Density and trace results in generalized fractal networks Serge - - PowerPoint PPT Presentation

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

1

Introduction

2

p-adic trees and Sobolev spaces

3

Density results

4

Trace results

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 H1

µ(T ) = H1 µ,0(T )?

H1

µ,0(T ) being the closure in H1 µ(T ) of compactly supported functions.

• 2. If H1

µ(T ) = H1 µ,0(T ), define a trace space (at infinity) of elements of

H1

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 N2: Ep =

• (ℓ, j) ∈ N2 such that 0 ≤ j ≤ pℓ − 1
• ,

Vp = (0, 0) ∪

• (ℓ, j) ∈ N2 such that ℓ ≥ 1 and 0 ≤ j ≤ pℓ−1 − 1
• .

Definition T is a p-adic tree if there exists two families E = (eℓ,j)(ℓ,j)∈Ep (set of edges) and V = (vℓ,j)(ℓ,j)∈Vp (set of nodes) such that: each vℓ,j is a point of Rd, each eℓ,j is a straight segment in Rd of length Lℓ,j, whose extremities are vℓ,⌊j/p⌋ and vℓ+1,j, (ℓ, j) = (ℓ′, j′) ⇒ vℓ,j = vℓ′,j′, (ℓ, j) = (ℓ′, j′) ⇒ eℓ,j ∩ eℓ′,j′ = ∅.

Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 4 / 25

p-adic trees and Sobolev spaces

An example

e0,0 e3,6 e3,7 e3,5 e3,4 e2,0 e3,1 e2,1 e3,0 e3,2 e3,3 e2,3 e1,1 e1,0 e2,2 v0,0 v2,0 v3,0 v3,1 v3,2 v3,3 v1,0 v2,1

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.

e0,0 e3,6 e3,7 e3,5 e3,4 e2,0 e3,1 e2,1 e3,0 e3,2 e3,3 e2,3 e1,1 e2,2 e1,0 v0,0 v3,0 v3,1 v3,2 v3,3 v1,0 v2,1 v2,0

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 : Ep → R. One says that µ is a weight on T if and only if 0 < µℓ,j := µ(ℓ, j) < ∞, ∀(ℓ, j) ∈ Ep. 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) = µℓ,j, ∀x ∈ eℓ,j.

Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 7 / 25

p-adic trees and Sobolev spaces

Weighted L2 spaces

Definition Let T = (V, E, µ) be a weighted tree. A function u : E → R will be in L2

µ(T ) if and only if µ|u|2 ∈ L1(E):

u2

L2

µ(T ) =

• T

µ(x)|u(x)|2dx :=

• (ℓ,j)∈Ep
• eℓ,j

µ(x)|u(x)|2dx < ∞.

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. H1

µ(T ) =

• u ∈ L2

µ,loc(E) ∩ C(E) / u′ ∈ L2 µ(T )

• .

This space is an Hilbert space with associated norm u2

H1

µ(T ) =

• u(v0,0)
• 2 + |u|2

H1

µ(T ),

|u|H1

µ(T ) =

• u′
• L2

µ(T ) .

(1)

• Rk. 1 ∈ H1

µ(T ).

Definition Let T = (E, V, µ) be a weighted tree. H1

µ,c(T ) = subset of functions u ∈ H1 µ(T ) whose support is

compact, H1

µ,0(T ) the closure of H1 µ,c(T ) in H1 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), H1

µ(T ) = H1 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) H1

µ(T ) = H1 µ,0(T )

⇐ ⇒ 1 ∈ H1

µ,0(T )

(2) Proof. ⇒: trivial. ⇐: By assumption, ∃vn ∈ H1

µ,c(T ) s.t. vn → 1 in H1 µ(T ).

For any u ∈ H1

µ(T ), we build up a sequence un ∈ H1 µ,c(T ) (using vn

and u) s.t. un → u in H1

µ(T ).

Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 11 / 25

Density results

Auxiliary Dirichlet problems

(PD) Find uD ∈ H1

µ,0(T ) such that uD(v0,0) = 1 and

• T

µ(x)u′

D(x)φ′(x)dx = 0,

∀φ ∈ H1

µ,0(T ), φ(v0,0) = 0.

For all n ∈ N, introduce the following spaces: H1,n

µ,c(T ) =

• u ∈ H1

µ,c(T ) such that supp u ⊂ T n

, H1,n

µ,c,0(T ) =

• u ∈ H1

µ,c(T ) such that u(v0,0) = 0 and supp u ⊂ T n

. (PD,n) Find un ∈ H1,n

µ,c(T ) such that un(v0,0) = 1 and

• T

µ(x)(un)′(x)φ′(x)dx = 0, ∀φ ∈ H1,n

µ,c,0(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 is solution of (PD) ⇐ ⇒ 1 ∈ H1

µ,0(T )

(3)

Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 13 / 25

Density results

Some properties of un

un → uD in H1

µ(T ) as n → ∞.

0 ≤ un ≤ 1 (maximum principle). (un

ℓ,j)′ ≤ 0, ∀ℓ, j.

|un|2

H1

µ(T ) = −µ0,0(un)′(v0,0).

(4)

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

• nly 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 uD = 1 ⇐ ⇒ T is a Liouville network. (5) Proof. The implication ⇒ is direct, since uD is harmonic and bounded (since 0 ≤ uD ≤ 1). ⇐: Fix a bounded harmonic function h on T . As the assumption is that uD = 1, by Prop. 2, this is equivalent to 1 ∈ H1

µ,0(T ). Hence let us fix a

sequence of functions (vn)n∈N ∈ H1

µ,c(T ) s. t.

1 − vnH1

µ(T ) → 0

as n → ∞. Then the result is based on (consequence of Green’s formula)

• T µ(x)(h′(x))2v2

n(x)dx = −2

• T µ(x)h′(x)vn(x)v′

n(x)h(x)dx

+µ0,0h′(v0,0)v2

n(v0,0)h(v0,0).

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 H1

µ(T ) = H1 µ,0(T )

⇐ ⇒ limn→∞ |un|H1

µ(T ) = 0.

Proof. ⇐: Since |un|H1

µ(T ) → 0 and un(v0,0) = 1:

limn→∞ un − 1H1

µ(T ) = 0.

As un ∈ H1

µ,c(T ), one gets that 1 ∈ H1 µ,0(T ), and H1 µ(T ) = H1 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

On each edge eℓ,j, we introduce the resistance Rℓ,j =

• eℓ,j

dx µ(x), and the

new unknowns Un

ℓ,j = un(vℓ+1,j) − un(vℓ,⌊p−1j⌋), In ℓ,j = µℓ,j(un)′(vℓ+1,j).

This new set of unknowns (Un

ℓ,j, In ℓ,j) allows us to re-write (PD,n) in the

following equivalent form:

• n each edge eℓ,j, In

ℓ,j is constant and Un ℓ,j = Rℓ,jIℓ,j,

for any j ∈ {0, . . . , pn − 1}, we have n

ℓ=0 Un ℓ,⌊pℓ−nj⌋ = −1,

for any 0 ≤ ℓ ≤ n − 1, we have In

ℓ,j = p−1 k=0 In ℓ+1,pj+k (Kirchoff law).

We have actually rewritten problem (PD,n) as a general electrical problem. If Rn is the equivalent resistance of the finite tree T n, Ohm law ⇒ − 1 = RnIn

0,0.

The definition of In

0,0 and relation (4) ⇒

|un|2

H1

µ(T ) = (Rn)−1 .

(6)

Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 18 / 25

Density results

An explicit NSC

Theorem H1

µ(T ) = H1 µ,0(T )

⇐ ⇒ limn→∞ Rn = +∞. Proof. We use Thm 3 and the relation (6). Proposition (Equivalent resistance) Rn = R0,0 +    p−1

j1=0

 R1,j1 +

• p−1

j2=0

• R2,pj1+j2 + . . .

p−1

jn=0 R−1 n,n

k=1 pn−kjk

−1−1−1 Proof. Induction on n, all resistances of the last generation are in parallel.

Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 19 / 25

Density results

An example

Theorem Let us assume that the ratios µℓ,j/Lℓ,j are constant and equal on the same generation, i. e., for all ℓ ∈ N there exists νℓ > 0 s. t. µℓ,j/Lℓ,j = νℓ, ∀j = 0, · · · , pℓ − 1. Then H1

µ(T ) = H1 µ,0(T )

⇐ ⇒ ∞

ℓ=0 p−ℓν−1 ℓ

diverges. Rk In the case p = 2, we obtain the continuous version of Theorem 2.12 of

• 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 20 / 25

Trace results

A recursive partition

Definition We call p-recursive partition of the unit interval a sequence of real numbers γn

p,j ∈ [0, 1] defined for n ≥ 1 and 0 ≤ j ≤ pn−1 such that

γn

p,0 = 0, γn p,pn−1 = 1 for any n ≥ 1,

γn

p,j ≤ γn p,j+1, for all 0 ≤ j < pn−1,

γn+1

p,pj = γn p,j for any n ≥ 1 and for any 0 ≤ j ≤ pn−1.

Rk • the intervals ]γn

p,j, γn p,j+1[, 0 ≤ j < pn−1: subdivision of ]0, 1[,

• the intervals ]γn+1

p,pj+k, γn+1 p,pj+k+1[, 0 ≤ k < p: subdivision of ]γn p,j, γn p,j+1[.

γ1

2,0 = γ2 2,0

γ1

2,1 = γ2 2,2

γ2

2,1

Figure: Example of a 2-recursive partition

Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 21 / 25

Trace results

Trace operators

Definition Given γ = (γn

p,j)n≥1,0≤j≤pn−1 a p-recursive partition of the unit interval,

we can define the following trace operators: the n-th trace operator T n

γ : for u ∈ H1 µ(T ), we define

T n

γ (u) ∈ L2(]0, 1[) as

T n

γ (u)(x) = u(vn,j),

x ∈

• γn

p,j, γn p,j+1

• ,

(7) the trace operator T ∞

γ (u) defined as the limit of T n γ (u) in L2(]0, 1[),

as n goes to infinity (if it exists). Rk For any choose of γ, T ∞

γ (1) = 1.

Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 22 / 25

Trace results

A negative result

Proposition If 1 ∈ H1

µ,0(T ), then there exists no p-recursive partition γ of the unit

interval such that T ∞

γ

is continuous from H1

µ(T ) to L2(]0, 1[).

Proof. Let γ be any p-recursive partition of the unit interval. By hypothesis, there exists a sequence (un) ∈ H1

µ,c(T ) such that

1 − unH1

µ(T ) → 0,

as n → ∞. As for a given n, T ∞

γ (1 − un) = 1,

• T ∞

γ (1 − un)

• L2(]0,1[) = 1 → 0,

as n → ∞.

Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 23 / 25

Trace results

A trace theorem

Theorem Given a weighted p-adic tree T = (V, E, µ), and assume that R = limn→∞ Rn < ∞. Then ∃ a p-recursive partition γ of the unit interval such that T ∞

γ

is continuous from H1

µ(T ) to L2(]0, 1[).

Proof. Based on the identity

• T n µ(x)u′

D(x)(u2)′(x)dx =

pn−1

j=0 µn,ju′ D,n,j|u(vn+1,j)|2 − µ0,0u′ D(v0,0)|u(v0,0)|2.

consequence of Green’s formula. Then we take ℓn

p,j = −Rµn−1,ju′ D,n−1,j, γn p,0 = 0, γn p,j =

• k<j

ℓn

p,k, ∀j ≥ 1.

Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 24 / 25

Trace results Serge Nicaise (LAMAV) Density and trace results in networks WDPJ, August 2017 25 / 25