Eigenvalues and eigenfunctions of measure-geometric Laplacians - - PowerPoint PPT Presentation

Eigenvalues and eigenfunctions of measure-geometric Laplacians

Hendrik Weyer (joint work with M. Kesseböhmer and T. Samuel) Winter School on Diffusion on Fractals and Non-linear Dynamics March 24, 2015

Dynamical systems and geometry Department 03 Mathmatics/Computer science

Table of contents

1 Differentiation with respect to measures 2 Eigenvalues and eigenfunctions of measure-geometric Laplacians 3 Examples 4 Outlook

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Differentiation with respect to measures

Let a, b ∈ R with a < b; I := [a, b], µ finite atomless Borel measure on I with a, b in the support and Fµ denote the (continuous) distribution function of µ.

Definition Dµ

1

Dµ

1 :=

• f ∈ L2(µ)
• ∃ g ∈ L2(µ) s.t. ∀x ∈ I : f (x) = f (a) +

x

a

g(y) dµ(y)

• .

It is shown, that for f ∈ Dµ

1 the function g ∈ L2(µ) is unique and that every

function in Dµ

1 is continuous on [a, b].

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Dynamical systems and geometry Department 03 Mathmatics/Computer science

Differentiation with respect to measures

Let a, b ∈ R with a < b; I := [a, b], µ finite atomless Borel measure on I with a, b in the support and Fµ denote the (continuous) distribution function of µ.

Definition Dµ

1

Dµ

1 :=

• f ∈ L2(µ)
• ∃ g ∈ L2(µ) s.t. ∀x ∈ I : f (x) = f (a) +

x

a

g(y) dµ(y)

• .

It is shown, that for f ∈ Dµ

1 the function g ∈ L2(µ) is unique and that every

function in Dµ

1 is continuous on [a, b].

3 / 20 ∇µ and ∆µ Eigenvalues and eigenfunctions Examples Outlook

Dynamical systems and geometry Department 03 Mathmatics/Computer science

Differentiation with respect to measures

Let a, b ∈ R with a < b; I := [a, b], µ finite atomless Borel measure on I with a, b in the support and Fµ denote the (continuous) distribution function of µ.

Definition Dµ

1

Dµ

1 :=

• f ∈ L2(µ)
• ∃ g ∈ L2(µ) s.t. ∀x ∈ I : f (x) = f (a) +

x

a

g(y) dµ(y)

• .

It is shown, that for f ∈ Dµ

1 the function g ∈ L2(µ) is unique and that every

function in Dµ

1 is continuous on [a, b].

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Definition ∇µ

Let f ∈ Dµ

1 and g be as above. Then

∇µ : Dµ

1 → L2(µ),

f → g is called the µ-derivative operator. In the case that µ is the Lebesgue measure Λ on [a, b], ∇Λ coincides with the weak derivative and DΛ

1 with the Sobolev space W 1,2(]a, b[).

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Definition ∇µ

Let f ∈ Dµ

1 and g be as above. Then

∇µ : Dµ

1 → L2(µ),

f → g is called the µ-derivative operator. In the case that µ is the Lebesgue measure Λ on [a, b], ∇Λ coincides with the weak derivative and DΛ

1 with the Sobolev space W 1,2(]a, b[).

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Definition Dµ

2

We define Dµ

2 ⊆ Dµ 1 by

Dµ

2 := {f ∈ Dµ 1 : ∇µf ∈ Dµ 1 } .

Definition ∆µ

Let f ∈ Dµ

2 . Then the operator

∆µ : Dµ

2 → L2(µ),

f → ∇µ (∇µf ) is called the µ-Laplace operator.

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Definition Dµ

2

We define Dµ

2 ⊆ Dµ 1 by

Dµ

2 := {f ∈ Dµ 1 : ∇µf ∈ Dµ 1 } .

Definition ∆µ

Let f ∈ Dµ

2 . Then the operator

∆µ : Dµ

2 → L2(µ),

f → ∇µ (∇µf ) is called the µ-Laplace operator.

