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 Department 03 and geometry Mathmatics/Computer science Table of contents 1 Differentiation with respect to measures 2 Eigenvalues and eigenfunctions of measure-geometric Laplacians 3 Examples 4 Outlook ∇ µ and ∆ µ Eigenvalues and eigenfunctions Examples Outlook 2 / 20
Dynamical systems Department 03 and geometry 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 F µ denote the (continuous) distribution function of µ . Definition D µ 1 � x � � � f ∈ L 2 ( µ ) � ∃ g ∈ L 2 ( µ ) s.t. ∀ x ∈ I : f ( x ) = f ( a ) + D µ 1 := g ( y ) d µ ( y ) . � a 1 the function g ∈ L 2 ( µ ) is unique and that every It is shown, that for f ∈ D µ function in D µ 1 is continuous on [ a , b ] . ∇ µ and ∆ µ Eigenvalues and eigenfunctions Examples Outlook 3 / 20
Dynamical systems Department 03 and geometry 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 F µ denote the (continuous) distribution function of µ . Definition D µ 1 � x � � � f ∈ L 2 ( µ ) � ∃ g ∈ L 2 ( µ ) s.t. ∀ x ∈ I : f ( x ) = f ( a ) + D µ 1 := g ( y ) d µ ( y ) . � a 1 the function g ∈ L 2 ( µ ) is unique and that every It is shown, that for f ∈ D µ function in D µ 1 is continuous on [ a , b ] . ∇ µ and ∆ µ Eigenvalues and eigenfunctions Examples Outlook 3 / 20
Dynamical systems Department 03 and geometry 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 F µ denote the (continuous) distribution function of µ . Definition D µ 1 � x � � � f ∈ L 2 ( µ ) � ∃ g ∈ L 2 ( µ ) s.t. ∀ x ∈ I : f ( x ) = f ( a ) + D µ 1 := g ( y ) d µ ( y ) . � a 1 the function g ∈ L 2 ( µ ) is unique and that every It is shown, that for f ∈ D µ function in D µ 1 is continuous on [ a , b ] . ∇ µ and ∆ µ Eigenvalues and eigenfunctions Examples Outlook 3 / 20
Dynamical systems Department 03 and geometry Mathmatics/Computer science Definition ∇ µ Let f ∈ D µ 1 and g be as above. Then ∇ µ : D µ 1 → L 2 ( 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 D Λ 1 with the Sobolev space W 1 , 2 (] a , b [) . ∇ µ and ∆ µ Eigenvalues and eigenfunctions Examples Outlook 4 / 20
Dynamical systems Department 03 and geometry Mathmatics/Computer science Definition ∇ µ Let f ∈ D µ 1 and g be as above. Then ∇ µ : D µ 1 → L 2 ( 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 D Λ 1 with the Sobolev space W 1 , 2 (] a , b [) . ∇ µ and ∆ µ Eigenvalues and eigenfunctions Examples Outlook 4 / 20
Dynamical systems Department 03 and geometry Mathmatics/Computer science Definition D µ 2 We define D µ 2 ⊆ D µ 1 by D µ 2 := { f ∈ D µ 1 : ∇ µ f ∈ D µ 1 } . Definition ∆ µ Let f ∈ D µ 2 . Then the operator ∆ µ : D µ 2 → L 2 ( µ ) , f �→ ∇ µ ( ∇ µ f ) is called the µ -Laplace operator . ∇ µ and ∆ µ Eigenvalues and eigenfunctions Examples Outlook 5 / 20
Dynamical systems Department 03 and geometry Mathmatics/Computer science Definition D µ 2 We define D µ 2 ⊆ D µ 1 by D µ 2 := { f ∈ D µ 1 : ∇ µ f ∈ D µ 1 } . Definition ∆ µ Let f ∈ D µ 2 . Then the operator ∆ µ : D µ 2 → L 2 ( µ ) , f �→ ∇ µ ( ∇ µ f ) is called the µ -Laplace operator . ∇ µ and ∆ µ Eigenvalues and eigenfunctions Examples Outlook 5 / 20
Dynamical systems Department 03 and geometry Mathmatics/Computer science 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 ∆ µ f ≡ 0) is equal to { x �→ A + B · F µ ( x ): A , B ∈ R } . ∇ µ and ∆ µ Eigenvalues and eigenfunctions Examples Outlook 6 / 20
