Characterization of solutions of a generalized Helmholtz problem

TL;DR

This paper classifies all distributional solutions to a generalized Helmholtz equation involving Bernstein functions, revealing that their Fourier transforms are single-layer distributions on the sphere, thus extending classical solutions to a broader class of operators.

Contribution

It provides a complete classification of solutions for a generalized Helmholtz problem with Bernstein functions, including explicit examples like fractional Laplacians and logarithmic operators.

Findings

Fourier transform of solutions is a single-layer distribution on the sphere.

Includes operators like fractional Laplacians and logarithmic variants.

Extends classical Helmholtz solution characterization to generalized operators.

Abstract

In this article, we classify all distributional solutions of $f(-\Delta)u=f(1)u$ where $f$ is a non-constant Bernstein function. Specifically, we show that the Fourier transform of $u$ is a single-layer distribution on the unit sphere. Examples of such operators include $(-\Delta)^\sigma$ (for $\sigma \in (0,1]$), $\log(1-\Delta)$ and $(-\Delta)^\frac{1}{2}\text{tanh}((-\Delta)^\frac{1}{2})$.