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Freiberg and Zähle showed in 2002 analytic properties of ∆µ: The µ-Laplace operator is linear, it fulfils Green’s identities and when additionally assuming homogeneous Dirichlet or von Neumann boundary conditions, ∆µ is symmetric and non-positive.

Proposition

The set of µ-harmonic functions (these are the functions f for which ∆µf ≡ 0) is equal to {x → A + B · Fµ(x): A, B ∈ R}.

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Freiberg and Zähle showed in 2002 analytic properties of ∆µ: The µ-Laplace operator is linear, it fulfils Green’s identities and when additionally assuming homogeneous Dirichlet or von Neumann boundary conditions, ∆µ is symmetric and non-positive.

Proposition

The set of µ-harmonic functions (these are the functions f for which ∆µf ≡ 0) is equal to {x → A + B · Fµ(x): A, B ∈ R}.

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Eigenvalues and eigenfunctions of measure-geometric Laplacians

First results about spectral properties by Freiberg and Zähle:

Theorem (Freiberg, Zähle 2002)

Considering self-similar measures µ living on Cantor-like sets, one can obtain −λn ≍ n2, as n → ∞, where {λn} are the eigenvalues of the µ-Laplacian ∆µ on Dµ

2 under

homogeneous Dirichlet or von Neumann boundary conditions, such that 0 ≥ λ1 ≥ λ2 ≥ · · · .

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Main Theorem (Kesseböhmer, Samuel, W. 2014)

Let µ be a atomless Borel probability measure with distribution function Fµ and set λn := −(πn)2, for n ∈ N0. (i) The eigenvalues of ∆µ on Dµ

2 under homogeneous Dirichlet boundary

conditions are λn, for n ∈ N, with corresponding eigenfunctions f µ

n (x) := sin(πnFµ(x)),

for x ∈ [a, b]. (ii) The eigenvalues of ∆µ on Dµ

2 under homogeneous von Neumann

boundary conditions are λn, for n ∈ N0, with corresponding eigenfunctions gµ

n (x) := cos(πnFµ(x)),

for x ∈ [a, b].

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Main Theorem (Kesseböhmer, Samuel, W. 2014)

Let µ be a atomless Borel probability measure with distribution function Fµ and set λn := −(πn)2, for n ∈ N0. (i) The eigenvalues of ∆µ on Dµ

2 under homogeneous Dirichlet boundary

conditions are λn, for n ∈ N, with corresponding eigenfunctions f µ

n (x) := sin(πnFµ(x)),

for x ∈ [a, b]. (ii) The eigenvalues of ∆µ on Dµ

2 under homogeneous von Neumann

boundary conditions are λn, for n ∈ N0, with corresponding eigenfunctions gµ

n (x) := cos(πnFµ(x)),

for x ∈ [a, b].

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Main Theorem (Kesseböhmer, Samuel, W. 2014)

Let µ be a atomless Borel probability measure with distribution function Fµ and set λn := −(πn)2, for n ∈ N0. (i) The eigenvalues of ∆µ on Dµ

2 under homogeneous Dirichlet boundary

conditions are λn, for n ∈ N, with corresponding eigenfunctions f µ

n (x) := sin(πnFµ(x)),

for x ∈ [a, b]. (ii) The eigenvalues of ∆µ on Dµ

2 under homogeneous von Neumann

boundary conditions are λn, for n ∈ N0, with corresponding eigenfunctions gµ

n (x) := cos(πnFµ(x)),

for x ∈ [a, b].

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Proof of Main Theorem

Dirichlet case:

• 1. If µ = Λ, the Lebesgue measure supported on the unit interval, then ∆Λ is

the weak Laplacian and the result is well-known (e.g. classical Sturm-Liouville theory).

• 2. Show that f µ

n are eigenfunctions (for a general µ), with the definition of

µ-derivative and the measure identities µ ◦ F −1

µ

= Λ and Λ ◦ Fµ = µ.

