What Is Not in the Domain of the Laplacian on Sierpinski Gasket Type Fractals

Oren Ben-Bassat; Robert S. Strichartz; Alexander Teplyaev

doi:10.1006/jfan.1999.3431

What Is Not in the Domain of the Laplacian on Sierpinski Gasket Type Fractals

Oren Ben-Bassat, Robert S. Strichartz, Alexander Teplyaev

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Abstract

We consider the analog of the Laplacian on the Sierpinski gasket and related fractals, constructed by Kigami. A function f is said to belong to the domain of Δ if f is continuous and Δf is defined as a continuous function. We show that if f is a nonconstant function in the domain of Δ, then f² is not in the domain of Δ. We give two proofs of this fact. The first is based on the analog of the pointwise identity Δf²-2fΔf=∇f², where we show that ∇f² does not exist as a continuous function. In fact the correct interpretation of Δf² is as a singular measure, a result due to Kusuoka; we give a new proof of this fact. The second is based on a dichotomy for the local behavior of a function in the domain of Δ, at a junction point x₀ of the fractal: in the typical case (nonvanishing of the normal derivative) we have upper and lower bounds for f(x)-f(x₀) in terms of d(x, x₀)^β for a certain value β, and in the nontypical case (vanishing normal derivative) we have an upper bound with an exponent greater than 2. This method allows us to show that general nonlinear functions do not operate on the domain of Δ.

Original language	English
Pages (from-to)	197-217
Number of pages	21
Journal	Journal of Functional Analysis
Volume	166
Issue number	2
DOIs	https://doi.org/10.1006/jfan.1999.3431
State	Published - 20 Aug 1999
Externally published	Yes

Bibliographical note

Funding Information:
We consider the analog of the Laplacian on the Sierpinski gasket and related fractals, constructed by Kigami. A function f is said to belong to the domain of 2 if f is continuous and 2f is defined as a continuous function. We show that if f is a nonconstant function in the domain of 2, then f2 is not in the domain of 2. We give two proofs of this fact. The first is based on the analog of the pointwise identity 2f2&2f2f=|{f|2,whereweshowthat|{f|2doesnotexistasacontinuousfunc-tion. In fact the correct interpretation of 2f2 is as a singular measure, a result due to Kusuoka; we give a new proof of this fact. The second is based on a dichotomy for the local behavior of a function in the domain of 2, at a junction point x0 of the fractal: in the typical case (nonvanishing of the normal derivative) we have upper and lower bounds for |f(x)&f(x0)| in terms of d(x,x0); for a certain value ;, and in the nontypical case (vanishing normal derivative) we have an upper bound with an exponent greater than 2. This method allows us to show that general nonlinear functions do not operate on the domain of 2. ν 1999 Academic Press * Research supported by the National Science Foundation through the Research Experiences for Undergraduates Program at the Cornell Mathematics Department. Current address: Department of Physics, University of Pennsylvania, Philadelphia, PA 19104-6396. -Research supported in part by the National Science Foundation, Grant DMS-9623250. Research supported in part by an Alfred P. Sloan Doctoral Dissertation Fellowship and the National Science Foundation. Current address: Department of Mathematics, McMaster University, Hamilton, ON L8S 4K1, Canada.

ASJC Scopus subject areas

Analysis

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10.1006/jfan.1999.3431

Cite this

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title = "What Is Not in the Domain of the Laplacian on Sierpinski Gasket Type Fractals",

abstract = "We consider the analog of the Laplacian on the Sierpinski gasket and related fractals, constructed by Kigami. A function f is said to belong to the domain of Δ if f is continuous and Δf is defined as a continuous function. We show that if f is a nonconstant function in the domain of Δ, then f2 is not in the domain of Δ. We give two proofs of this fact. The first is based on the analog of the pointwise identity Δf2-2fΔf=∇f2, where we show that ∇f2 does not exist as a continuous function. In fact the correct interpretation of Δf2 is as a singular measure, a result due to Kusuoka; we give a new proof of this fact. The second is based on a dichotomy for the local behavior of a function in the domain of Δ, at a junction point x0 of the fractal: in the typical case (nonvanishing of the normal derivative) we have upper and lower bounds for f(x)-f(x0) in terms of d(x, x0)β for a certain value β, and in the nontypical case (vanishing normal derivative) we have an upper bound with an exponent greater than 2. This method allows us to show that general nonlinear functions do not operate on the domain of Δ.",

author = "Oren Ben-Bassat and Strichartz, {Robert S.} and Alexander Teplyaev",

note = "Funding Information: We consider the analog of the Laplacian on the Sierpinski gasket and related fractals, constructed by Kigami. A function f is said to belong to the domain of 2 if f is continuous and 2f is defined as a continuous function. We show that if f is a nonconstant function in the domain of 2, then f2 is not in the domain of 2. We give two proofs of this fact. The first is based on the analog of the pointwise identity 2f2&2f2f=|{f|2,whereweshowthat|{f|2doesnotexistasacontinuousfunc-tion. In fact the correct interpretation of 2f2 is as a singular measure, a result due to Kusuoka; we give a new proof of this fact. The second is based on a dichotomy for the local behavior of a function in the domain of 2, at a junction point x0 of the fractal: in the typical case (nonvanishing of the normal derivative) we have upper and lower bounds for |f(x)&f(x0)| in terms of d(x,x0); for a certain value ;, and in the nontypical case (vanishing normal derivative) we have an upper bound with an exponent greater than 2. This method allows us to show that general nonlinear functions do not operate on the domain of 2. ν 1999 Academic Press * Research supported by the National Science Foundation through the Research Experiences for Undergraduates Program at the Cornell Mathematics Department. Current address: Department of Physics, University of Pennsylvania, Philadelphia, PA 19104-6396. -Research supported in part by the National Science Foundation, Grant DMS-9623250. Research supported in part by an Alfred P. Sloan Doctoral Dissertation Fellowship and the National Science Foundation. Current address: Department of Mathematics, McMaster University, Hamilton, ON L8S 4K1, Canada.",

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N2 - We consider the analog of the Laplacian on the Sierpinski gasket and related fractals, constructed by Kigami. A function f is said to belong to the domain of Δ if f is continuous and Δf is defined as a continuous function. We show that if f is a nonconstant function in the domain of Δ, then f2 is not in the domain of Δ. We give two proofs of this fact. The first is based on the analog of the pointwise identity Δf2-2fΔf=∇f2, where we show that ∇f2 does not exist as a continuous function. In fact the correct interpretation of Δf2 is as a singular measure, a result due to Kusuoka; we give a new proof of this fact. The second is based on a dichotomy for the local behavior of a function in the domain of Δ, at a junction point x0 of the fractal: in the typical case (nonvanishing of the normal derivative) we have upper and lower bounds for f(x)-f(x0) in terms of d(x, x0)β for a certain value β, and in the nontypical case (vanishing normal derivative) we have an upper bound with an exponent greater than 2. This method allows us to show that general nonlinear functions do not operate on the domain of Δ.

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What Is Not in the Domain of the Laplacian on Sierpinski Gasket Type Fractals

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