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Helmholtz solutions for the fractional Laplacian and other related operators Guan, Vincent
Abstract
We show that the bounded solutions to the fractional Helmholtz equation, (−∆)ˢ u = u for 0 < s < 1 in ℝⁿ, are given by the bounded solutions to the classical Helmholtz equation (−∆)u = u in ℝⁿ for n ≥ 2 when u is additionally assumed to be vanishing at ∞. When n = 1, we show that the bounded fractional Helmholtz solutions are again given by the classical solutions A cos(x) + B sin(x). We show that this classification of fractional Helmholtz solutions extends for 1 < s ≤ 2 and s ∈ ℕ when u ∈ C∞(ℝⁿ). Finally, we prove that the classical solutions are the unique bounded solutions to the more general equation ψ(−∆)u = ψ(1)u in ℝⁿ, when ψ is complete Bernstein and certain regularity conditions are imposed on the associated weight a(t).
Item Metadata
Title |
Helmholtz solutions for the fractional Laplacian and other related operators
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Creator | |
Supervisor | |
Publisher |
University of British Columbia
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Date Issued |
2022
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Description |
We show that the bounded solutions to the fractional Helmholtz equation, (−∆)ˢ u = u for 0 < s < 1 in ℝⁿ, are given by the bounded solutions to the classical Helmholtz equation (−∆)u = u in ℝⁿ for n ≥ 2 when u is additionally assumed to be vanishing at ∞. When n = 1, we show that the bounded fractional Helmholtz solutions are again given by the classical solutions A cos(x) + B sin(x). We show that this classification of fractional Helmholtz solutions extends for 1 < s ≤ 2 and s ∈ ℕ when u ∈ C∞(ℝⁿ). Finally, we prove that the classical solutions are the unique bounded solutions to the more general equation ψ(−∆)u = ψ(1)u in ℝⁿ, when ψ is complete Bernstein and certain regularity conditions are imposed on the associated weight a(t).
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Genre | |
Type | |
Language |
eng
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Date Available |
2022-04-06
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0412628
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
2022-05
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Campus | |
Scholarly Level |
Graduate
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Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International