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 α = { α 1 , … , α n } ∈ N 0 n , {\displaystyle \alpha =\{\alpha _{1},\dots ,\alpha _{n}\}\in \mathbb {N} _{0}^{n},} , with N 0 {\displaystyle \mathbb {N} _{0}} standing for the nonnegative integers; denote | α | = α 1 + ⋯ + α n {\displaystyle |\alpha |=\alpha _{1}+\cdots +\alpha _{n}} and
∂ x α = ( ∂ ∂ x 1 ) α 1 ⋯ ( ∂ ∂ x n ) α n {\displaystyle \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 = ∑|α| ≤m Aα(x)∂αx 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 {\displaystyle \Omega } be a connected open neighborhood in R n {\displaystyle \mathbb {R} ^{n}} , and let Σ {\displaystyle \Sigma } be an analytic hypersurface in Ω {\displaystyle \Omega } , such that there are two open subsets Ω + {\displaystyle \Omega _{+}} and Ω − {\displaystyle \Omega _{-}} in Ω {\displaystyle \Omega } , nonempty and connected, not intersecting Σ {\displaystyle \Sigma } nor each other, such that Ω = Ω − ∪ Σ ∪ Ω + {\displaystyle \Omega =\Omega _{-}\cup \Sigma \cup \Omega _{+}} . Let P = ∑ | α | ≤ m A α ( x ) ∂ x α {\displaystyle P=\sum _{|\alpha |\leq m}A_{\alpha }(x)\partial _{x}^{\alpha }} be a differential operator with real-analytic coefficients. Assume that the hypersurface Σ {\displaystyle \Sigma } is noncharacteristic with respect to P {\displaystyle P} at every one of its points:
C h a r P ∩ N ∗ Σ = ∅ {\displaystyle \mathop {\rm {Char}} P\cap N^{*}\Sigma =\emptyset } . 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 {\displaystyle \mathop {\rm {Char}} P=\{(x,\xi )\subset T^{*}\mathbb {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 {\displaystyle P} .
N ∗ Σ {\displaystyle N^{*}\Sigma } is a conormal bundle to Σ {\displaystyle \Sigma } , defined as
N ∗ Σ = { ( x , ξ ) ∈ T ∗ R n : x ∈ Σ , ξ | T x Σ = 0 } {\displaystyle N^{*}\Sigma =\{(x,\xi )\in T^{*}\mathbb {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 {\displaystyle u} be a distribution in Ω {\displaystyle \Omega } such that P u = 0 {\displaystyle Pu=0} in Ω {\displaystyle \Omega } . If u {\displaystyle u} vanishes in Ω − {\displaystyle \Omega _{-}} , then it vanishes in an open neighborhood of Σ {\displaystyle \Sigma } .
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 , {\displaystyle \partial _{t}^{m}u=F(t,x,\partial _{x}^{\alpha }\,\partial _{t}^{k}u),\quad \alpha \in \mathbb {N} _{0}^{n},\quad k\in \mathbb {N} _{0},\quad |\alpha |+k\leq m,\quad k\leq m-1,}
with the Cauchy data
∂ t k u | t = 0 = ϕ k ( x ) , 0 ≤ k ≤ m − 1 , {\displaystyle \partial _{t}^{k}u|_{t=0}=\phi _{k}(x),\qquad 0\leq k\leq m-1,}
Assume that F ( t , x , z ) {\displaystyle F(t,x,z)} is real-analytic with respect to all its arguments in the neighborhood of t = 0 , x = 0 , z = 0 {\displaystyle t=0,x=0,z=0}
and that ϕ k ( x ) {\displaystyle \phi _{k}(x)} are real-analytic in the neighborhood of x = 0 {\displaystyle x=0} .
Theorem (Cauchy–Kowalevski) There is a unique real-analytic solution u ( t , x ) {\displaystyle u(t,x)} in the neighborhood of ( t , x ) = ( 0 , 0 ) ∈ ( R × R n ) {\displaystyle (t,x)=(0,0)\in (\mathbb {R} \times \mathbb {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 ) {\displaystyle F(t,x,z)} is polynomial of order one in z {\displaystyle 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 , {\displaystyle \partial _{t}^{m}u=F(t,x,\partial _{x}^{\alpha }\,\partial _{t}^{k}u)=\sum _{\alpha \in \mathbb {N} _{0}^{n},0\leq k\leq m-1,|\alpha |+k\leq m}A_{\alpha ,k}(t,x)\,\partial _{x}^{\alpha }\,\partial _{t}^{k}u,}
Holmgren's theorem states that the solution u {\displaystyle u} is real-analytic and hence, by the Cauchy–Kowalevski theorem, is unique.
See also Cauchy–Kowalevski theorem FBI transform
References