partial\_t^m\; u\; =\; F(t,x,\backslash partial\_x^\backslash alpha\backslash ,\backslash partial\_t^k\; u)$

= \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.

• Cauchy–Kowalevski theorem
• FBI transform

Category:Partial differential equations

Category:Theorems in mathematical analysis

Category:Uniqueness theorems