Holmgren's uniqueness theorem

Simple form of Holmgren's theorem

We will use the multi-index notation: Let α = { α 1 , … , α n } ∈ N 0 n , , with N 0 standing for the nonnegative integers; denote | α | = α 1 + ⋯ + α n and

∂ x α = ( ∂ ∂ x 1 ) α 1 ⋯ ( ∂ ∂ x n ) α n .

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

Assume that P = ∑_|α| ≤m A_α(x)∂α
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:

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 Ω be a connected open neighborhood in R n , and let Σ be an analytic hypersurface in Ω , such that there are two open subsets Ω + and Ω − in Ω , nonempty and connected, not intersecting Σ nor each other, such that Ω = Ω − ∪ Σ ∪ Ω + .

Let P = ∑ | α | ≤ m A α ( x ) ∂ x α be a differential operator with real-analytic coefficients.

Assume that the hypersurface Σ is noncharacteristic with respect to P at every one of its points:

C h a r P ∩ N ∗ Σ = ∅ .

Above,

C h a r P = { ( x , ξ ) ⊂ T ∗ R n ∖ 0 : σ p ( P ) ( x , ξ ) = 0 } ,  with  σ p ( x , ξ ) = ∑ | α | = m i | α | A α ( x ) ξ α

the principal symbol of P . N ∗ Σ is a conormal bundle to Σ , defined as N ∗ Σ = { ( x , ξ ) ∈ T ∗ R n : x ∈ Σ , ξ | T x Σ = 0 } .

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

Holmgren's theorem Let u be a distribution in Ω such that P u = 0 in Ω . If u vanishes in Ω − , then it vanishes in an open neighborhood of Σ .

Relation to the Cauchy–Kowalevski theorem

Consider the problem

∂ t m u = F ( t , x , ∂ x α ∂ t k u ) , α ∈ N 0 n , k ∈ N 0 , | α | + k ≤ m , k ≤ m − 1 ,

with the Cauchy data

∂ t k u | t = 0 = ϕ k ( x ) , 0 ≤ k ≤ 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 ϕ 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 ) ∈ ( R × 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

∂ t m u = F ( t , x , ∂ x α ∂ t k u ) = ∑ α ∈ N 0 n , 0 ≤ k ≤ m − 1 , | α | + k ≤ m A α , k ( t , x ) ∂ x α ∂ t k u ,

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