Dynamical systems Department 03 and geometry Mathmatics/Computer science 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 ∆ µ f ≡ 0) is equal to { x �→ A + B · F µ ( x ): A , B ∈ R } . ∇ µ and ∆ µ Eigenvalues and eigenfunctions Examples Outlook 6 / 20
Dynamical systems Department 03 and geometry Mathmatics/Computer science 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 − λ n ≍ n 2 , as n → ∞ , where { λ n } are the eigenvalues of the µ -Laplacian ∆ µ on D µ 2 under homogeneous Dirichlet or von Neumann boundary conditions, such that 0 ≥ λ 1 ≥ λ 2 ≥ · · · . ∇ µ and ∆ µ Eigenvalues and eigenfunctions Examples Outlook 7 / 20
Dynamical systems Department 03 and geometry Mathmatics/Computer science Main Theorem (Kesseböhmer, Samuel, W. 2014) Let µ be a atomless Borel probability measure with distribution function F µ and set λ n := − ( π n ) 2 , for n ∈ N 0 . (i) The eigenvalues of ∆ µ on D µ 2 under homogeneous Dirichlet boundary conditions are λ n , for n ∈ N , with corresponding eigenfunctions f µ n ( x ) := sin ( π nF µ ( x )) , for x ∈ [ a , b ] . (ii) The eigenvalues of ∆ µ on D µ 2 under homogeneous von Neumann boundary conditions are λ n , for n ∈ N 0 , with corresponding eigenfunctions g µ n ( x ) := cos ( π nF µ ( x )) , for x ∈ [ a , b ] . ∇ µ and ∆ µ Eigenvalues and eigenfunctions Examples Outlook 8 / 20
Dynamical systems Department 03 and geometry Mathmatics/Computer science Main Theorem (Kesseböhmer, Samuel, W. 2014) Let µ be a atomless Borel probability measure with distribution function F µ and set λ n := − ( π n ) 2 , for n ∈ N 0 . (i) The eigenvalues of ∆ µ on D µ 2 under homogeneous Dirichlet boundary conditions are λ n , for n ∈ N , with corresponding eigenfunctions f µ n ( x ) := sin ( π nF µ ( x )) , for x ∈ [ a , b ] . (ii) The eigenvalues of ∆ µ on D µ 2 under homogeneous von Neumann boundary conditions are λ n , for n ∈ N 0 , with corresponding eigenfunctions g µ n ( x ) := cos ( π nF µ ( x )) , for x ∈ [ a , b ] . ∇ µ and ∆ µ Eigenvalues and eigenfunctions Examples Outlook 8 / 20
Dynamical systems Department 03 and geometry Mathmatics/Computer science Main Theorem (Kesseböhmer, Samuel, W. 2014) Let µ be a atomless Borel probability measure with distribution function F µ and set λ n := − ( π n ) 2 , for n ∈ N 0 . (i) The eigenvalues of ∆ µ on D µ 2 under homogeneous Dirichlet boundary conditions are λ n , for n ∈ N , with corresponding eigenfunctions f µ n ( x ) := sin ( π nF µ ( x )) , for x ∈ [ a , b ] . (ii) The eigenvalues of ∆ µ on D µ 2 under homogeneous von Neumann boundary conditions are λ n , for n ∈ N 0 , with corresponding eigenfunctions g µ n ( x ) := cos ( π nF µ ( x )) , for x ∈ [ a , b ] . ∇ µ and ∆ µ Eigenvalues and eigenfunctions Examples Outlook 8 / 20
Dynamical systems Department 03 and geometry Mathmatics/Computer science 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 = Λ and Λ ◦ F µ = µ . µ 3. Obtain a one-to-one correspondence between the eigenfunctions of ∆ Λ and ∆ µ via the pseudoinverse of the distribution function F − 1 ˇ µ ( x ) := inf { y ∈ [ a , b ]: F µ ( y ) ≥ x } and the measure identities. ∇ µ and ∆ µ Eigenvalues and eigenfunctions Examples Outlook 9 / 20
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