Helmholtz Solutions for the Fractional Laplacian and Other Related Operators

TL;DR

This paper classifies bounded solutions to the fractional Helmholtz equation, showing they coincide with classical solutions under certain conditions, and extends these results to more general operators using Bernstein functions.

Contribution

It provides a comprehensive classification of bounded solutions for fractional Helmholtz equations and extends the results to general operators defined by Bernstein functions.

Findings

Bounded fractional Helmholtz solutions match classical solutions in multiple dimensions.

Classification extends to $1<s extless 2$ and integer $s$ with smoothness assumptions.

Classical solutions are unique bounded solutions for a broad class of operators involving Bernstein functions.

Abstract

We show that the bounded solutions to the fractional Helmholtz equation, $(-\Delta)^s u= u$ for $0<s<1$ in $\mathbb{R}^n$, are given by the bounded solutions to the classical Helmholtz equation $(-\Delta)u= u$ in $\mathbb{R}^n$ for $n \ge 2$ when $u$ is additionally assumed to be vanishing at $\infty$. 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 $1<s \le 2$ and $s\in \mathbb{N}$ when $u \in C^\infty(\mathbb{R}^n)$. Finally, we prove that the classical solutions are the unique bounded solutions to the more general equation $\psi(-\Delta) u= \psi(1)u$ in $\mathbb{R}^n$, when $\psi$ is complete Bernstein and certain regularity conditions are imposed on the associated weight $a(t)$.