An approximation theorem of Runge type for kernels of certain - PowerPoint PPT Presentation

An approximation theorem of Runge type for kernels of certain non-elliptic partial differential operators Thomas Kalmes TU Chemnitz Workshop on Fourier Analysis and Partial Differential Equations Ferrara, September 10-11, 2018 Thomas Kalmes An approximation theorem of Runge type 1 / 15

Introduction 1 A general approximation theorem for kernels of differential operators 2 A Runge type approximation theorem for certain non-elliptic differential 3 operators Thomas Kalmes An approximation theorem of Runge type 2 / 15

Introduction Thomas Kalmes An approximation theorem of Runge type 3 / 15

Runge’s Approximation Theorem For X 1 ⊆ X 2 ⊆ C open the following are equivalent. i) For every g ∈ H ( X 1 ) , for every compact K ⊆ X 1 , and for every ε > 0 there is f ∈ H ( X 2 ) such that ε > sup | f ( z ) − g ( z ) | =: � f − g � 0 ,K , z ∈ K i.e. r : H ( X 2 ) → H ( X 1 ) , f �→ f | X 1 has dense range when H ( X 1 ) is equipped with the compact-open topology. ii) X 2 does not contain a compact connected component of C \ X 1 . Thomas Kalmes An approximation theorem of Runge type 4 / 15

For X ⊆ C = R 2 we have H ( X ) = { f ∈ C ∞ ( X ); 1 � � ∂ 1 + i∂ 2 f = 0 } . 2 Thomas Kalmes An approximation theorem of Runge type 5 / 15

For X ⊆ C = R 2 we have H ( X ) = { f ∈ C ∞ ( X ); 1 � � ∂ 1 + i∂ 2 f = 0 } . 2 X ⊆ R d open, E ( X ) := C ∞ ( X ) equipped with its natural locally convex topology which is generated by the seminorms | ∂ α f ( x ) | ( f ∈ C ∞ ( X )) ∀ K ⊆ X compact , l ∈ N 0 : � f � l,K := sup sup x ∈ K | α |≤ l and which makes E ( X ) a Fr´ echet space. Thomas Kalmes An approximation theorem of Runge type 5 / 15

For X ⊆ C = R 2 we have H ( X ) = { f ∈ C ∞ ( X ); 1 � � ∂ 1 + i∂ 2 f = 0 } . 2 X ⊆ R d open, E ( X ) := C ∞ ( X ) equipped with its natural locally convex topology which is generated by the seminorms | ∂ α f ( x ) | ( f ∈ C ∞ ( X )) ∀ K ⊆ X compact , l ∈ N 0 : � f � l,K := sup sup x ∈ K | α |≤ l and which makes E ( X ) a Fr´ echet space. For P ∈ C [ X 1 , . . . , X d ] non-constant E P ( X ) := { f ∈ C ∞ ( X ); P ( ∂ ) f = 0 in X } . Thomas Kalmes An approximation theorem of Runge type 5 / 15

For X ⊆ C = R 2 we have H ( X ) = { f ∈ C ∞ ( X ); 1 � � ∂ 1 + i∂ 2 f = 0 } . 2 X ⊆ R d open, E ( X ) := C ∞ ( X ) equipped with its natural locally convex topology which is generated by the seminorms | ∂ α f ( x ) | ( f ∈ C ∞ ( X )) ∀ K ⊆ X compact , l ∈ N 0 : � f � l,K := sup sup x ∈ K | α |≤ l and which makes E ( X ) a Fr´ echet space. For P ∈ C [ X 1 , . . . , X d ] non-constant E P ( X ) := { f ∈ C ∞ ( X ); P ( ∂ ) f = 0 in X } . E P ( X ) is a closed subspace of E ( X ) , hence a Fr´ echet space, too. In the sequel E P ( X ) is always equipped with the subspace topology. Thomas Kalmes An approximation theorem of Runge type 5 / 15