• 3. Obtain a one-to-one correspondence between the eigenfunctions of ∆Λ and

∆µ via the pseudoinverse of the distribution function ˇ F −1

µ (x) := inf{y ∈ [a, b]: Fµ(y) ≥ x} and the measure identities.

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Proof of Main Theorem

Dirichlet case:

• 1. If µ = Λ, the Lebesgue measure supported on the unit interval, then ∆Λ is

the weak Laplacian and the result is well-known (e.g. classical Sturm-Liouville theory).

• 2. Show that f µ

n are eigenfunctions (for a general µ), with the definition of

µ-derivative and the measure identities µ ◦ F −1

µ

= Λ and Λ ◦ Fµ = µ.

• 3. Obtain a one-to-one correspondence between the eigenfunctions of ∆Λ and

∆µ via the pseudoinverse of the distribution function ˇ F −1

µ (x) := inf{y ∈ [a, b]: Fµ(y) ≥ x} and the measure identities.

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Proof of Main Theorem

Dirichlet case:

• 1. If µ = Λ, the Lebesgue measure supported on the unit interval, then ∆Λ is

the weak Laplacian and the result is well-known (e.g. classical Sturm-Liouville theory).

• 2. Show that f µ

n are eigenfunctions (for a general µ), with the definition of

µ-derivative and the measure identities µ ◦ F −1

µ

= Λ and Λ ◦ Fµ = µ.

• 3. Obtain a one-to-one correspondence between the eigenfunctions of ∆Λ and

∆µ via the pseudoinverse of the distribution function ˇ F −1

µ (x) := inf{y ∈ [a, b]: Fµ(y) ≥ x} and the measure identities.

9 / 20 ∇µ and ∆µ Eigenvalues and eigenfunctions Examples Outlook

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Proof of Main Theorem

Dirichlet case:

• 1. If µ = Λ, the Lebesgue measure supported on the unit interval, then ∆Λ is

the weak Laplacian and the result is well-known (e.g. classical Sturm-Liouville theory).

• 2. Show that f µ

n are eigenfunctions (for a general µ), with the definition of

µ-derivative and the measure identities µ ◦ F −1

µ

= Λ and Λ ◦ Fµ = µ.

• 3. Obtain a one-to-one correspondence between the eigenfunctions of ∆Λ and

∆µ via the pseudoinverse of the distribution function ˇ F −1

µ (x) := inf{y ∈ [a, b]: Fµ(y) ≥ x} and the measure identities.

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Corollary

Apart from the constant function, the µ-derivatives of von Neumann eigenfunctions are Dirichlet eigenfunctions and vice versa. This holds since ∆µ(∇µf ) = ∇µ(∆µf ) = ∇µ(λf ) = λ(∇µf ), and the corresponding boundary conditions are fulfilled by the definition of the µ-derivative.

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Inhomogeneous Cantor measures

Take the unit interval and define the contractions s1 : x → x/2 and s2 : x → x/3 + 2/3. Let E ⊂ [0, 1] be the attractor of the IFS Φ = ([0, 1]; s1, s2). We set p := (0.7, 0.3) and denote the associated self-similar measure by µ = µ(Φ, p). This is the atomless Borel probability measure supported on E satisfying µ(A) =

2

• i=1

piµ(s−1

i

(A)), for all Borel measurable sets A.

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Inhomogeneous Cantor measures

Take the unit interval and define the contractions s1 : x → x/2 and s2 : x → x/3 + 2/3. Let E ⊂ [0, 1] be the attractor of the IFS Φ = ([0, 1]; s1, s2). We set p := (0.7, 0.3) and denote the associated self-similar measure by µ = µ(Φ, p). This is the atomless Borel probability measure supported on E satisfying µ(A) =

2

• i=1

piµ(s−1

i

(A)), for all Borel measurable sets A.

