Characterization of solutions of a generalized Helmholtz problem

Daniel Hauer; David Lee

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arXiv · 2306.17562

Characterization of solutions of a generalized Helmholtz problem

Daniel Hauer·David Lee

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Abstract

In this article, we classify all distributional solutions of $f(-Δ)u=f(1)u$ 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 $(-Δ)^σ$ (for $σ\in (0,1]$), $\log(1-Δ)$ and $(-Δ)^\frac{1}{2}\text{tanh}((-Δ)^\frac{1}{2})$.

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Daniel Hauer, David Lee. 2023-06-30. Characterization of solutions of a generalized Helmholtz problem. https://arxiv.org/abs/2306.17562

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