Helmholtz solutions for the fractional Laplacian and other related operators

Vincent Guan
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 ∈
more » ... ⁿ). 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).
doi:10.14288/1.0412628 fatcat:hg7niooygfao5d6qr76gq5xytu