For X ⊆ C = R 2 we have H ( X ) = { f ∈ C ∞ ( X ); 1 � � ∂ 1 + i∂ 2 f = 0 } . 2 X ⊆ R d open, E ( X ) := C ∞ ( X ) equipped with its natural locally convex topology which is generated by the seminorms | ∂ α f ( x ) | ( f ∈ C ∞ ( X )) ∀ K ⊆ X compact , l ∈ N 0 : � f � l,K := sup sup x ∈ K | α |≤ l and which makes E ( X ) a Fr´ echet space. For P ∈ C [ X 1 , . . . , X d ] non-constant E P ( X ) := { f ∈ C ∞ ( X ); P ( ∂ ) f = 0 in X } . E P ( X ) is a closed subspace of E ( X ) , hence a Fr´ echet space, too. In the sequel E P ( X ) is always equipped with the subspace topology. D ′ ( X ) equipped with the strong dual topology, D ′ P ( X ) := { u ∈ D ′ ( X ); P ( ∂ ) u = 0 } equipped with subspace topology. Thomas Kalmes An approximation theorem of Runge type 5 / 15

For X ⊆ C = R 2 we have H ( X ) = { f ∈ C ∞ ( X ); 1 � � ∂ 1 + i∂ 2 f = 0 } . 2 X ⊆ R d open, E ( X ) := C ∞ ( X ) equipped with its natural locally convex topology which is generated by the seminorms | ∂ α f ( x ) | ( f ∈ C ∞ ( X )) ∀ K ⊆ X compact , l ∈ N 0 : � f � l,K := sup sup x ∈ K | α |≤ l and which makes E ( X ) a Fr´ echet space. For P ∈ C [ X 1 , . . . , X d ] non-constant E P ( X ) := { f ∈ C ∞ ( X ); P ( ∂ ) f = 0 in X } . E P ( X ) is a closed subspace of E ( X ) , hence a Fr´ echet space, too. In the sequel E P ( X ) is always equipped with the subspace topology. D ′ ( X ) equipped with the strong dual topology, D ′ P ( X ) := { u ∈ D ′ ( X ); P ( ∂ ) u = 0 } equipped with subspace topology. P hypoelliptic : ⇔ ∀ X open ∀ u ∈ D ′ ( X ) : P ( ∂ ) u = 0 ⇒ u ∈ C ∞ ( X ) � � Then E P ( X ) = D ′ P ( X ) as locally convex spaces and therefore: topology of E P ( X ) is generated by the seminorms {� · � 0 ,K ; K ⊆ X compact } . Thomas Kalmes An approximation theorem of Runge type 5 / 15

For X ⊆ C = R 2 we have H ( X ) = { f ∈ C ∞ ( X ); 1 � � ∂ 1 + i∂ 2 f = 0 } . 2 X ⊆ R d open, E ( X ) := C ∞ ( X ) equipped with its natural locally convex topology which is generated by the seminorms | ∂ α f ( x ) | ( f ∈ C ∞ ( X )) ∀ K ⊆ X compact , l ∈ N 0 : � f � l,K := sup sup x ∈ K | α |≤ l and which makes E ( X ) a Fr´ echet space. For P ∈ C [ X 1 , . . . , X d ] non-constant E P ( X ) := { f ∈ C ∞ ( X ); P ( ∂ ) f = 0 in X } . E P ( X ) is a closed subspace of E ( X ) , hence a Fr´ echet space, too. In the sequel E P ( X ) is always equipped with the subspace topology. D ′ ( X ) equipped with the strong dual topology, D ′ P ( X ) := { u ∈ D ′ ( X ); P ( ∂ ) u = 0 } equipped with subspace topology. P hypoelliptic : ⇔ ∀ X open ∀ u ∈ D ′ ( X ) : P ( ∂ ) u = 0 ⇒ u ∈ C ∞ ( X ) � � Then E P ( X ) = D ′ P ( X ) as locally convex spaces and therefore: topology of E P ( X ) is generated by the seminorms {� · � 0 ,K ; K ⊆ X compact } . | α |≤ m a α ξ α elliptic : ⇔ ∀ ξ ∈ R d \{ 0 } : 0 � = P m ( ξ ) := � | α | = m a α ξ α P ( ξ ) = � P elliptic ⇒ P hypoelliptic Thomas Kalmes An approximation theorem of Runge type 5 / 15