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0.25 0.5 0.75 1 0.1 0.3 0.5 0.7 0.9 1

Figure: Distribution function of µ being the self-similar measure associated to Φ = ([0, 1]; s1, s2) and p = (0.7, 0.3)

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0.25 0.5 0.75 1

• 1
• 0.5

0.5 1 0.25 0.5 0.75 1

• 1
• 0.5

0.5 1 0.25 0.5 0.75 1

• 1
• 0.5

0.5 1 0.25 0.5 0.75 1

• 1
• 0.5

0.5 1

Figure: Eigenfunctions of ∆µ, where µ is the self-similar measure associated to Φ = ([0, 1]; s1, s2) and p = (0.7, 0.3), under Dirichlet boundary conditions.

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Chebychev Polynomials

The Chebychev polynomials are used in numerical analysis, e.g. for polynomial

• interpolation. They are given inductively by

T0(x) := 1, T1(x) := x and Tn+1(x) := 2xTn(x) − Tn−1(x)

• r equivalently by

Tn(x) = cos(n arccos(x)).

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Chebychev Polynomials

The Chebychev polynomials are used in numerical analysis, e.g. for polynomial

• interpolation. They are given inductively by

T0(x) := 1, T1(x) := x and Tn+1(x) := 2xTn(x) − Tn−1(x)

• r equivalently by

Tn(x) = cos(n arccos(x)).

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Let Λ denote the Lebesgue measure on [-1, 1] and let µ be the measure given by dµ dΛ(x) = 1 π √ 1 − x 2 .

• 1
• 0.5

0.5 1 0.5 1

Figure: Distribution function Fµ(x) = 1

π arccos(−x)

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The von Neumann eigenfunctions of ∆µ are for n ∈ N0 and x ∈ [−1, 1], gµ

n (x) = cos(n arccos(−x)) = Tn(−x) = (−1)nTn(x).

Namely, these are gµ

0 (x) = 1,

gµ

1 (x) = −x,

gµ

2 (x) = 2x 2 − 1,

gµ

3 (x) = −4x 3 + 3x,

. . .

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gµ

0 , gµ 1 , gµ 2 and gµ 3

• 1
• 0.5

0.5 1

• 1
• 0.5

0.5 1

∇µgµ

0 , ∇µgµ 1 , ∇µgµ 2 and ∇µgµ 3

• 1
• 0.5

0.5 1

• :
• :/2

:/2 :

Figure: Eigenfunctions of ∆µ, where dµ/dΛ = 1/(π √ 1 − x2), under von Neumann boundary conditions, and their µ-derivatives.

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Further examples for measures for which our Main Theorem holds: Finite, absolutely continuous measures (with respect to the Lebesgue measure) One-dimensional Hausdorff-measure restricted to an interval [a, b] Markov measures Salem measures ...

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Outlook

• 1. We want to extend the class of measures by allowing them to have atoms

(purely atomic and mixtures).

• 2. Let µ, ν be two finite (atomless) Borel measures supported on the interval

[a, b]. We want to look at the operator ∆µ,ν : Dµ,ν

2

→ L2(µ) f → ∇µ(∇νf ) and try to obtain spectral results (see e.g. Freiberg 2003, Arzt 2014).

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Outlook

• 1. We want to extend the class of measures by allowing them to have atoms

(purely atomic and mixtures).

• 2. Let µ, ν be two finite (atomless) Borel measures supported on the interval

[a, b]. We want to look at the operator ∆µ,ν : Dµ,ν

2

→ L2(µ) f → ∇µ(∇νf ) and try to obtain spectral results (see e.g. Freiberg 2003, Arzt 2014).

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References

• 1. P. Arzt: Measure Theoretic Trigonometric Functions.

Pre-print: arXiv:1405.4693A, 2014.

• 2. U. Freiberg: A survey on measure geometric Laplacians on Cantor like sets.

AJSE 28, No. 1C (2003), 189198

• 3. U. Freiberg and M. Zähle: Harmonic calculus on fractals - A measure

geometric approach I. Potential Analysis 16(1) 265–277 (2002).

• 4. M. Kesseböhmer, T. Samuel and H. Weyer: A note on measure-geometric

Laplacians. Pre-print: arXiv:1411.2491, 2014

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