Density and trace results in generalized fractal networks Serge Nicaise Université de Valenciennes et du Hainaut-Cambresis Laboratoire de Mathematiques et leurs Applications de Valenciennes, LAMAV joint work with Adrien Semin Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 1 / 25
Outline Introduction 1 p -adic trees and Sobolev spaces 2 3 Density results Trace results 4 Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 2 / 25
Introduction Main questions If T is an infinite p -adic tree and µ a weight function, we are interested in the two following questions: 1. Find NSC such that H 1 µ ( T ) = H 1 µ, 0 ( T ) ? H 1 µ, 0 ( T ) being the closure in H 1 µ ( T ) of compactly supported functions. 2. If H 1 µ ( T ) � = H 1 µ, 0 ( T ) , define a trace space (at infinity) of elements of H 1 µ ( T ) . For some particular trees and weights in the finite difference version, see B. Maury, D. Salort, and C. Vannier. Trace theorems for trees, application to the human lungs. Network and Heteregeneous Media , 4(3):469 – 500, 2009. Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 3 / 25
p -adic trees and Sobolev spaces p -adic trees Given p in N ∗ , we denote the following set of indexes in N 2 : � ( ℓ, j ) ∈ N 2 such that 0 ≤ j ≤ p ℓ − 1 � = , E p � ( ℓ, j ) ∈ N 2 such that ℓ ≥ 1 and 0 ≤ j ≤ p ℓ − 1 − 1 � = ( 0 , 0 ) ∪ . V p Definition T is a p-adic tree if there exists two families E = ( e ℓ, j ) ( ℓ, j ) ∈ E p (set of edges) and V = ( v ℓ, j ) ( ℓ, j ) ∈ V p (set of nodes) such that: each v ℓ, j is a point of R d , each e ℓ, j is a straight segment in R d of length L ℓ, j , whose extremities are v ℓ, ⌊ j / p ⌋ and v ℓ + 1 , j , ( ℓ, j ) � = ( ℓ ′ , j ′ ) ⇒ v ℓ, j � = v ℓ ′ , j ′ , ( ℓ, j ) � = ( ℓ ′ , j ′ ) ⇒ e ℓ, j ∩ e ℓ ′ , j ′ = ∅ . Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 4 / 25
p -adic trees and Sobolev spaces An example e 3 , 7 e 3 , 6 v 0 , 0 v 3 , 3 e 3 , 5 e 2 , 3 v 3 , 2 e 0 , 0 e 2 , 2 e 3 , 4 v 2 , 1 e 1 , 1 e 3 , 3 v 1 , 0 v 3 , 1 e 1 , 0 e 2 , 1 e 3 , 2 v 2 , 0 e 2 , 0 v 3 , 0 e 3 , 1 e 3 , 0 Figure: A dyadic tree. We circle nodes and we color edges in blue. Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 5 / 25
p -adic trees and Sobolev spaces Subtrees T ℓ = subtree of T made of the edges up to the ℓ -th generation. e 3 , 7 e 3 , 6 v 0 , 0 v 3 , 3 e 3 , 5 e 2 , 3 v 3 , 2 e 0 , 0 e 2 , 2 e 3 , 4 v 2 , 1 e 1 , 1 e 3 , 3 v 1 , 0 v 3 , 1 e 1 , 0 e 2 , 1 e 3 , 2 v 2 , 0 e 2 , 0 v 3 , 0 e 3 , 1 e 3 , 0 Figure: The dyadic subtree T 1 in red. Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 6 / 25
p -adic trees and Sobolev spaces Weight on a p -adic tree Definition Let us consider a p-adic tree T = ( V , E ) and a function µ : E p → R . One says that µ is a weight on T if and only if 0 < µ ℓ, j := µ ( ℓ, j ) < ∞ , ∀ ( ℓ, j ) ∈ E p . In this case, we denote the weighted p-adic tree T = ( V , E , µ ) . By abuse of notation, we also denote by µ the function from E to R defined by µ ( x ) = µ ℓ, j , ∀ x ∈ e ℓ, j . Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 7 / 25
p -adic trees and Sobolev spaces Weighted L 2 spaces Definition Let T = ( V , E , µ ) be a weighted tree. A function u : E → R will be in µ ( T ) if and only if µ | u | 2 ∈ L 1 ( E ) : L 2 � � � u � 2 µ ( x ) | u ( x ) | 2 d x := � µ ( x ) | u ( x ) | 2 d x < ∞ . µ ( T ) = L 2 T e ℓ, j ( ℓ, j ) ∈ E p Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 8 / 25
p -adic trees and Sobolev spaces Sobolev spaces Definition Let T = ( V , E , µ ) be a weighted tree. � µ, loc ( E ) ∩ C ( E ) / u ′ ∈ L 2 � u ∈ L 2 H 1 µ ( T ) = µ ( T ) . This space is an Hilbert space with associated norm � 2 + | u | 2 � u � 2 � u ′ � � � � µ ( T ) = � u ( v 0 , 0 ) µ ( T ) , | u | H 1 µ ( T ) = µ ( T ) . (1) H 1 H 1 � L 2 Rk. 1 ∈ H 1 µ ( T ) . Definition Let T = ( E , V , µ ) be a weighted tree. H 1 µ, c ( T ) = subset of functions u ∈ H 1 µ ( T ) whose support is compact, H 1 µ, 0 ( T ) the closure of H 1 µ, c ( T ) in H 1 µ ( T ) for the norm (1). Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 9 / 25
