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An annulus multiplier and applications to the limiting absorption principle for Helmholtz equations with a step potential

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Abstract

We consider the Helmholtz equation \(-\Delta u+V \, u - \lambda \, u = f \) on \({\mathbb {R}}^n\) where the potential \(V:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) is constant on each of the half-spaces \({\mathbb {R}}^{n-1}\times (-\infty ,0)\) and \({\mathbb {R}}^{n-1}\times (0,\infty )\). We prove an \(L^p-L^q\)-Limiting Absorption Principle for frequencies \(\lambda >\max \, V\) with the aid of Fourier Restriction Theory and derive the existence of nontrivial solutions of linear and nonlinear Helmholtz equations. As a main analytical tool we develop new \(L^p-L^q\) estimates for a singular Fourier multiplier supported in an annulus.

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Acknowledgements

Funded by the Deutsche Forschungsgemeinschaft (DFG,German Research Foundation)—Project-ID 258734477—SFB 1173.

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  1. Karlsruhe Institute of Technology, Institute for Analysis, Englerstraße 2, 76131, Karlsruhe, Germany

    Rainer Mandel & Dominic Scheider

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  1. Rainer Mandel
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  2. Dominic Scheider
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Correspondence to Dominic Scheider.

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Communicated by Loukas Grafakos.

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Mandel, R., Scheider, D. An annulus multiplier and applications to the limiting absorption principle for Helmholtz equations with a step potential. Math. Ann. 379, 865–907 (2021). https://doi.org/10.1007/s00208-020-02093-3

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