Lax-Malgrange Theorem ([4], [5]) For X 1 ⊆ X 2 ⊆ R d open and P elliptic the following are equivalent. i) The restriction map r E : E P ( X 2 ) → E P ( X 1 ) , f �→ f | X 1 has dense range. ii) X 2 does not contain a compact connected component of R d \ X 1 . Thomas Kalmes An approximation theorem of Runge type 6 / 15

Lax-Malgrange Theorem ([4], [5]) For X 1 ⊆ X 2 ⊆ R d open and P elliptic the following are equivalent. i) The restriction map r E : E P ( X 2 ) → E P ( X 1 ) , f �→ f | X 1 has dense range. ii) X 2 does not contain a compact connected component of R d \ X 1 . d = 2 , P ( ξ 1 , ξ 2 ) = 1 2 ( ξ 1 + iξ 2 ) gives Runge’s Approximation Theorem. Thomas Kalmes An approximation theorem of Runge type 6 / 15

Lax-Malgrange Theorem ([4], [5]) For X 1 ⊆ X 2 ⊆ R d open and P elliptic the following are equivalent. i) The restriction map r E : E P ( X 2 ) → E P ( X 1 ) , f �→ f | X 1 has dense range. ii) X 2 does not contain a compact connected component of R d \ X 1 . d = 2 , P ( ξ 1 , ξ 2 ) = 1 2 ( ξ 1 + iξ 2 ) gives Runge’s Approximation Theorem. Objective: Given P non-constant, find conditions ensuring that the restriction map r E : E P ( X 2 ) → E P ( X 1 ) , f �→ f | X 1 resp. r D ′ : D ′ P ( X 2 ) → D ′ P ( X 1 ) , u �→ u | X 1 has dense range. Thomas Kalmes An approximation theorem of Runge type 6 / 15

A general approximation theorem for kernels of differential operators Thomas Kalmes An approximation theorem of Runge type 7 / 15

Given P � = 0 , X ⊆ R d open. Recall that the following are equivalent i) P ( ∂ ) : E ( X ) → E ( X ) is surjective. ii) X is P -convex for supports, i.e. ∀ u ∈ E ′ ( X ) : dist ( supp ˇ P ( ∂ ) u, R d \ X ) = dist ( supp u, R d \ X ) where ˇ P ( ξ ) := P ( − ξ ) . Thomas Kalmes An approximation theorem of Runge type 8 / 15

Given P � = 0 , X ⊆ R d open. Recall that the following are equivalent i) P ( ∂ ) : E ( X ) → E ( X ) is surjective. ii) X is P -convex for supports, i.e. ∀ u ∈ E ′ ( X ) : dist ( supp ˇ P ( ∂ ) u, R d \ X ) = dist ( supp u, R d \ X ) where ˇ P ( ξ ) := P ( − ξ ) . If P is elliptic, every open X ⊆ R d is P -convex for supports. Thomas Kalmes An approximation theorem of Runge type 8 / 15

Given P � = 0 , X ⊆ R d open. Recall that the following are equivalent i) P ( ∂ ) : E ( X ) → E ( X ) is surjective. ii) X is P -convex for supports, i.e. ∀ u ∈ E ′ ( X ) : dist ( supp ˇ P ( ∂ ) u, R d \ X ) = dist ( supp u, R d \ X ) where ˇ P ( ξ ) := P ( − ξ ) . If P is elliptic, every open X ⊆ R d is P -convex for supports. f : X → R satisfies the minimum principle in a closed subset H of R d if for every compact set K ⊆ H ∩ X we have x ∈ K f ( x ) = inf inf ∂ H K f ( x ) , where ∂ H K denotes the boundary of K in H . Thomas Kalmes An approximation theorem of Runge type 8 / 15