Density results The question On which condition over the triplet ( E , V , µ ) , H 1 µ ( T ) = H 1 µ, 0 ( T ) ? For some particular trees and weights in the finite difference version, see again B. Maury, D. Salort, and C. Vannier. Trace theorems for trees, application to the human lungs. Network and Heteregeneous Media , 4(3):469 – 500, 2009. Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 10 / 25
Density results A first implicit NSC Theorem (Thm 1) H 1 µ ( T ) = H 1 1 ∈ H 1 µ, 0 ( T ) ⇐ ⇒ µ, 0 ( T ) (2) Proof. ⇒ : trivial. ⇐ : By assumption, ∃ v n ∈ H 1 µ, c ( T ) s.t. v n → 1 in H 1 µ ( T ) . For any u ∈ H 1 µ ( T ) , we build up a sequence u n ∈ H 1 µ, c ( T ) (using v n and u ) s.t. u n → u in H 1 µ ( T ) . Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 11 / 25
Density results Auxiliary Dirichlet problems Find u D ∈ H 1 ( P D ) µ, 0 ( T ) such that u D ( v 0 , 0 ) = 1 and � µ ( x ) u ′ ∀ φ ∈ H 1 D ( x ) φ ′ ( x ) d x = 0 , µ, 0 ( T ) , φ ( v 0 , 0 ) = 0 . T For all n ∈ N , introduce the following spaces: � µ, c ( T ) such that supp u ⊂ T n � H 1 , n u ∈ H 1 µ, c ( T ) = , � µ, c ( T ) such that u ( v 0 , 0 ) = 0 and supp u ⊂ T n � H 1 , n u ∈ H 1 µ, c , 0 ( T ) = . Find u n ∈ H 1 , n µ, c ( T ) such that u n ( v 0 , 0 ) = 1 and ( P D,n ) � µ ( x )( u n ) ′ ( x ) φ ′ ( x ) d x = 0 , ∀ φ ∈ H 1 , n µ, c , 0 ( T ) . T Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 12 / 25
Density results A reformulation of the first NSC Proposition (Prop 2) We have the following equivalence 1 ∈ H 1 1 is solution of ( P D ) ⇐ ⇒ µ, 0 ( T ) (3) Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 13 / 25
Density results Some properties of u n u n → u D in H 1 µ ( T ) as n → ∞ . 0 ≤ u n ≤ 1 (maximum principle). ℓ, j ) ′ ≤ 0 , ∀ ℓ, j . ( u n | u n | 2 µ ( T ) = − µ 0 , 0 ( u n ) ′ ( v 0 , 0 ) . (4) H 1 Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 14 / 25
Density results Relation with the Liouville property Definition A weighted p-adic tree T = ( V , E , µ ) is called a Liouville network if and only if every bounded harmonic function on T is constant. For µ = 1, see J. von Below and J. A. Lubary. Harmonic functions on locally finite networks. Results Math. , 45(1-2):1–20, 2004. Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 15 / 25
Density results Proposition We have the following equivalence u D = 1 ⇐ ⇒ T is a Liouville network. (5) Proof. The implication ⇒ is direct, since u D is harmonic and bounded (since 0 ≤ u D ≤ 1). ⇐ : Fix a bounded harmonic function h on T . As the assumption is that u D = 1, by Prop. 2, this is equivalent to 1 ∈ H 1 µ, 0 ( T ) . Hence let us fix a sequence of functions ( v n ) n ∈ N ∈ H 1 µ, c ( T ) s. t. � 1 − v n � H 1 µ ( T ) → 0 n → ∞ . as Then the result is based on (consequence of Green’s formula) � T µ ( x )( h ′ ( x )) 2 v 2 � T µ ( x ) h ′ ( x ) v n ( x ) v ′ n ( x ) d x = − 2 n ( x ) h ( x ) d x + µ 0 , 0 h ′ ( v 0 , 0 ) v 2 n ( v 0 , 0 ) h ( v 0 , 0 ) . Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 16 / 25
Density results A third implicit NSC Theorem (Thm 3) One has H 1 µ ( T ) = H 1 lim n →∞ | u n | H 1 µ, 0 ( T ) ⇐ ⇒ µ ( T ) = 0 . Proof. ⇐ : Since | u n | H 1 µ ( T ) → 0 and u n ( v 0 , 0 ) = 1: lim n →∞ � u n − 1 � H 1 µ ( T ) = 0 . As u n ∈ H 1 µ, c ( T ) , one gets that 1 ∈ H 1 µ, 0 ( T ) , and H 1 µ ( T ) = H 1 µ, 0 ( T ) by using Thm 1. ⇒ : We use a contradiction argument and Prop 2. Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 17 / 25
Density results An electrical problem d x � On each edge e ℓ, j , we introduce the resistance R ℓ, j = µ ( x ) , and the e ℓ, j new unknowns U n ℓ, j = u n ( v ℓ + 1 , j ) − u n ( v ℓ, ⌊ p − 1 j ⌋ ) , I n ℓ, j = µ ℓ, j ( u n ) ′ ( v ℓ + 1 , j ) . This new set of unknowns ( U n ℓ, j , I n ℓ, j ) allows us to re-write ( P D,n ) in the following equivalent form: on each edge e ℓ, j , I n ℓ, j is constant and U n ℓ, j = R ℓ, j I ℓ, j , for any j ∈ { 0 , . . . , p n − 1 } , we have � n ℓ = 0 U n ℓ, ⌊ p ℓ − n j ⌋ = − 1 , ℓ, j = � p − 1 for any 0 ≤ ℓ ≤ n − 1, we have I n k = 0 I n ℓ + 1 , pj + k (Kirchoff law). We have actually rewritten problem ( P D,n ) as a general electrical problem. If R n is the equivalent resistance of the finite tree T n , Ohm law ⇒ − 1 = R n I n 0 , 0 . The definition of I n 0 , 0 and relation (4) ⇒ µ ( T ) = ( R n ) − 1 . | u n | 2 (6) H 1 Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 18 / 25
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